**3.3 Multiple internal rate of return**

Multiple internal rate of return is probably the best known flaw of the internal rate of return method. The problem is often experienced when the internal rate of return equation yields two (or more) results. Of course, in that case the mathematical result isn't meaningful for investment decision making purposes.

The reason for two or more results from the equation lies in its structure. The internal rate of return equation is a polynomial. Lumby and Jones (1999) explain that the fifteenth-century mathematician Descartes proved with his "rule of sign", that there are possible solutions to polynomial equation for each change of sign. Thus, any particular investment project may have more than one internal rate of return (i.e., there may be more than one discount rate that will reduce a project's cash flow to a zero net present value), or it may not have any internal rate of return at all.

In projects, where the cash flow can change the sign more than once among different periods, the internal rate of return method yields more than one root of the equation, which means more than one internal rate of return (Figure 1)

Fig. 1. Two roots of polynomial equation – multiple internal rate of return.

Several internal rates of return cannot form the basis for a justified decision, since the internal rates of return may differ several times. In cases of multiple internal rates of return, financial calculators and spreadsheet processing programs offer a choice of guessing the right internal rate of return; however, without knowing the theoretical process of avoiding multiple internal rates of return, this may not be the perfect solution either.

In theory, there are projects with conventional cash flows and projects with nonconventional cash flows. Puxty and Dodds (1991) explain that a conventional cash flow project is one where a cash outflow, or series of outflows, is followed by a cash inflow or series of inflows. A projects with a conventional cash flow has only one change in sign (+, – )

Multiple internal rate of return is probably the best known flaw of the internal rate of return method. The problem is often experienced when the internal rate of return equation yields two (or more) results. Of course, in that case the mathematical result isn't meaningful for

The reason for two or more results from the equation lies in its structure. The internal rate of return equation is a polynomial. Lumby and Jones (1999) explain that the fifteenth-century mathematician Descartes proved with his "rule of sign", that there are possible solutions to polynomial equation for each change of sign. Thus, any particular investment project may have more than one internal rate of return (i.e., there may be more than one discount rate that will reduce a project's cash flow to a zero net present value), or it may not have any

In projects, where the cash flow can change the sign more than once among different periods, the internal rate of return method yields more than one root of the equation, which

**3.3 Multiple internal rate of return** 

investment decision making purposes.

means more than one internal rate of return (Figure 1)

internal rates of return (*IRR*)

Fig. 1. Two roots of polynomial equation – multiple internal rate of return.

discount rate (*r*)

multiple internal rates of return, this may not be the perfect solution either.

Several internal rates of return cannot form the basis for a justified decision, since the internal rates of return may differ several times. In cases of multiple internal rates of return, financial calculators and spreadsheet processing programs offer a choice of guessing the right internal rate of return; however, without knowing the theoretical process of avoiding

In theory, there are projects with conventional cash flows and projects with nonconventional cash flows. Puxty and Dodds (1991) explain that a conventional cash flow project is one where a cash outflow, or series of outflows, is followed by a cash inflow or series of inflows. A projects with a conventional cash flow has only one change in sign (+, – )

internal rate of return at all.

 net present value (*NPV*) between the time periods. Non-conventional cash flows can therefore be defined as those that involve more than one change in sign. Such projects are due to modifications, reconstructions and overhauls, which require intensive investments that often cause negative cash flow, quite common in engineering projects.

The problem of more than one change in sign can be overcome with the elimination of second and further changes in sign by discounting such part of equation to the article of the equation with the same sign in cash flow as the discounted one.

A further possibility for overcoming the multiple internal rate of return problem, according Puxty and Dodds (1991), involves the net present value rule. It would have no difficulty in giving the correct advice: to reject the project because it has a negative net present value or to accept it because it has a positive one at the given discount rate.

In connection with the internal rate of return, it is worth mentioning at least one further problem: a non-existent internal rate of return. Since the problem is very unlikely to appear in practice, we will not give it any in depth attention.

According to Pšunder and Ferlan's (2007) research among Slovenian project managers, only 37.5 percent of project managers with an education in mechanical engineering know about the multiple internal rate of return problem.
