Appendix 5A Tests for Multiple IRRs
When the IRR method is used to evaluate projects, we have to test for multiple IRRs. If there are undetected multiple IRRs, an IRR might be calculated that seems correct, but is in error. We consider three tests for multiple IRRs, forming essentially a threestep procedure. In the first test, the signs of the cash flows are examined to see if the project is a simple investment. In the second test, the present worth of the project is plotted against the interest rate to search for interest rates at which the present value is zero. In the third test, project balances are calculated. Each of these tests has three possible outcomes.
1. There is definitely a unique IRR and there is no possibility for multiple IRRs.
2. There are definitely multiple IRRs because two or more IRRs have been found.
3. The test outcome is inconclusive; a unique IRR or multiple IRRs are both possible.
CHAPTER 5 Comparison Methods Part 2
The tests are applied sequentially. The second test is applied only if the outcome of the first test is not clear. The third test is applied only if the outcomes of the first two are not clear. Keep in mind that, even after all three tests have been applied, the test outcomes may remain inconclusive.
The first test examines whether the project is simple. Recall that most projects consist of one or more periods of outflows at the start, followed only by one or more periods of inflows; these are called simple investments. Although simple investments guarantee a single IRR, a project that is not simple may have a single IRR or multiple IRRs. Some investment projects have large cash outflows during their lives or at the ends of their lives that cause net cash flows to be negative after years of positive net cash flows. For example, a project that involves the construction of a manufacturing plant may involve a planned expansion of the plant requiring a large expenditure some years after its initial operation. As another example, a nuclear electricity plant may have planned large cash outflows for disposal of spent fuel at the end of its life. Such a project may have a unique IRR, but it mav also have multiple IRRs. Consequently we must examine such projects further.
Where a project is not simple, we go to the second test. The second test consists of making a graph plotting present worth against interest rate. Points at which the present worth crosses or just touches the interestrate axis (i.e., where present worth = 0) are IRRs. (We assume that there is at least one IRR.) If there is more than one such point, we know that there is more than one IRR. A convenient way to produce such a graph is using a spreadsheet. See Example 5A.1.
EXAMPLE 5 A . 1 (EXAMPLE 5.7 RESTATED)
A project pays S1000 today, costs S5000 a year from now, and pays S6000 in two years. Are there multiple IRRs?
Table 5A. 1 was obtained by computing the present worth of the cash flows in Example 5.7 for a variety of interest rates. Figure 5A. 1 shows the graph of the values in Table 5A. 1.
While finding multiple IRRs in a plot ensures that the project does indeed have multiple IRRs, failure to find multiple IRRs does not necessarily mean that multiple IRRs do not exist. Any plot will cover only a finite set of points. There may be values of the interest rate for which the present worth of the project is zero that are not in the range of interest rates used.
Interest Rate, i 
Present Worth (S) 
0.6 
218.8 
0.8 
74.1 
1.0 
0.0 
1.2 
33.1 
1.4 
41.7 
1.6 
35.5 
1.8 
20.4 
2.0 
0.0 
2.2 
23.4 
2.4 
48.4 
Figure 5A.1 Illustration of Two IRRs for Example 5A.1
200
Where the project is not simple and a plot does not show multiple IRRs, we apply the third test. The third test entails calculating the project balances. As we mentioned earlier, project balances refer to the cumulative net cash flows at the end of each time period. For an IRR to be unique, there should be no time when the project balances, computed using that IRR, are positive. This means that there is no extra cash not reinvested in the project. This is a sufficient condition for there to be a unique IRR. (Recall that it is the cash generated by a project, but not reinvested in the project, that creates the possibility of multiple IRRs.)
We now present three examples. All three examples involve projects that are not simple investments. In the first, a plot shows multiple IRRs. In the second, the plot shows only a single IRR. This is inconclusive, so project balances are computed. None of the project balances is positive, so we know that there is a single IRR. In the third example, the plot shows only one IRR, so the project balances are computed. One of these is positive, so the results of all tests are inconclusive.^
Wellington Woods is considering buying land that they will log for three years. In the second year, they expect to develop the area that they clear as a residential subdivision that will entail considerable costs. Thus, in the second year, the net cash flow will be negative. In the third year, they expect to sell the developed land at a profit. The net cash flows that are expected for the project are:
EXAMPLE 5 A.2
End of Year
Cash Flow
100 000
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