Internal rate of return (IRR)

[Lorenzo Incarnato s220426]

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Abstract

The Internal Rate of Return (IRR) is a powerful discounted cash flow method used in capital budgeting and corporate finance to estimate and evaluate the profitability of potential investments. Keeping in mind the formula for the Net present value (NPV), the IRR is defined as the discount rate that makes the present value of the costs (negative cash flows) of an investment equal to the present value of the benefits (positive cash flows). In other words, the IRR is the discount rate that gives a net present value of zero when applied to the expected cash flow of a project. This rate of return is called internal because the formula predicts a rate that depends only on the project, more precisely on the project's cash flows, and does not depend on external factors such as market interest rates or inflation. One of the main advantages in using the internal rate of return to evaluate project investments, compared to other methods such as the Payback period or the Benefit-cost ratio, is that IRR considers the time value of money. In general, due to the relationship between NPV and IRR, the higher the Internal Rate of Return of a project, the more desirable the investment to be made. This article shows also the limitations of the internal rate of return; however, when the IRR is unique, it provides relevant information about the return on investment and is also used as a measure of investment efficiency. In fact, according to academic research, three-quarters of Chief Financial Officers use the IRR method to evaluate capital projects.

Time value of money

A key consideration regarding discounted cash flow methods is the Time Value of Money (TVM). This paragraph explains why money tomorrow is not as valuable as money today. In finance, an amount of money available today or in the future has a different value. In other words, $ 100 today is not $ 100 tomorrow and this difference is due to the time value of money.

The reason is that today you have the opportunity to invest the money and thus grow, according to the time horizon and the interest rate. Another reason for the time value of money is the purchasing power of money which changes over time due to inflation or deflation. Hence, time has a monetary value and we have to consider it if we want to compare money available in different moments . It is, therefore, necessary to briefly underline three fundamental terms about investments to clearly understand the time value of money:

• The Present value PV : the sum of money available today that can be invested [$];
• The Interest rate (or opportunity cost if we decide not to invest) r : the amount of interest due per period, as a proportion of the amount deposited or borrowed [%]
• The Future value FV : is the value of a current asset (present value) at a future date based on an assumed rate of growth (interest rate) [$].

Present to Future	$FV = PV (1 + r)^n$
Future to Present	$PV = FV / (1 + r)^n$

Example 1: Would you rather have $ 100,000 today or the same amount in 1 year from now? To calculate how much money will be $ 100,000 in a year from now, supposing an interest rate r = 3%, we can apply the formula for the future value:

$FV = PV * (1 + r)^n = $ 100,000 * (1 + 0.03)^1 = $ 103,000$

Hence, having $ 100 000 now corresponds to having $ 103 000 in the future considering an interest rate of 3%. If we agreed to receive $ 100,000 in a year instead of now, we would have $ 3,000 less. As shown, the present value and the future value differ by a factor $(1+r)^n$; this difference represents the time value of money. It can be concluded that, in order to compare money available in two different moments, it is needed to consider the time value of money, which enable us to discount money available in the future and consider them in the present or vice versa; only then we can compare the two amounts.

Example 2: Considering an interest rate of 9%, is it better to receive $ 150 000 today or $ 180 000 in three years? Which amount of money do you have indifference of choice? If we invest the present value at an interest rate of 9%, we will have in three years a future value of:

$FV = PV * (1 + r)^n = $ 150,000 * (1 + 0.09)^3 = $ 194,254$

It is better to have $ 150,000 today because by investing them we will have $ 194,254 in three years, which is $ 14,254 higher than $ 180,000. To have indifference, hence, the amount of money available in three years should be $ 194,254 .

IRR: definition and formula

$N P V (irr) =\sum_{n=0}^{N} \frac{\left(C F_{n}\right)}{(1+irr)^{n}}$