Internal Rate of Return (IRR)

The internal rate of return (IRR) is a widespread used figure to evaluate investment projects. But it is a very dangerous figure that can lead to wrong decisions easily. The figure is only valid if the investment project fulfills special conditions.

IRR Definition

I want to give you a clear derivation of the IRR definition to point out some very critical issues. The internal rate of return is defined to be the zero of a polynomial that calculates the net present value (NPV) of an investment project. The NPV is the sum of all discounted incremental cash flows to the firm after taxes, c_i. The discount factors \gamma_i can be different for each cach flow c_i.

    \[NPV=\sum_{i}^{}\gamma _i c_i\]

Let us assume that we can sum up the products in each period t, because all cash flows one period t have the same discount factor \gamma _t (uniformity). Let C_t be the sum of the cash flows in period t. Hence we can simplify the NPV calculation by summing over all time periods instead of over all cash flows:

    \[NPV=\sum_{t}^{}\gamma _t C_t\]

Further we assume that the discount factor \gamma_t has polynomial character in t (flatness). In this case we can rewrite the NPV as a polynomial:

    \[NPV=\sum_{t}^{}C_t\gamma ^t\]

Let i be the annual interest rate to discount cash flows. Setting \gamma=\left( 1+i \right)^{-1} we get the well-known formula:

    \[NPV=\sum_{t}^{}C_t\left( 1+i \right) ^{-t}\]

The internal rate of return i^{IRR} is now defined to be the values of i for which the NPV is zero:

    \[NPV=\sum_{t}^{}C_t\left( 1+i^{IRR} \right) ^{-t}=0\]

I want to point out that we require two restrictive premises to get an expression for the IRR, flatness and uniformity.

Flat and uniform discount rates

To give the NPV the form of a polynomial we required two important assumptions. At first the investment project must have a uniform discount rate. That means that the discount rate has to be the same for all cash flows in one period. All cash flows are assumed to have the same equity risk premium and the same risk free rate. In almost all investment projects this assumption does not hold. Secondly, the discount rate structure must be flat. That means that you have the same discount rate for each period. The risk free rate and the equity risk premiums are assumed to be constant over time. In most investment projects this is also not true.

We can state the the premises uniformity and flatness to let the NPV get a polynomial cannot be applied in almost all investment projects. Especially in complex projects with many different cash flows, currencies and risks these assumptions are not valid.

Ambiguity

The IRR are the zero points of a polynomial. A polynomial of order n can have n different zeros. That means that you get n different IRR in general, the solution for possible IRR can be ambiguous. If you get more than one IRR, which one is the right one? Or you can even get no solution. But even when you get no solution for the IRR, there exists a contribution of the project to the company’s value. NPV (or eNPV) can evaluate all kinds of value contributions to the firm and must be preferred.

IRR decision rule

According to the decision rule of the IRR method an investment should be realized, if the IRR is bigger than the WACC. That is only true, if the the NPV has a negative derivative at the IRR. A zero point of a n-order polynomial can have a positive or negative first derivation at zero points. Accordingly you cannot say that the NPV decreases with increasing discount rate in general. That means that the IRR decision rule does not hold when the first derivative is positive.

Mutually exclusive investment projects

Firms often have to choose between several alternative ways of doing the same job or using the same facility. In other words, they need to choose between mutually exclusive projects. The IRR decision rule can be misleading in that case as well. An alternative can have a bigger IRR but a smaller NPV than another alternative at the same time. The investors of the project (who are the relevant deciders) are interested in a maximum increase of value of the firm. That is represented by the NPV. Hence comparing alternatives by their IRR can lead to false decisions.

Conclusion

My conclusion is the the IRR method with its decision rule should only be applied at very simple projects with one investment in the beginning and uniform / flat discount rates. It should not be applied as decision figure in a standardized investment valuation concept of a company. In all complex projects the NPV (or eNPV) is the much better concept. Additionally the NPV / eNPV gives the investors the information they demand, namely the contribution of the investment project to the value of the firm and hence the value of their shares.

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