DTU ProjectLab - User contributions [en-gb]

Risk Profile in Turnkey Projects

2015-10-04T16:09:37Z

Andkamp: /* Conclution */

As a result of innovations in the partnering relationships between the client and the project coalition actors, there has been a transition in the construction industry. There is a number of different ways to form construction projects, and the different structures come with unique risks for actors involved in the project. The currently trend is use of the integrated project coalition, also called turnkey, characterised by a single contract for both execution and design of a project. Benefiting from this is the client, who pay for low risk and project uncertainty. <ref>[''Managing Construction Projects''] ''Winch, G.M. (2010) Chapter 5.4.5, Second edition, Wiley-Blackwell'' </ref>On the other hand, this provides more uncertainties for actors as contractors, designers and consultants. [[Managing Uncertainty and Risk on the Project]] is especially important in turnkey.

This type of project management approach requires a thorough risk profile for the project coalition actors to avoid damaging missteps. It will always be numerous risks in construction contracts, and without a risk profile it could leave the contractor, designer or the consultant dangerously exposed to unexpected responsibilities and risks. However, without risks there will not be innovation. Risks and opportunities are related and you need a balance between the two. This wiki-article will look at the leading nordic contractual risk factors and risk profile for actors in turnkey contracting in the construction industry.

=Turnkey Projects=
== Definition ==
Turnkey is also named integrated coalitions or single–point responsibility. This type of project is constructed as a complete product for a client. This means that within the clients order there is both design and execution. <ref>[''Managing Construction Projects''] ''Winch, G.M. (2010), Chapter 5.4.2, Second edition, Wiley-Blackwell'' </ref> With design it means engineering and architectural design. In turnkey the goal of the contractor is to satisfy the clients performance specifications. Where in general contracting the contractor address designer's plans and specifications. Turnkey contracting is getting more used in the building industry because the client transfer maximum risk to the contractor.

==Structure==
First of all, it is the client who decides the contract structure of the project. The relationships between the actors in a turnkey is illustrated at the figure below:

[[File:1.png|600px|]]
<br clear=all> '''Fig. 1''': Turnkey contract structur <ref>[''Photography''] ''From: Managing Construction Projects'',Winch, G.M. (2010), Second edition, Wiley-Blackwell '' </ref>.

As you can see in the figure above, the main responsibility is centralized with the turnkey contractor. Normally they guarantee compensation for delays and take full responsibility for overall project cost. <ref>[''Reliability through Experience''] ''"Full Turnkey Construction(Internet)", ''Available from: http://www.dsengineers.com/en/contracting-services/full-turnkey-epc-construction.html'', Read: 10.09.15'' '' </ref> In a general contract it can be a three-party arrangement between the client, designer and the contractor. Here, the client must relate to two parties and get responsible for impact caused by design defects <ref>[''Shapiro, B.S''] "Design/build and turnkey contracts - advantages and disadvantages(Internet)",''Shapiro Hankinson & Knutson'', ''Available from: http://www.shk.ca/wp/wp-content/uploads/2013/02/Design-Build-and-Turnkey-Contracts-Advantages-and-Disadvantages.pdf'', Read: 10.09.15'' '' </ref> .

The responsibility that lays with the contractor does not only include risks and finishing the project. They are also responsible for the sub-contractors. Meaning social-dumping and the health and safety executive's.

Often, the client hires an independent consultant to advise in the pre-contract phase, and sometimes throughout the project. This way they make shore the constructor holds the requirements of the project. The third-party can be a project manager, engineer or another kind of consultant. As there often occur conflict of interest when a single contractor is responsible for the overall project on behalf of the client, a consultant could be wise. The constructor wants to complete the work as fast and cheap as possible, with a good quality of the finished project. Unfortunately, these goals often collide.

The turnkey structure is most applicable when there is a large amount of information early in the process, the building is repeated or it is a simple type. With these requirements, there is often less conflicts.

==Benefits==
One of the main benefits with this type of contracts is the possibility of savings in production cost. The client pay the contractor to take on the risks of the whole project. Further, because the contractor has responsibility for both design and construction, the building process can start before the design specifications are completed. This way, the contractor saves time and money. It is also more efficient when they are responsible for law related and local neighbour problems.

Turnkey contracts are most of all benefiting the client of the project in a risk-prospective. First of all, this type of contracting provides the client with just one source of responsibility. In contrast, the contractor alone has the responsibility of defects caused by both design and construction. Meaning that the client does not have to determine if a defect is caused by a misstep in design or construction - nevertheless, it is not their responsibility to fix it. The contractor must pay for any additional cost because of defective design or construction. Nothwithstanding, if the client authorized the construction plan in the start, then they are responsible of extra costs caused by insufficient plans.

Additionally, turnkey contracts will be attractive for client without proper technical knowledge. Owners do not need to have the detailed qualifications, that will be one of the constructor’s prerequisites.

On the other hand, the turnkey structure benefits contractors as well. Here, the contractor can choose their own team based on experiences. With good relationships to the sub-contractors, they have better overall control when they price the project. This way they limit the possibility of risks and increases the chance of profit.

==Limitations==
The construction business is a competitive industry and compared to other sectors there is high risks and low profit. In turnkey contracts tendering is the normal way to get the job. Then there is a fixed price and a packaged deal for the client. Therefore, the client decides the criteria. Sometimes they want specific firms or qualifications, but the lowest price is often the winner. A consequent of this is how the constructor price and handle the risks involved in the project. Often, the constructions are simple to get the cost-effectiveness at a level so they achieve an attractive price for the project. Another result can be deficient design, and without detailed checking from a third-part, this can be expensive.

Where there is not a fixed price, the client has to negotiate the price after hiring a contractor based on design. This way the contractor make sure they do not waste money on a proposal design. On the other hand the owner may not get the lowest possible price for the project.

Further, nordic standards for turnkey contracts, for example ABT 93 in Denmark, there is only a general risk distribution. There is a lot of uncertainties for the turnkey contractor and the sub-contactors, and it is a need for a safety net in the contract so that they have controll over variations under the project. A way to solve this could be payment at fixed milestones. Where there is a lump sum price for the project, the uncertainties are even bigger. This is mostly because it is hard to know the precise timeline and costs of a building project. Therefor the contract must be specified carefully.

With turnkey the client of the project will have a limitation to the amount of influence on design details. Examples of limitation can be knowledge of delays, defective work and extension of the time schedule. Overall they will have less control over the process. Since the client is not part of the construction part and have little verification options, they are exposed in a turnkey. This could result in improper payment and non-conforming work. In the traditional turnkey there is no party that protect the client’s interests, so they have to be extra aware of liquidity and responsibilities in different situations. Overall, the client have to relay on accurate information from the contractor about the progress of the project.

=Risk Profile as a tool=
Problems facing risks are often hard to clarify because it is not visible or definite. In every industry there is a form for risk, and there is various definitions of the term. In common they all sum up that risk is an unwanted uncertainty. How it is handled depends on the projects internal and external environment. By making a risk profile the parties involved can manage the risks effectively.

==Definition==
Before looking into risk profiling, it is important to define risk as a term. Risk will be determined differently depending on what industry your project is related to. Marc and Shapira (1987)<ref>[''Kwak, Y.H. & LaPlaced, K.S.(2005)''] "Examining risk tolerance in project-driven organization",''Project Management Program, The George Washington University', ''Available from: http://www.synergybusiness.com/files/PDF/White_Papers/Examining-Risk-Tolerance-in-Projectdriven-Organization.pdf'' '' </ref> described risk in technology-driven organizations as “distribution of possible outcomes, their likelihood, and their subjective values”. Meaning that risk will be individual for each firm in a specified project. Stakeholders will have another view at the risks and the outcome of a project compared to the client, the constructor or the consultant.

Risk profiling is an evaluation process of an organizations accept of risk, the level of risk to which they are exposed and the required risk finishing the job. It is an analytical tool used to plan [[Risk Management]]. This type of analytical techniques is used to define the total risk management situation of the project<ref>[''Project Management Body of Knowledge''] '''Project Management Institute (2013) "Chapter 11.2", "Fifth edition", ''Project Management Institute'' '' </ref>. For example, the profile will tell the clients, stakeholder or the consultants risk acceptance and enthusiasm according to the project. The profile will be specific for an individual company associated with a specific project.

It is important to have the risk profile tool separated from the risk assessment tool<ref>[''RiskBuisness''] ''"Risk Profiling Service(Internet 2011)", ''Available from: http://www.riskbusiness.com/Services/RiskBusiness%20Risk%20Profiling%20Service.pdfl'', Read: 12.09.15'' '' </ref>. The two develop dissimilar and provide different information for the project manager. The purpose of the risk assessment is to give a detailed picture of risks within single processes in a company, risk profiling is risk identification on a higher level.

==Structure==
As said, there are three aspects in risk profiling that must be evaluated and compared. Figure nr.2 shows the relationship between them:

[[File:RiskProfile.png|600px|]]
<br clear=all> '''Fig. 2''': The relationship between the three aspects in risk profiling<ref>[''Internet photography''] ''Available from: https://www.google.no/search?q=turnkey&biw=1264&bih=751&source=lnms&tbm=isch&sa=X&ved=0CAYQ_AUoAWoVChMIzvzTy7qQyAIVQ4wsCh0WcgCE#tbm=isch&q=risk+profile&imgrc=oIwLWc3-fhIZOM%3A
'', Found: 24.09.15'' '' </ref>.

===== Risk Tolerance =====
This is the level of risk that the organization is willing to take. It will affect the corporation’s decision making and the project itself. The firm´s acceptable risk must be coupled with the correct definition of risk.

Because of its human dynamic, you can say that risk tolerance is a subjective concept. To quantify risk tolerance you have to figure out the probabilities for occurrences and the impact of it. Too measure the impact and this psychological characteristic, you have to look at the circumstances, values, preferences, attitudes and investment and financial risk of the project for a specific firm.

When a firm knows the risks together with the likelihood of occurring, they can focus their resources towards the risks that lie above their acceptable risk level. This can be visualised as in the figure of probability and impact below:

[[File:risk1.png|600px|]]
<br clear=all> '''Fig. 3''': Visualization of probability and impact <ref>[''Photography''] ''From: Managing Construction Projects'',Winch, G.M. (2010), Second edition, Wiley-Blackwell '' </ref>.

At the figure you can see that the company's risk level drops when the impact increases. And for this example there is one risk factor over the acceptable level, and this must be addressed by the relevant company.

Another tool for understanding and visualizing risk tolerance is through a utility curve:

[[File:risk2.png|400px|]]
<br clear=all> '''Fig. 4''': Visualizing through utility curve <ref>[''Photography''] ''From: Managing Construction Projects'',Winch, G.M. (2010), Second edition, Wiley-Blackwell '' </ref>.

Figure 4 shows the relationship between cost and utility for a risk taking company. The more benefit they get, the more it costs.

Awareness of risk tolerance will give the project managers a better overlook on how they should use their recourses. By locating the most crucial risks they can be handled and eased to an acceptable level. The result will be better decision-making, shorter length of the project and avoidance of unexpected costs.

It also important to communicate the acceptable level of risk throughout the firm so everyone knows how innovation it is acceptable to be. Because it is important to remember that there is a close connection between risk tolerating and innovation.

Further, it is important to remember that risk tolerance is dynamic and that the acceptance of risk will change with time and through the project. Additionally, when you have bigger firms with portfolio management structure, it is important to recall that lowering the overall risks will not automatically work for a single project. It is a challenge to find a common risk tolerance for all participants for the whole life cycle of the project. The level of risk tolerance should be set already in the start when the purpose of the project is being clarified.

===== Risk Capacity =====
This is the measure of financial risk the firm can afford to take. The factor is based on the firm’s investment strategy and how the measured risks for the task required will negatively affect the company’s financial goal.

===== Risk Requirement =====
This is the actual risk the project requires to be done within the budget, schedule and the requirements of the client. Unlike the risk tolerance level, this is risk the firm has to take to reach the goals for the project. When a client espect more in return then the risk they are willing to take supply, you have to adjust the balance between the tow and the risk required will follow.

== Benefits ==
The success of a job comes easier with an in-depth understanding of the companies needs in accordance to their expectation of the outcome of the project. This way companies choose more wisely and avoid dangerous risks. In projects it is impossible to avoid risks, but when you understand them you can benefit from them.

== Limitations ==
There is no accurate way to measure a firms risk capacity and risk tolerance because it is unique for each company and exclusive for every single project<ref>[''Stammers, R''] "Beyond the Questionnaire: New Tools for Risk Profiling(Internet 2015)",''CFA institute'', ''Available from: https://annual.cfainstitute.org/2015/04/27/beyond-the-questionnaire-new-tools-for-risk-profiling/'', Read: 10.09.15'' '' </ref>.

Very often after a risk profiling, the client see that the risk tolerance and the two financial aspects, required and capacity of risk, are not the same. The result is often not reaching the goal of the task. In these cases adjustments must be done to make a balance between the risks taken and the goal of the task.

In the classic risk profiling you only look at the current situation of the firm in a particular project. With this method the history of the company will not be considered. By looking at a firm’s decision-making over time, their experience in the industry and their overall attitude to risk, it will give a better risk profile.

= Risk Profile in Turnkey Projects =
Risk and uncertainty are damaging factors in the construction industry. The industry is characterized by complexity, time pressure and a competitive marked. It can be very costly for actors not to account for risk. The chance of failure in turnkey shows that project coalition actors, especially contractors, need to have a framework for risk analysis. As the risks are subjective, risk profile in turnkey vil be different from [[Risk Profile in General Contracting]].

Ling (2005)<ref>[''Taylor, S. & Mbachu, J.(2014)''] "Profiling and mitigation risks in construction contracts",''School of Engineering and Advanced Technology, Massey University, Auckland'', ''Available from: http://construction.massey.ac.nz/NZBERS-2014_proc_fp_Taylor-S.pdf'' '' </ref> identified contractual global risk factors:
*The nature of the work.
*Combination of the workload and the constructor’s need of the project.
*The level of realistic pricing of the project.
*The amount of competition of the project.

These factors can be modified and used in analysis of turnkey contracts:
*Pricing of the project; client’s unreasonable expectations.
*The amount and use of subcontractors; inexperience, lac of skilled labour.
*The contractor; poor project management, incompetent team.
*The external risks.
*The internal risks of the different coalition actors.
*The unforeseen site conditions.

When you make a companies risk profile you need to look at the factors above. To be able to establish the level of risk tolerance and risk capacity you have to identify all risk required completing the firm’s task within the project. As explained, the risk tolerance is a psychological characteristic. Knowing the level and amount of risks will shape a company’s risk profile connected to a specific project.

As explained under the section about turnkey contracts, the use of tendering forces contractors to deliver as low price as possible for projects. A way to cut down on prices is to shorten the length of the project. It is not easy to know the amount of work or the exact timeline of a construction project. With this [[Management of risk]], the project manager can make a more likely picture of worst-case scenario of the project. This way the cost will match the amount of work to finish.

Further, in turnkey contracts the client pay contractors to take on the risk included in the responsibility of a overall project. If a project is successful or not depends on the contractors ability to take on the economic risk he agrees to. The client knows that unforeseen risks costs, that is why they prefer turnkey.

As a result of long experience and innovation, new technical solutions and building methods are being developed. Risk caused by this development are most often the clients responsibility. Meaning that the contractors responsible are based on result and effort<ref>[''Byggejuss''] "Kort om utviklingsrisikoen i entrepriseretten", ''Available from: http://byggejuss.no/kort-om-utviklingsrisikoen-i-entrepriseretten/'' ''Read 25.09.15'''' </ref>.

Baloi & Price (2003)<ref>[''Taylor, S. & Mbachu, J.(2014)''] "Profiling and mitigation risks in construction contracts",''School of Engineering and Advanced Technology, Massey University, Auckland'', ''Available from: http://construction.massey.ac.nz/NZBERS-2014_proc_fp_Taylor-S.pdf'' '' </ref>. found that there is a close relationship between the use of virtuous risk management tools and successful projects. This is logic; risk is defined after its impact on the objects set for the project. It is important that, for example, the consultants knows the risk they are taking on and their capacity so that they can price their risk and protect their interests in the contract. Still, many contactors rely on their own experiences and similar jobs to price the project. When doing this, it is important not to avoid innovation in the process.

The process from [[Risk Identification]] till pricing is crucial in a turnkey contract. It is important that the firm has a realistic expectation for the outcome of a project, combined with their risk tolerance. The contractor, consultant, owner or sub-contractor need to process the risks identified and construct preventing measures.

There is many companies that offers risk profile generators, such as OGC(www.ogc.gov.uk) or RiskBusiness. This way companies can look at their risk profile over time and see how it changes over a project. Another benefit is the opportunity to compare with similar companies in the industry or standardised risk profiles.

====A real life example====

A possible scenario would be when a construction consultant company evaluates if they want to apply for an advertised job by a constructor. Example of consultant in this case is COWI.

As explained under structure of turnkey contracts, the project has already started as the client has hired the constructor in form of turnkey. Example of the constructor is NCC and the client is Copenhagen Commune.

NCC has been hired to build a new public school at Vestrebrogade. To evaluate the possibility of success and profitability, COWI executes a risk profile for the project:
{| class="wikitable"
|+ style="caption-side:bottom; "|Table 1: An example of risk profiling as a tool.
! Analytique technique
! Approach
|-
|Risk Tolerance
|''. COWI has to look at the risks that NCC has taken on when they confirmed the project, and if they are transferred to them as consultants. Are these acceptable risks to COWI - can they handle a worst-case scenario? There is often more pressure for a consultant in a turnkey situation as the constructor is in a tense situation.
|-
|Risk Capacity
|''. After COWI has established the risk required and their tolerance level, they can evaluate if they have the necessary financial capacity to handle the risk-picture.
|-
| Risk Required
|'' . This is the first step COWI needs to address. Here, COWI needs to figure out the qualitative and quantitative risks of the project. After they have made a overall picture of the risks required to complete their task from NCC, they can evaluate their tolerance towards them.
|}

=Limitation=

Risk tolerance, as explained, is a crucial part of risk profiling.<ref>[''Kwak, Y.H. & LaPlaced, K.S.(2005)''] "Examining risk tolerance in project-driven organization",''Project Management Program, The George Washington University', ''Available from: http://www.synergybusiness.com/files/PDF/White_Papers/Examining-Risk-Tolerance-in-Projectdriven-Organization.pdf'' '' </ref> In big and complex project like turnkey, bad channels of communication results in miscommunication of risk acceptance level. Without [[Effective Communication in Project Management]], a normal outcome would be delays, overspending and displeased stakeholders. There are very few communication tools for project managers, and without a communication strategy the risk profiling has little value. Therefor the risk profile is depended on [[Communication in Project Management]]. Without this, the collation actors waste a lot of time and money on risk profiles that are not getting through to the right persons.

Further, a valid risk profile depends on the risk-taking and innovating culture of the workers in the company. It has no effect if the head of the firm has a high acceptance of risk if the workers are not included and work with the same innovating goal.

=Conclusion=

The main characteristic of turnkey is the financial aspect where the contractor is the center of risk responsible. Further, if the contract does not state otherwise, the contractor handles all risk included in the start-phase of the project, design and construction.

As discussed in the article, the risk in turnkey basically frames the contractor and the sub-contractors. Therefore for the contractor, the barrier for use of turnkey should be high. With for example program management of several turnkey project, a constructor face high probability of deficit. There need to be a balance between the risks identified and measures to handle it. On the other hand, a successful turnkey project will be time saving and consequently profitable.

Risk profiling basically paint a picture of the risk situation within a company related to a specific project. After defining turnkey and the risks contractors face, it is clear that consultants, sub-contractors and contractors will benefit from risk profiling. When all parties communicate their risk profile it will be a better overall understanding of the projects realistic time perspective and costs. This is benefiting pre-work for all parties involved in the project. The coalition actors now has a tool to pick suitable projects.

=References=
<references />

= Annotated Bibliography=

''Managing Construction Projects - Winch, G.M. (2010), Second edition, Wiley-Blackwell''

This book is first of all about “managing construction projects in an information processing approach”. It is intended for master students and other that want to develop their managing skills. The theories are supported by case studies, which make it easy to understand the basic. Additionally, it is updated after current standards and practice in the industry. Further, this book is used as class material at DTU at the Civil Engineer Department, which make it a legit and secure source. In my study, this book was highly relevant, especially for the sections about turnkey structure and the risks involved.

''Taylor, S. & Mbachu, J.(2014) - "Profiling and mitigation risks in construction contracts",School of Engineering and Advanced Technology, Massey University, Auckland, Available from: http://construction.massey.ac.nz/NZBERS-2014_proc_fp_Taylor-S.pdf''

This article is made as a case study of “profiling and mitigating risks in construction contracts”, mainly looking at New Zealand. General terms and risk factors were presented in an enlightening way. The text is easy to read and the structure fits the content. I used the article as a base for my explanation on tender biding and pricing on risky projects. The study is written on behalf of “New Zealand built environment research symposium,” and therefore a legit source.

''Shapiro, B.S - "Design/build and turnkey contracts - advantages and disadvantages(Internet)",Shapiro Hankinson & Knutson, Available from: http://www.shk.ca/wp/wp-content/uploads/2013/02/Design-Build-and-Turnkey-Contracts-Advantages-and-Disadvantages.pdf, Read: 10.09.15''

The law corporation “Shapiro Hankinson & Knutsen” from Vancouver, Canada, publishes this article. In the study there is a discussion about advantages and disadvantages of design/build and turnkey contracts. It has a good structure where the definitions come first, then benefits and limitations, conclusion and in the end alternatives for contractual arrangements. It gave me a good overall picture over turnkey contracting and how it is benefitting the different actors involved in a turnkey project.

Financial Portfolio Optimization Methods

2015-10-04T16:07:28Z

Andkamp: /* Annotated Bibliography */

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>r_{ti}</math> is the [[wikipedia:rate_of_return|rate of return]] of asset <math>i</math> on day <math>t</math>, and <math>T</math> number of time intervals. Objective function aims to minimize the square of the [[wikipedia:Downside_risk|downside semi-variance]] <math>z_{t}</math> This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector <ref>Embrechts, P., Resnick, S. I., & Samorodnitsky, G. (1999). Extreme value theory as a risk management tool. North American Actuarial Journal, 3(2), 30-41.</ref>. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value of <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref> Sharpe, W. F. (1963). A simplified model for portfolio analysis. Management science, 9(2), 277-293. </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios<ref>Wilding, T. (2003). Using genetic algorithms to construct portfolios. Advances in portfolio construction and implementation, 135-160.</ref>. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature, it is reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]
=Annotated Bibliography=
Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2009). Modern portfolio theory and investment analysis. John Wiley & Sons.
A book that examines the characteristics and analysis of individual securities as well as the theory and practice of optimally combining securities into portfolios. It stresses the economic intuition behind the subject matter while presenting advanced concepts of investment analysis and portfolio management. Readers will also discover the strengths and weaknesses of modern portfolio theory as well as the latest breakthroughs.

Schwalbe, K. (2013). Information technology project management. Cengage Learning. This book demonstrates the principles distinctive to managing information technology (IT) projects that extend beyond standard project management requirements. The book weaves today's theory with successful practices for an understandable, integrated presentation that focuses on the concepts, tools, and techniques that are most effective today.

=References=
<references/>

Financial Portfolio Optimization Methods

2015-10-04T16:06:56Z

Andkamp:

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>r_{ti}</math> is the [[wikipedia:rate_of_return|rate of return]] of asset <math>i</math> on day <math>t</math>, and <math>T</math> number of time intervals. Objective function aims to minimize the square of the [[wikipedia:Downside_risk|downside semi-variance]] <math>z_{t}</math> This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector <ref>Embrechts, P., Resnick, S. I., & Samorodnitsky, G. (1999). Extreme value theory as a risk management tool. North American Actuarial Journal, 3(2), 30-41.</ref>. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value of <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref> Sharpe, W. F. (1963). A simplified model for portfolio analysis. Management science, 9(2), 277-293. </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios<ref>Wilding, T. (2003). Using genetic algorithms to construct portfolios. Advances in portfolio construction and implementation, 135-160.</ref>. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature, it is reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]
=Annotated Bibliography=
Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2009). Modern portfolio theory and investment analysis. John Wiley & Sons.
A book that examines the characteristics and analysis of individual securities as well as the theory and practice of optimally combining securities into portfolios. It stresses the economic intuition behind the subject matter while presenting advanced concepts of investment analysis and portfolio management. Readers will also discover the strengths and weaknesses of modern portfolio theory as well as the latest breakthroughs.
Schwalbe, K. (2013). Information technology project management. Cengage Learning. This book demonstrates the principles distinctive to managing information technology (IT) projects that extend beyond standard project management requirements. The book weaves today's theory with successful practices for an understandable, integrated presentation that focuses on the concepts, tools, and techniques that are most effective today.

=References=
<references/>

Talk:Financial Portfolio Optimization Methods

2015-09-29T13:24:16Z

Andkamp: /* Reviewer 2 - Biankajuh */

Josef: Thank you for an interesting, and already rather detailed, Wiki article.

What I struggle with is the relationship of your article with the management of project portfolios. There are in fact serious limitations to the applicability of financial portfolio theory to project portfolios, for example the assumptions that you can invest/divest into options without changing their risk/return balance, or the assumption that you can actually divest from options ("selling" a failing project will almost always be impossible, as I am not aware of a secondary market for projects).
I am not sure how we can "salvage" all the details you have already produced. What I would suggest is to focus on what part of financial portfolio management theory is applicable to project portfolio management, or better, why it is not applicable.

== Reviewer 1 – User: s141938 ==

+

* Methods are really thoroughly described. Nothing is left out. I am wondering if you could specify which ones are popular and which ones have been left out*
* When you’re describing the formulas, you describe also the meaning of the variables. Perfect! But some have been left out like in CVar and semi-variance - the z and T and r
* I like the application part at the end of the article, showing where this optimization is still in use. But then again, could you give more details about which method is more popular
* References are as they should be : quick description, well formatted, good sources
* Links to other articles. Keep it that way
* see also section is nice

-
* You could write a quick abstract at the beginning to give a general overview of the article:Fixed it
* Like I have already mentioned check all of the variables in each formulas (CVar, semi-variance - the z and T and r, ):Fixed it
* I was a bit confused by the usage of different words sometimes. For instance in the mean variance method you are talking about bonds, then in other methods it’s about portfolios. I think that you could explain the key concepts more thoroughly in the first block and make it a separate paragraph, so that it is easy to spot when going through the article quickly:Bonds was changed into assets, recheck the term <math>w_{i}</math> in every model. You'll see that this refers to each asset of a portfolio
* Try to structure things a bit. For instance separate the first block into „Introduction" and "main concepts and principles"
* Elaborate the idea of financial portfolio, unless there is nothing more to add:Nothing more to add,you can check the reference
* Divide the text into paragraphs -> might be easier to read or at least scan
* Could you be a bit more precise about the assumptions - who made them ?:Two references on the assumptions state the persons that made these assumptions
* I think it would be nice to have a comparison of the effectiveness of the different methods:That should be a whole different article as it is more of a matter of taste what is more efficient in different areas and circumstances. There is a lot of literature on that, you can chack it through the references
* SOmetimes I think you can drop some of the references like in the following example - mixed integer programming for portfolio selection with pragmatic characteristics [24] [25] [26] [27] [28]:Fixed it
* COuld you write the advantages of the methods - it would stick more to the structure of a method article (unless there are no advantages):No specific advantages except the obvious advantages of linear against quadratic problems in terms of solvability
* chech the "in order to” cause there are plenty of missing „to” like in "In order a business to minimize”:Done it, thanks!

Conclusion : Nice article, lots of information but try to structure things to make it more reader-friendly.

== Reviewer 2 - Biankajuh ==
# I have find the description of the models very accurate. Also commendable that you mention the implementations and the limitation/disadvantages of the certain Financial Portfolio Optimization Methods. Furthermore, I really like the usage of direct links to other articles and websites. It shows a very precise job and makes it easy to follow and read up in the certain topic.
# Structural suggestions:
## I would suggest to write a brief abstract section for the beginning of the article. That would help the reader to find a short few sentences summary about the main aspect and purpose of this work.:did it,thanks for the info!
## Did I assume correctly that the section of “Financial Portfolio Optimization Methods in PPM” is more about the history? Would it make more sense to refer on it rather as History perhaps also in the title of the section?:Fixed it, I think it is more clear now
# As for me, you could make the article easier to read by defining some expressions such as “assets” in the given circumstances (similarly like you defined “investor” in the same paragraph). Please also consider to define abbreviation such as MPT. (Financial Portfolio Optimization Methods in PPM):Fixed it, and recheck the definition of assets, given as " Consider "assets" to be financial, physical, or information"
# The article is nicely illustrated with the pictures which help the understanding. Although, I would suggest to name them as ‘’Figure 1, 2, 3, …etc.’’ which would allow to refer to the pictures at the relevant place of the text. For example: “Figure shows an example where all the possible portfolios which are formed based on the expected return and risk relations.” - Here you could specify by saying Figure 2.:Fixed that,thanks!
# At the first picture entitled as “Translation of MPT criterias to PPM criterias”, you use MPT in the title and also in the article above it, however, it says MPM in the figure itself. Are MPT and MPM referring to the same subject? Could you define what they are?:Fixed the figure,my mistake.!
# Grammatical/Formatting hints:
## “Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects” (Financial Portfolio Optimization Methods in PPM) —> Dot is missing from the end of the sentence. Plus I would suggest to change the word order for the following: “Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking the interaction and influence of other projects in consideration.”

'''Reviewer 3 – User: s113735'''

Feedback:

Formal Aspects:
*The article clearly follows the “method or tool” requirement for the wiki article
*I recognize very few spelling errors. Things I were able to find are small mistakes like omitting a word or missing a large letter in the beginning of a new sentence:
“In order [for] a business to minimize the danger of exposure to a failed project… “:Fixed,thanks.!
“… strategic alignment and resource levelling. [A]<strike>a</strike>pplication of such methods…”
*I am not sure I fully understand figure 1: “Tranlation of MPT criterias to PPM criterias”, maybe this can be elaborated better? You second figure is well explained and easy to understand.
*You make great use of the <nowiki><math></math></nowiki> tool in your article. And although the content is quite complex, it gives the article a wiki-“esque” feeling, which is good.

Content Aspects:

*The article is clearly within the topic scope, as it directly relates to a “portfolio” topic
*You clearly have extensive knowledge of the subject and your very rigid use of references show me that you back up every statement you make. Sadly, (and this is of no fault of yours or a drawback to the content of the article) most of your references are hard-copy (aka. Book) references so I do not currently have to possibility to review your sources.
*“Northwestern” part of the curve… Why not “Top left?”:Fixed!Thanks!
*I have no idea what “[καινουριο paper CVaR]” (in the CVaR model section) is or means, maybe this is something you left by mistake in the article?:Some greek over there..Fixed it.!
*The size, quality and references of the article are definitely up to par. It is very clear that you know much more about the topic than I ever will.

Overall conclusion:

As mentioned, it is very clear by the content and quality of the article, that you have extensive knowledge of the tools you present. The only, albeit minor, drawback is that I feel the article is not very engaging. I think the article could benefit a lot from a few “bridging” sentences about the '''general''' purpose of the models you present – some more example uses or sentences of how exactly they relate to portfolio management. I think that would really bind the article together.

Talk:Financial Portfolio Optimization Methods

2015-09-29T13:20:20Z

Andkamp: /* Reviewer 1 – User: s141938 */

Josef: Thank you for an interesting, and already rather detailed, Wiki article.

What I struggle with is the relationship of your article with the management of project portfolios. There are in fact serious limitations to the applicability of financial portfolio theory to project portfolios, for example the assumptions that you can invest/divest into options without changing their risk/return balance, or the assumption that you can actually divest from options ("selling" a failing project will almost always be impossible, as I am not aware of a secondary market for projects).
I am not sure how we can "salvage" all the details you have already produced. What I would suggest is to focus on what part of financial portfolio management theory is applicable to project portfolio management, or better, why it is not applicable.

== Reviewer 1 – User: s141938 ==

+

* Methods are really thoroughly described. Nothing is left out. I am wondering if you could specify which ones are popular and which ones have been left out*
* When you’re describing the formulas, you describe also the meaning of the variables. Perfect! But some have been left out like in CVar and semi-variance - the z and T and r
* I like the application part at the end of the article, showing where this optimization is still in use. But then again, could you give more details about which method is more popular
* References are as they should be : quick description, well formatted, good sources
* Links to other articles. Keep it that way
* see also section is nice

-
* You could write a quick abstract at the beginning to give a general overview of the article:Fixed it
* Like I have already mentioned check all of the variables in each formulas (CVar, semi-variance - the z and T and r, ):Fixed it
* I was a bit confused by the usage of different words sometimes. For instance in the mean variance method you are talking about bonds, then in other methods it’s about portfolios. I think that you could explain the key concepts more thoroughly in the first block and make it a separate paragraph, so that it is easy to spot when going through the article quickly:Bonds was changed into assets, recheck the term <math>w_{i}</math> in every model. You'll see that this refers to each asset of a portfolio
* Try to structure things a bit. For instance separate the first block into „Introduction" and "main concepts and principles"
* Elaborate the idea of financial portfolio, unless there is nothing more to add:Nothing more to add,you can check the reference
* Divide the text into paragraphs -> might be easier to read or at least scan
* Could you be a bit more precise about the assumptions - who made them ?:Two references on the assumptions state the persons that made these assumptions
* I think it would be nice to have a comparison of the effectiveness of the different methods:That should be a whole different article as it is more of a matter of taste what is more efficient in different areas and circumstances. There is a lot of literature on that, you can chack it through the references
* SOmetimes I think you can drop some of the references like in the following example - mixed integer programming for portfolio selection with pragmatic characteristics [24] [25] [26] [27] [28]:Fixed it
* COuld you write the advantages of the methods - it would stick more to the structure of a method article (unless there are no advantages):No specific advantages except the obvious advantages of linear against quadratic problems in terms of solvability
* chech the "in order to” cause there are plenty of missing „to” like in "In order a business to minimize”:Done it, thanks!

Conclusion : Nice article, lots of information but try to structure things to make it more reader-friendly.

== Reviewer 2 - Biankajuh ==
# I have find the description of the models very accurate. Also commendable that you mention the implementations and the limitation/disadvantages of the certain Financial Portfolio Optimization Methods. Furthermore, I really like the usage of direct links to other articles and websites. It shows a very precise job and makes it easy to follow and read up in the certain topic.
# Structural suggestions:
## I would suggest to write a brief abstract section for the beginning of the article. That would help the reader to find a short few sentences summary about the main aspect and purpose of this work.
## Did I assume correctly that the section of “Financial Portfolio Optimization Methods in PPM” is more about the history? Would it make more sense to refer on it rather as History perhaps also in the title of the section?
# As for me, you could make the article easier to read by defining some expressions such as “assets” in the given circumstances (similarly like you defined “investor” in the same paragraph). Please also consider to define abbreviation such as MPT. (Financial Portfolio Optimization Methods in PPM)
# The article is nicely illustrated with the pictures which help the understanding. Although, I would suggest to name them as ‘’Figure 1, 2, 3, …etc.’’ which would allow to refer to the pictures at the relevant place of the text. For example: “Figure shows an example where all the possible portfolios which are formed based on the expected return and risk relations.” - Here you could specify by saying Figure 2.
# At the first picture entitled as “Translation of MPT criterias to PPM criterias”, you use MPT in the title and also in the article above it, however, it says MPM in the figure itself. Are MPT and MPM referring to the same subject? Could you define what they are?
# Grammatical/Formatting hints:
## “Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects” (Financial Portfolio Optimization Methods in PPM) —> Dot is missing from the end of the sentence. Plus I would suggest to change the word order for the following: “Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking the interaction and influence of other projects in consideration.”

'''Reviewer 3 – User: s113735'''

Feedback:

Formal Aspects:
*The article clearly follows the “method or tool” requirement for the wiki article
*I recognize very few spelling errors. Things I were able to find are small mistakes like omitting a word or missing a large letter in the beginning of a new sentence:
“In order [for] a business to minimize the danger of exposure to a failed project… “
“… strategic alignment and resource levelling. [A]<strike>a</strike>pplication of such methods…”
*I am not sure I fully understand figure 1: “Tranlation of MPT criterias to PPM criterias”, maybe this can be elaborated better? You second figure is well explained and easy to understand.
*You make great use of the <nowiki><math></math></nowiki> tool in your article. And although the content is quite complex, it gives the article a wiki-“esque” feeling, which is good.

Content Aspects:

*The article is clearly within the topic scope, as it directly relates to a “portfolio” topic
*You clearly have extensive knowledge of the subject and your very rigid use of references show me that you back up every statement you make. Sadly, (and this is of no fault of yours or a drawback to the content of the article) most of your references are hard-copy (aka. Book) references so I do not currently have to possibility to review your sources.
*“Northwestern” part of the curve… Why not “Top left?”
*I have no idea what “[καινουριο paper CVaR]” (in the CVaR model section) is or means, maybe this is something you left by mistake in the article?
*The size, quality and references of the article are definitely up to par. It is very clear that you know much more about the topic than I ever will.

Overall conclusion:

As mentioned, it is very clear by the content and quality of the article, that you have extensive knowledge of the tools you present. The only, albeit minor, drawback is that I feel the article is not very engaging. I think the article could benefit a lot from a few “bridging” sentences about the '''general''' purpose of the models you present – some more example uses or sentences of how exactly they relate to portfolio management. I think that would really bind the article together.

Financial Portfolio Optimization Methods

2015-09-28T18:08:06Z

Andkamp:

Financial Portfolio Optimization Methods

2015-09-28T18:07:52Z

Andkamp:

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>r_{ti}</math> is the [[wikipedia:rate_of_return|rate of return]] of asset <math>i</math> on day <math>t</math>, and <math>T</math> number of time intervals. Objective function aims to minimize the square of the [[wikipedia:Downside_risk|downside semi-variance]] <math>z_{t}</math> This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector <ref>Embrechts, P., Resnick, S. I., & Samorodnitsky, G. (1999). Extreme value theory as a risk management tool. North American Actuarial Journal, 3(2), 30-41.</ref>. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value of <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref> Sharpe, W. F. (1963). A simplified model for portfolio analysis. Management science, 9(2), 277-293. </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios<ref>Wilding, T. (2003). Using genetic algorithms to construct portfolios. Advances in portfolio construction and implementation, 135-160.</ref>. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature, it is reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-27T19:14:05Z

Andkamp: /* Mean Absolute Deviation Model */

Financial Portfolio Optimization Methods

2015-09-25T08:12:22Z

Andkamp: /* Disadvantages */

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>r_{ti}</math> is the [[wikipedia:rate_of_return|rate of return]] of asset <math>i</math> on day <math>t</math>, and <math>T</math> number of time intervals. Objective function aims to minimize the square of the [[wikipedia:Downside_risk|downside semi-variance]] <math>z_{t}</math> This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector <ref>Embrechts, P., Resnick, S. I., & Samorodnitsky, G. (1999). Extreme value theory as a risk management tool. North American Actuarial Journal, 3(2), 30-41.</ref>. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value of <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios<ref>Wilding, T. (2003). Using genetic algorithms to construct portfolios. Advances in portfolio construction and implementation, 135-160.</ref>. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature, it is reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T19:08:14Z

Andkamp: /* Additional Constraints */

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>r_{ti}</math> is the [[wikipedia:rate_of_return|rate of return]] of asset <math>i</math> on day <math>t</math>, and <math>T</math> number of time intervals. Objective function aims to minimize the square of the [[wikipedia:Downside_risk|downside semi-variance]] <math>z_{t}</math> This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector <ref>Embrechts, P., Resnick, S. I., & Samorodnitsky, G. (1999). Extreme value theory as a risk management tool. North American Actuarial Journal, 3(2), 30-41.</ref>. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value of <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios<ref>Wilding, T. (2003). Using genetic algorithms to construct portfolios. Advances in portfolio construction and implementation, 135-160.</ref>. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T18:52:08Z

Andkamp: /* Financial Portfolio Optimization Methods in PPM */

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>r_{ti}</math> is the [[wikipedia:rate_of_return|rate of return]] of asset <math>i</math> on day <math>t</math>, and <math>T</math> number of time intervals. Objective function aims to minimize the square of the [[wikipedia:Downside_risk|downside semi-variance]] <math>z_{t}</math> This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector <ref>Embrechts, P., Resnick, S. I., & Samorodnitsky, G. (1999). Extreme value theory as a risk management tool. North American Actuarial Journal, 3(2), 30-41.</ref>. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value of <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios<ref>Wilding, T. (2003). Using genetic algorithms to construct portfolios. Advances in portfolio construction and implementation, 135-160.</ref>. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T18:41:58Z

Andkamp: /* Cardinality constraints */

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>r_{ti}</math> is the [[wikipedia:rate_of_return|rate of return]] of asset <math>i</math> on day <math>t</math>, and <math>T</math> number of time intervals. Objective function aims to minimize the square of the [[wikipedia:Downside_risk|downside semi-variance]] <math>z_{t}</math> This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector <ref>Embrechts, P., Resnick, S. I., & Samorodnitsky, G. (1999). Extreme value theory as a risk management tool. North American Actuarial Journal, 3(2), 30-41.</ref>. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value of <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios<ref>Wilding, T. (2003). Using genetic algorithms to construct portfolios. Advances in portfolio construction and implementation, 135-160.</ref>. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T18:34:11Z

Andkamp: /* Mean Semi-Variance Model */

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>r_{ti}</math> is the [[wikipedia:rate_of_return|rate of return]] of asset <math>i</math> on day <math>t</math>, and <math>T</math> number of time intervals. Objective function aims to minimize the square of the [[wikipedia:Downside_risk|downside semi-variance]] <math>z_{t}</math> This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector <ref>Embrechts, P., Resnick, S. I., & Samorodnitsky, G. (1999). Extreme value theory as a risk management tool. North American Actuarial Journal, 3(2), 30-41.</ref>. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value of <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T18:26:45Z

Andkamp: /* CVaR Model */

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>r_{ti}</math> is the [[wikipedia:rate_of_return|rate of return]] of asset <math>i</math> on day/month/year <math>t</math>, and <math>T</math> number of time intervals. This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector <ref>Embrechts, P., Resnick, S. I., & Samorodnitsky, G. (1999). Extreme value theory as a risk management tool. North American Actuarial Journal, 3(2), 30-41.</ref>. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value of <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T18:24:00Z

Andkamp: /* Mean Semi-Variance Model */

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>r_{ti}</math> is the [[wikipedia:rate_of_return|rate of return]] of asset <math>i</math> on day/month/year <math>t</math>, and <math>T</math> number of time intervals. This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value of <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T18:19:28Z

Andkamp: /* CVaR Model */

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value of <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T18:18:53Z

Andkamp: /* CVaR Model */

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& \eta+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Term <math>\beta</math> is the investors confidence level as defined in [[wikipedia:Value_at_risk|VaR]]. The term <math>\eta</math> is a real variable taking the value οφ <math>\beta</math>-[[wikipedia:quantile|quantile]] of the decision variables vector at the optimum<ref>Ogryczak, W., and A. Ruszczy´nski (2002). Dual stochastic dominance and related mean-risk models. SIAM
Journal on Optimization, 13, 60–78.</ref>. Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T08:13:10Z

Andkamp:

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management.
= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T08:12:49Z

Andkamp: /* Financial Portfolio Optimization Methods in PPM */

= Financial Portfolio Optimization Methods in PPM =

In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

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Andkamp: /* Financial Portfolio Optimization Methods in PPM */

Financial Portfolio Optimization Methods

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Andkamp: /* Background */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure 2]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T07:56:20Z

Andkamp: /* Background */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| '''Figure 2:''' [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]

==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T07:55:54Z

Andkamp: /* Financial Portfolio Optimization Methods in PPM */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource leveling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T07:53:17Z

Andkamp: /* Financial Portfolio Optimization Methods in PPM */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb|'''Figure 1:''' Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T07:52:34Z

Andkamp: /* Financial Portfolio Optimization Methods in PPM */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order for a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T07:49:53Z

Andkamp: /* Financial Portfolio Optimization Methods in PPM */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:Pm.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

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Financial Portfolio Optimization Methods

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= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects.<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T07:23:31Z

Andkamp: /* Mean Variance Model */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> assets. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-23T07:00:16Z

Andkamp: /* Financial Portfolio Optimization Methods in PPM */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in Modern Portfolio Theory (MPT ) were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-22T17:19:38Z

Andkamp: /* Financial Portfolio Optimization Methods in PPM */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in MPT theory were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. Application of methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-22T17:18:36Z

Andkamp: /* Financial Portfolio Optimization Methods in PPM */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in MPT theory were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. Application of such methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-22T17:17:34Z

Andkamp: /* Financial Portfolio Optimization Methods in PPM */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Each of Markowitz’ principles in MPT theory were translated into a criterion for project prioritization that aids in the success of project portfolio management (PPM). In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. application of such methods of financial portfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Talk:Managing Uncertainty and Risk on the Project

2015-09-22T17:12:02Z

Andkamp:

Review1, Buurbuur:
- I found the topic very interesting and very relevant for this course.
- Good that many point are illustrated with an appropriate figure.
- However I find the scope very broad, maybe choose only to focus on project or portofolio management.
- In the ''definition'' section i find it difficult to understand and navigate between all the different definitions, maybe it could make it more easy and understandable if you make a grid to show the difference. Or list them up on bulletspoints
- Secure that the article fit to one of the two article types i relation to the required topics
- However I find the figures clear and understandable
- Good that there are references on every figure

Anna: It is an interesting and very relevant topic, however, I would suggest that you try to narrow down the scope and focus your article on one method/tool that can be applied for managing uncertainty and risk in a project. Remember that your article has to fit one of the two article types and the required structure :)

Reviewer 3:Andkamp
*Great use of English and nice sentences structure
*Great subject but, very broad, needs to be narrowed a little bit
*Nice figures, but the matrix with the papers is something I expected to be illustrated in matrix and not a Figure, in order to maintain the wiki style of the article.
*Nice Referencing of the figures, although it hadn't any inter-wiki links or in text links.
*The interest of a practitioner could be greater if there was more focusing in aspects of this subject.
*Profound relation with project management
*The good structure assures logical flow
*Extremely poor reference material for such extended article, that could lead someone with bad intensions to think that there's some amount of plagiarism
*Expected to see pros/cons
*Implementations would narrow the whole subject and make it more interesting
Generally speaking a more wiki style text, with hyperlinks,more bibliography and see also section would be more appropriate to this great text.

Talk:SCRUM Method

2015-09-22T15:38:52Z

Andkamp:

Mette: Hey, I like your idea. Perhaps you need to be more specific of what exactly it is you want to look at in your article, what you want to study and discuss in your article. Maybe you have a particular aspect to look at, study in your article.
Reviewer 2 :Andkamp
Formal aspects:
*Great interest on the method described on the article, with nice structure and points
*Nice style, with precise sentences, but maybe it needs a little improvement in English as it has some minor mistakes (for example: insure the team working on highest valued features->to ensure that the team is...)
*Nice figures, but better quality and explanation of them would be appreciated
*Better structure as a wiki article would be preferable as there are no inter-wiki or other in text hyperlinks
*Do not know if the author has the right to use those images as there is no reference
*Pictures are nicely placed
*Scrum method seems highly interesting for a practitioner, although no real implementation were presented in the article
Content aspects:
*Firmly related to the topic of project management
*Appropriate length, although points 2.1.1-2.2.5 could be a little more extensive
*More things could be added in the introduction section
*One sided sources and only sites, no proper literature
*Could be connected with other articles through a see also section
*Could add some real examples of implementation of this method
All in all it is a nice article that gave me some understanding of SCRUM method but the lack of bibliography and implementations,as well as the small extent of points 2.1.1-2.2.5 made me a have more questions than answers after reading it.

Financial Portfolio Optimization Methods

2015-09-22T15:31:20Z

Andkamp:

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Markowitz’ principles in MPT theory were translated into a criterion for project prioritization that aids in the success of project portfolio management. In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. application of such methods of financial protfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in upper left part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-22T15:30:40Z

Andkamp:

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Markowitz’ principles in MPT theory were translated into a criterion for project prioritization that aids in the success of project portfolio management. In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. application of such methods of financial protfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in "northwestern" part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Talk:SCRUM Method

2015-09-22T15:29:11Z

Andkamp:

Mette: Hey, I like your idea. Perhaps you need to be more specific of what exactly it is you want to look at in your article, what you want to study and discuss in your article. Maybe you have a particular aspect to look at, study in your article.
Reviewer 2 :Andkamp

*Great interest on the method described on the article, with nice structure and points
*Nice style, with precise sentences, but maybe it needs a little improvement in English as it has some minor mistakes
*Nice figures, but better quality and explanation of them would be appreciated
*Better structure as a wiki article would be preferable as there are no inter-wiki or other in text hyperlinks
*Do not know if the author has the right to use those images as there is no reference
*Pictures could nicely placed
*Scrum method seems highly interesting for a practitioner, although no real implementation were presented in the article
*Firmly related to the topic of project management
*Appropriate length, although points 2.1.1-2.2.5 could be a little more extensive
*More things could be added in the introduction section
*One sided sources and only sites, no proper literature
*Could be connected with other articles through a see also section
*Could add some real examples of implementation of this method

Talk:Project Execution Model (PEM)

2015-09-22T14:37:27Z

Andkamp:

Mette: Very nice topic choice that fits the requirements for the type of article. Remember the structure of a "method article". Look forward to reading more about this tool.

Reviewer 1:Andkamp

*Describes an interesting method developed by Novo Nordisk, a nice blend between case study and method description
*Minor English mistakes, that can be corrected through second time reading
*Good presentation of the topics, however some minor grammar mistakes make it a little more difficult to be read
*Nice figures, but missing captions and misplacement could provide a better outlook.
*Main points are clear but not extensively described, in order the reader to get a better understanding of the topic.
*No figure reference
*More effort in inter-wiki links and hyperlinks in the text should be done
*Better matrix of plan communication of stakeholders
*Missing references and bibliography
*Interesting subject with various aspects
*Length of the article is small, as probably it is unfinished, but there is a clear enthusiasm on that subject
*Phases makes a concrete structure of the article but the sections should be better presented in the table of contents.
*It would be nice to see implementations on other business, as well as the pros and cons of this method
*All in all, a nice subject that needs more effort in order to be useful and be connected with subjects such as Project Evaluation and Selection for the Formation of the Optimal Portfolio

Talk:Project Execution Model (PEM)

2015-09-22T14:35:11Z

Andkamp:

Mette: Very nice topic choice that fits the requirements for the type of article. Remember the structure of a "method article". Look forward to reading more about this tool.

Reviewer 1:Andkamp

• Describes an interesting method developed by Novo Nordisk, a nice blend between case study and method description
• Minor English mistakes, that can be corrected through second time reading
• Good presentation of the topics, however some minor grammar mistakes make it a little more difficult to be read
• Nice figures, but missing captions and misplacement could provide a better outlook.
• Main points are clear but not extensively described, in order the reader to get a better understanding of the topic.
• No figure reference
• More effort in inter-wiki links and hyperlinks in the text should be done
• Better matrix of plan communication of stakeholders
• Missing references and bibliography
• Interesting subject with various aspects
• Length of the article is small, as probably it is unfinished, but there is a clear enthusiasm on that subject
• Phases makes a concrete structure of the article but the sections should be better presented in the table of contents.
• It would be nice to see implementations on other business, as well as the pros and cons of this method
• All in all, a nice subject that needs more effort in order to be useful and be connected with subjects such as Project Evaluation and Selection for the Formation of the Optimal Portfolio

Financial Portfolio Optimization Methods

2015-09-21T14:39:04Z

Andkamp: /* Cardinality constraints */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Markowitz’ principles in MPT theory were translated into a criterion for project prioritization that aids in the success of project portfolio management. In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. application of such methods of financial protfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in "northwestern" part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR[καινουριο paper CVaR]. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that these constraints are a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-21T14:38:08Z

Andkamp: /* Mean Variance Model */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Markowitz’ principles in MPT theory were translated into a criterion for project prioritization that aids in the success of project portfolio management. In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. application of such methods of financial protfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in "northwestern" part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of this quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR[καινουριο paper CVaR]. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that the above limitation is a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-21T14:37:12Z

Andkamp: /* Mean Semi-Variance Model */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Markowitz’ principles in MPT theory were translated into a criterion for project prioritization that aids in the success of project portfolio management. In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. application of such methods of financial protfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in "northwestern" part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of the above quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance.<ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

This formulation led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR[καινουριο paper CVaR]. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that the above limitation is a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-21T14:34:27Z

Andkamp: /* Cardinality constraints */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Markowitz’ principles in MPT theory were translated into a criterion for project prioritization that aids in the success of project portfolio management. In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. application of such methods of financial protfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in "northwestern" part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of the above quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance such as discussed above <ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR[καινουριο paper CVaR]. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the [[#Mean Variance Model|models]] can be done by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that the above limitation is a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-21T14:32:13Z

Andkamp: /* Transaction roundlots */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Markowitz’ principles in MPT theory were translated into a criterion for project prioritization that aids in the success of project portfolio management. In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. application of such methods of financial protfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in "northwestern" part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of the above quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance such as discussed above <ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR[καινουριο paper CVaR]. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an[[#Mean Variance Model| above-mentioned portfolio optimization model ]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the above models by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that the above limitation is a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
<references/>

Financial Portfolio Optimization Methods

2015-09-21T14:30:16Z

Andkamp: /* Transaction roundlots */

= Financial Portfolio Optimization Methods in PPM =

In today's globalized market, financial risk and treatment of it that has gained great importance, especially after the [[Wikipedia:Financial_Crisis_of_2008|Financial crisis of 2008]], where factors which may affect the fragile global economy proved to be thousands and often unconnected to each other. [http://www.telegraph.co.uk/news/worldnews/europe/greece/11705720/European-debt-crisis-Its-not-just-Greece-thats-drowning-in-debt.html Nations fail to pay their debts] and [http://www.telegraph.co.uk/finance/financialcrisis/6173145/The-collapse-of-Lehman-Brothers.html giants of the finance industry bailed out] <ref> McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. </ref>. These financial institutions have developed various quantitative methods which can give a prediction of this risk level in financial portfolios. A financial portfolio is considered the summary of investments owned by an investor ( company or individual )<ref>[http://www.investopedia.com/terms/p/portfolio.asp] Investopedia, Portfolio definition and explanation, Retrieved September 2015</ref>. The first step for the quantitative measurement of risk in portfolios was made by Harry Markowitz in 1952 <ref name="Harry1">Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.</ref>, with the development of the mean-variance model as risk measurement, which shows interest until today and it is used by investors. Thereafter, various other methods were developed, focusing on alternative risk measures that could lead to linearization of the portfolio optimization problem <ref>Sharpe, W. F. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(05), 1263-1275.</ref> . The main principles of risk handing and best profit can be applied in project management. In order a business to minimize the danger of exposure to a failed project, financial portfolio methods can be applied. Consider "assets" to be financial, physical, or information that combined with other assets in a project will increase value. The "group" of assets is designed to achieve the growth of value at acceptable levels of risk over the longer term. The "investor" is a business manager whose job it is to put assets to function efficiently as a portfolio <ref>Nikonov, O.V, (2007). Efficient Project Portfolio as a tool for Enterprise Risk Management. ERM Symposia </ref>. Markowitz’ principles in MPT theory were translated into a criterion for project prioritization that aids in the success of project portfolio management. In modern project portfolio management, other than risk and return, there are elements such as benefits maximization, balance, strategic alignment and resource levelling. application of such methods of financial protfolio optimization can help a project manager to evaluate projects taking in consideration the interaction and influence of other projects<ref>Thorp, J. (2003). The information paradox: realizing the business benefits of information technology. McGraw-Hill Ryerson.</ref>
[[File:PPMMPT.png|1000px|center|middle|thumb| Translation of MPT criterias to PPM criterias <ref>Bonham, S. S. (2005). IT project portfolio management. Artech House.</ref>]]

= Models of Optimization =

==Background==
The term efficient portfolios was developed in the 1950's by Harry Markowitz <ref name="Harry1"/>, <ref name="Harry2">Markowitz, H. M. (1968). Portfolio selection: efficient diversification of investments (Vol. 16) Yale university press</ref>. An efficient portfolio is one that at given level of risk provides the greatest return and at given performance holds the less amount of risk. According to this definition, an investor will choose from a set of possible portfolios, the portfolio which:
# offers the maximum expected return for different levels of risk
# offers the lowest risk for different levels of expected return.
Those portfolios that meet the before-mentioned requirements are considered as effective ones. [[:Image:EF.png|Figure]] shows an example where all the possible portfolios which are formed based on the expected return and risk relations. The set of efficient portfolios has the form of a parabola in between the axes of the expected return (vertical axis) and the level of risk (horizontal axis). Points A, B, C, D, E, F, show some of the possible portfolios. Of all the available portfolios, the most efficient ones are those found in "northwestern" part of the curve of efficient portfolios between A and F. All other portfolios are regarded as ineffective. For example, the A portfolio excels E because it offers the same performance with less risk. Similarly the C portfolio excels D because it offers more return at the same risk level.

[[File:EF.png|1000px|center|middle|thumb| [[Wikipedia:Efficient_Frontier|Efficient Frontier]]'' Example '']]
==Assumptions==
A series of assumptions regarding the market and investors were made in order to quantify risk <ref>Xidonas, P., Mavrotas, G., Krintas, T., Psarras, J., & Zopounidis, C. (2012). Multicriteria portfolio management (pp. 5-21). Springer New York.</ref> <ref>Reilly, F., & Brown, K. (2011). Investment analysis and portfolio management. Cengage Learning.</ref>
* Investors are "reasonable" and behave in such a way to maximize their benefits according to the available capital.
* Investors have free access to a fair and accurate information concerning risk and expected returns.
* Markets are efficient and absorb information quickly and perfect.
* Investors avoid risk through the effort to maximize the profit and minimize the danger of investments.
* Investors base their decisions in accordance with the expected performance and a mathematically defined risk measure.
* Investors prefer higher returns than lower ones at a given risk levels.

== Mean Variance Model==
This model of optimization financial portfolios was the beginning of [[Wikipedia:Modern_Portfolio_Theory|Modern Portfolio Theory]] and was the cornerstone for the study of risk in securities. It is still used because of its simplicity and its greater degree of physical understanding of the mathematical aspects, however its non-linear nature raises the complexity of solution in order to find the best portfolio <ref name="Mansini">Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2), 518-535.</ref> compared to models that are linear and will be further discussed below. In the case of this model it is assumed that there are <math>n</math> bonds. Each bond <math>i</math> has expected return <math>\mu_{i}</math>, the variation of performance is denoted as <math>\sigma_{i}^2</math>, and <math>\sigma_{ij}</math> is the co-variance of returns in relation to another security <math>j</math>. If <math>R</math> is the desired performance of the portfolio then:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{i=1}^n \sum\limits_{j=1}^n \sigma_{ij} w_{i} w_{j}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} \mu_{i}\geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1,\\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The term <math>w_{i}</math> represents the percentage of the capital that will be invested in each bond. This is a quadratic programming problem, and hence the solvability puzzled the financial industry for many years. Today, however, the available computational tools enable the solution of the above quadratic program even for large-scale data <ref name="Mansini"/>.

==Mean Semi-Variance Model==
An alternative to the problem of portfolio optimization using mean variance is to use the term of the average semi-variance, a proposal made again by Markowitz <ref name="Harry2"/>, <ref>[http://www.rand.org/content/dam/rand/pubs/research_memoranda/2009/RM1438.pdf]H. Markowitz. The optimization of quadratic functions subject to linear constraints. February 1955</ref>. Because an investor is more concerned about minimizing under-performance instead of over-performance, the semi-variance was considered as a more appropriate measure of the risk variance .The difference is that semi-variance measures only downward deflection, not both negative and positive deflections as described in Mean Variance model .
This model seeks to identify optimal portfolios considering the expected outcomes of each security and its the semi-variance such as discussed above <ref>Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45(1), 307-317 </ref>. The efficient portfolios of the solution of this problem have little semi-variance for a given expected return, and maximum expected return for a given semi-variance. The set of all efficient portfolios form the efficient frontier in returns / semi-variance graph. The linear minimization problem <ref name="Mansini"/> formulized by the above is:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T z_{t}^{2}, \\
\text{s.t} & \sum\limits_{i=1}^n w_{i} = 1 \\
& \frac{1}{\sqrt{T}}\sum_{i=1}^{n}(r_{ti}-\mu_i)-d_t+z_t=0,\quad t=1,2,\ldots,T, \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R, \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling led to the first form of utilization of risk of deterioration as a risk measure, which although it is computationally difficult in cases where further restrictions are added <ref>Gilli, M., & Këllezi, E. (2002). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In Computational methods in decision-making, economics and finance (pp. 167-183). Springer US</ref>, it is a quite popular method <ref>Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220.</ref>.

==CVaR Model==
The Conditional Value at Risk, (CVAR) is an extension of the term [[wikipedia:Value_at_risk|value at risk (VaR)]] , in order to create a better estimation of losses in extreme adverse conditions and to address certain theoretical problems of the value at risk <ref name="CVAR1">Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.</ref> . For [[Wikipedia:Probability_distribution#Continuous_probability_distribution|continuous distributions]], CVAR is defined as the expected loss in those cases where the loss of an investment position exceed the corresponding VaR. However, for general loss distributions, including [[Wikipedia:Uniform_distribution_(discrete)|discrete distributions]], the CVAR is defined as the weighted average of the VaR and losses that strictly exceed VaR <ref name="CVAR1"/>. For general distributions, the CVAR, has more attractive properties from VaR[καινουριο paper CVaR]. CVaR is [[Wikipedia:Subadditivity|subadditive]] and [[Wikipedia:Convex_function|convex]] <ref name="CVAR1"/>. In addition, the CVAR has all the essential qualities of a reasonable risk measure, which does not happen in the occasion of VaR, according to Artzner <ref>Artzner, P. (1997). Applebaum, D.(2004). Lévy Processes and Stochastic Calculus (Cambridge University Press). Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D.(1997). Thinking coherently, Risk 10, pp. 68–71. Risk, 10, 68-71.</ref>, <ref>Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (2002). Coherent Measures of Risk1. Risk management: value at risk and beyond, 145.</ref>. Although CVaR is not fully accepted in the financial industry, it is gaining ground in the insurance sector [12]. Mathematical formulation of CVaR is <ref name="Angelelli">Angelelli, E., Mansini, R., & Speranza, M. G. (2008). A comparison of MAD and CVaR models with real features. Journal of Banking & Finance, 32(7), 1188-1197.</ref>:

<math display="block">
\begin{align}
\min \quad &\eta + \frac{1}{T\beta} \sum\limits_{t=1}^T d_{t},\\
\text{s.t} &\sum\limits_{i=1}^n w_{i} = 1 \\
&\sum\limits_{i=1}^n w_{i} \mu_{i} \geq R,\\
& n+\sum_{i=1}^{n}{r_{ti}w_i}+d_{t}\ge 0,\\
&d_{t} \ge 0 \qquad t=1, 2, \ldots, T\\
&w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

Optimization of CVaR minimizes VaR, since <math> CVaR \geq VaR </math>
.Furthermore another advantage of CVaR against simple VaR is that it can be optimized by linear programming methods and thus creating optimal portfolio can become a process of solving a linear problem which is relatively simple to understand and easy to use as well as its implementation is undemanding <ref>Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4, 43-68.</ref>.

==Mean Absolute Deviation Model==
The difficulties of solving the model of Markowitz and variants <ref>A simplified model for portfolio analysis </ref> led Konno and Yamazaki in 1988 to present a model consisting exclusively of linear constraints <ref>Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.</ref>, which utilizes as a measure of risk the [[Wikipedia:Average_absolute_deviation|mean absolute deviation (MAD)]] . Mean absolute deviation is defined as the average of the absolute deviation from the mean of the data. So the following formula was proposed as risk measure:

<math display="block">
\begin{align}
MAD_{i} = \frac{1}{T} \sum\limits_{i=1}^n
\left\vert r_{it}-\mu_{i} \right\vert
\end{align}
</math>

Extending this risk measure to financial portfolios it is observable that because of the absolute nature of risk measurement, the generated problem can not be linear. But it can easily be transformed to one by adding a further variable. So the problem presented for solution is the following:

<math display="block">
\begin{align}
\min \quad & \sum\limits_{t=1}^T y_{t} , \\
\text{s.t} & \sum\limits_{i=1}^n w_{i}[r_{ti}-\mu_{i}]+y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i}[r_{it}-\mu_{i}]-y_{t} \geq 0 \\
& \sum\limits_{i=1}^n w_{i} \mu_{i} \geq R ,\\
& \sum\limits_{i=1}^n w_{i} = 1 \\
& y_{t} \geq 0 \quad t=1,2,3....T \\
& w_{i} \geq 0 , i= 1,2,3...n
\end{align}
</math>

The above modeling helps in examination of a larger number of securities due to the ease in solving a problem of linear constraints as well as the addition of constraints that make it realistic becomes more computationally viable <ref name="Angelelli"/>.

==Additional Constraints==
In real situations where investors desire portfolios that meet different realistic features like minimum lots of transactions and transaction costs, solvability of linear models is a critical aspect.
Even if the solution of quadratic models like Mean Variance and Semi-Variance has been treated for restrictions in the total number of shares and minimum transactions lots <ref>Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.</ref> <ref name="Angelelli"/>, the computational challenge to solve the big and realistic portfolio problems justifies the long tradition in literature of [[Wikipedia:Integer_programming|mixed integer programming]] for portfolio selection with pragmatic characteristics <ref>Bertsimas, D., Darnell, C., Soucy, R., & Darnellt, C. (1998). Portfolio construction through mixed integer programming.</ref> <ref>Chiodi, L., Mansini, R., & Speranza, M. G. (2003). Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research, 124(1-4), 245-265.</ref> <ref>Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1-4), 287-304.</ref> <ref>Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89(2), 233-250.</ref> <ref>Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE transactions, 37(10), 919-929.</ref>. The models extend the basic portfolio optimization models and result to increased complexity and difficulty of solution. However, the analyzes carried out are closer to reality and the parameters can be adjusted depending on the stock market and investor requirements. Such restrictions can be:
* Restriction in total number of shares. The investor chooses the maximum number of shares they want to invest.
* Limitation of minimum trading lots. The investor virtually round the percentages invested, due to limitations of the financial market.

===Transaction roundlots===
In any stock market transactions take place in predetermined units (pieces) of each security. Such restrictions are common transactions requirements which means that investing in a security should be expressed as a multiple of a predetermined unit of transaction . The monetary value of each transaction unit is expressed as a percentage of the value of a portfolio <ref name="Jobst"/> so that the composition of the portfolio ( percentage of participation of securities) shall be defined based on these percentages. Moreover limitation of budget becomes "flexible" by the introduction of deviation variables. These variables are denoted as <math> \epsilon^{-} </math> and <math> \epsilon^{+} </math> , which are minimized in order to limit the budget impbalances as little as possible at the optimal solution <ref name="Jobst">Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative finance, 1(5), 489-501.</ref> <ref name="Mansini"/> . To achieve this, the deviation variables should have very small values <ref name="Mansini"/>. In all models, without exception, the participation grade changes form and it is expressed as <math> w_{i} = f_{i}y_{i} </math> where <math> f_{i} </math> symbolizes the transaction module for the security <math> i </math> (as a percentage of available capital) and <math> y_{i} </math> is the number of units purchased of the security <math> i </math>. In order to implement this particular constraint, each portfolio optimization problem will be converted and the new formulation after the addition of variables mentioned above would be:

<math display="block">
\begin{align}
\min \quad & Z_{Lot}=f_{obj} +\gamma\epsilon^{+}+\gamma\epsilon^{-},\\
\text{s.t} \quad &\sum_{i=1}^{n}{w_i} = 1,\\
\quad &w_i-f_iy_i=0,\quad i=1, 2, \ldots, n, \\
\quad & w_i\in \mathcal{A},\quad i=1, 2, \ldots, n, \\
\quad & \epsilon^{+},\epsilon^{-}\geq 0, \\
\quad & y_i\in Z^*,\quad i=1, 2, \ldots, n
\end{align}
</math>

where as <math> f_{obj} </math> determined the objective function of an [[Models of Optimization|above-mentioned portfolio optimization model]] and <math> \mathcal{A}</math> is the set of feasible solutions.

===Cardinality constraints===
One of the basic assumptions of portfolio theory is that investors can hold well diversified portfolios. However, there are signs that investors typically hold only a small number of securities. Market imperfections such as fixed transaction costs, provide one possible explanation for the selection of undiversified portfolios[40]. Moreover, the need to avoid the costs of monitoring and re-weighting a portfolio leads investors to the common practice of limiting the number of investments (population portfolio securities) that can be selected in a portfolio. The restriction on the number of securities in a portfolio can be expressed either as a strict equality or inequality requiring that the number of selected titles can not be greater than a predetermined number. The addition of the above restriction in each of the above models by adding a variable <math>x_{i}</math> subject to the following limitations:

<math display="block">
\begin{align}
& \sum \limits_{i=1}^n x_{i} \leq k, \\
\quad & x_{i} =\begin{cases} 1\\0 \end{cases}, \\
\quad & l_{i}x_{i} \leq w_{i}\leq u_{i} x_{i}
\end{align}
</math>

The variable <math>x_{i}</math> is shown as a binary variable that indicates whether or not the stock participates in the portfolio . <math>K</math> indicates the maximum number of desired stocks and <math>l_{i}</math> and <math>u_{i}</math> symbolize the lower and upper boundaries respectively, of each participation percentage <math>w_{i}</math> of a security. It should be mentioned that the above limitation is a continuation of the restriction of minimum participation, as developed by Beale and Forrest <ref>Beale, E. M. L., & Forrest, J. J. H. (1976). Global optimization using special ordered sets. Mathematical Programming, 10(1), 52-69.</ref>.

=Implementations=

These basic models of financial portfolio optimization, that basically derive from the Modern Portofolio Theory, seem to have the eligibility to be implemented in more than one applications. The ability to choose the most appropriate set of projects and allocate the amount of in the most efficient way is a desired practical aspect of this mathematical theory. The utilization of MPT in water resource portfolios enabled the researchers to reduce drought problems <ref>Beuhler, M. (2006). Application of modern financial portfolio theory to water resource portfolios. Water Science & Technology: Water Supply, 6(5), 35-41.</ref> . Moreover, the same principles were implemented on intervening against the high nutrient loads at the river catchments and had as a main result better budget allocation on environmental investment decision processes <ref>Marinoni, O., & Adkins, P. Joint application of Cost-Utility Analysis and Modern Portfolio Theory to inform decision processes in a changing climate.</ref>. It should also be mentioned the contribution of these model in information retrieval <ref>Wang, J., & Zhu, J. (2009, July). Portfolio theory of information retrieval. In Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval (pp. 115-122). ACM.</ref> and energy marker <ref>Domingues, E. G., Arango, H., Abreu, P. G., Campinho, C. B., & Paulillo, G. (2001). Applying modern portfolio theory to investment projects in electric energy markets. In Power Tech Proceedings, 2001 IEEE Porto (Vol. 1, pp. 5-pp). IEEE.</ref>. Applications of financial portfolio optimization methods were also implemented in IT industry. The capability of rationalizing risks and maximize profits, is embedded in IBM's tool "[http://www-03.ibm.com/software/products/en/portfolio Rational Portfolio Manager], as well as in Palisde Corporation's project management tool [http://www.palisade.com/risk/?gclid=CL-PpYiYg8gCFcnUcgodnkYEFw @RISK]. Both of these tools, make use of [[Wikipedia : Monte_Carlo_Simulation|Monte Carlo Simulation]] to generate scenarios as input data for the mathematical models mentioned. Modern portfolio theory extensions were also used by [[Wikipedia:Schlumberger|Schlumberger]] , to make optimal portfolio choices of exploration projects (oil wells) having as key aspects the production, investments, cash flows, and geological chance of success<ref>Adams, T., Lund, J., Albers, J. A., Back, M., McVean, J., & Howell III, J. I. (1998). Portfolio management for strategic growth. Oil & Gas Journal, 96(48), 54-57.</ref>.

=Disadvantages=
In literature there has been reported that many of these assumptions do not seem realistic. Assumption of "reasonable investors" often seem to fall short as they generally prefer portfolios different from those resulting from analyzes <ref>Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of risk and uncertainty, 5(4), 325-370.</ref> <ref>Kroll, Y., Levy, H., & Rapoport, A. (1988). Experimental tests of the separation theorem and the capital asset pricing model. The American Economic Review, 500-519.</ref>. Moreover, the grade of complexity becomes greater as the problem grows and its solution becomes extremely difficult or even impossible. Finally, the assumptions mentioned do not take into account the uniqueness of each investor and consider everyone as a unified body, ignoring the behavior that each of them may present. So the difference of institutional and non-institutional investors can lead to values much higher than actual, due to herd behavior in the second category of investors, leading to systematic overvaluation of stock prices <ref>Maringer, D. G. (2006). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.</ref>. This kind of disadvantages seem to gave birth to various approaches in the problem of portfolio optimization. Fuzzy handling of the problem , seem to solve various of the issues made due to the assumptions such as the nonuniform character of the information among the investors<ref>Gupta, P., Mehlawat, M. K., & Saxena, A. (2008). Asset portfolio optimization using fuzzy mathematical programming. Information Sciences, 178(6), 1734-1755.</ref> <ref>Pandit, P. K. (2013, September). Portfolio optimization using fuzzy linear programming. In INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND STATISTICS 2013 (ICMSS2013): Proceedings of the International Conference on Mathematical Sciences and Statistics 2013 (Vol. 1557, No. 1, pp. 206-210). AIP Publishing.</ref> . Moreover multi-criteria analysis haw been implemented in order cope with the investor’s personal attitude towards risk and specific objectives he/she may have <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2010). Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study. International Journal of Operational Research, 8(1), 21-41.</ref> <ref>Xidonas, P., Mavrotas, G., & Psarras, J. (2009). A multicriteria methodology for equity selection using financial analysis. Computers & Operations Research, 36(12), 3187-3203.</ref>. Especially in the field of "non-financial" assets there has been a lot of concern in utilization of such models in order to optimize the project portfolio. First of all, the inability of non divisible allocation , makes the portfolio of projects far more inflexible. This inflexibility makes MPT almost useless. Projects either start or do not, but once they are started there should be an end too. A portfolio optimization method must consider this nature of projects in order to function properly. Also, assets of financial portfolios are liquid; assessment and re-assesement can be done at any point. However opportunity of starting a new project may be limited. Most of the projects that have already started cannot be ceased or sold without the loss of the sunk costs. More specifically, a semi-complete project seems to have no salvation "return", so all the cost of abandonment falls on the shoulders of the investor.<ref>Hubbard, D. W. (2014). How to measure anything: Finding the value of intangibles in business. John Wiley & Sons.</ref>
<ref>Sabbadini, Tony. "Manufacturing Portfolio Theory." International Institute for Advanced Studies in Systems Research and Cybernetics (2010): 120-160.</ref>. Finally, MPT mostly makes use of a mathematical defined risk measure, but portfolios consisting for example from major building projects do not have a firm one.

=See also=
*[[Wikipedia:Efficient_market_hypothesis|Efficient-Market Hypothesis]]
*[[Wikipedia:Post_Modern_Portfolio_Theory|Post Modern Portfolio Theory]]
*[[Wikipedia:Intertemporal_portfolio_choice|Intertempolar Portfolio Choice]]
*[[Wikipedia:Decision_theory|Decision Theory]]
*[[Wikipedia:Investment_strategy|Investment Strategy]]
*[[Wikipedia:Arbitrage_pricing_theory|Arbitrage Pricing Theory]]

=References=
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