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This is the multiple rates of return problem. Many financial computer packages (including a best-seller for personal computers) aren't aware of this problem and just report the first IRR that is found. Others report only the smallest positive IRR, even though this answer is no better than any other.

In our current example, the IRR rule breaks down completely. Suppose our required return is 10 percent. Should we take this investment? Both IRRs are greater than 10 percent, so, by the IRR rule, maybe we should. However, as Figure 9.7 shows, the NPV is negative at any discount rate less than 25 percent, so this is not a good investment. When should we take it? Looking at Figure 9.7 one last time, we see that the NPV is positive only if our required return is between 25 percent and 33% percent.

Nonconventional cash flows can occur in a variety of ways. For example, Northeast Utilities, owner of the Connecticut-located Millstone nuclear power plant, had to shut down the plant's three reactors in November 1995. The reactors were expected to be back on-line in January 1997. By some estimates, the cost of the shutdown would run about $334 million. In fact, all nuclear plants eventually have to be shut down for good, and the costs associated with "decommissioning" a plant are enormous, creating large negative cash flows at the end of the project's life.

The moral of the story is that when the cash flows aren't conventional, strange things can start to happen to the IRR. This is not anything to get upset about, however, because the NPV rule, as always, works just fine. This illustrates the fact that, oddly enough, the obvious question—What's the rate of return?—may not always have a good answer.

What's the IRR?

You are looking at an investment that requires you to invest $51 today. You'll get $100 in one year, but you must pay out $50 in two years. What is the IRR on this investment?

You're on the alert now for the nonconventional cash flow problem, so you probably wouldn't be surprised to see more than one IRR. However, if you start looking for an IRR by trial and error, it will take you a long time. The reason is that there is no IRR. The NPV is negative at every discount rate, so we shouldn't take this investment under any circumstances. What's the return on this investment? Your guess is as good as ours.

"I Think; Therefore, I Know How Many IRRs There Can Be."

We've seen that it's possible to get more than one IRR. If you wanted to make sure that you had found all of the possible IRRs, how could you do it? The answer comes from the great mathematician, philosopher, and financial analyst Descartes (of "I think; therefore I am" fame). Descartes's Rule of Sign says that the maximum number of IRRs that there can be is equal to the number of times that the cash flows change sign from positive to negative and/or negative to positive.7

In our example with the 25 percent and 33K percent IRRs, could there be yet another IRR? The cash flows flip from negative to positive, then back to negative, for a total of two sign changes. Therefore, according to Descartes's rule, the maximum number of IRRs is two and we don't need to look for any more. Note that the actual number of IRRs can be less than the maximum (see Example 9.5).

7To be more precise, the number of IRRs that are bigger than —100 percent is equal to the number of sign changes, or it differs from the number of sign changes by an even number. Thus, for example, if there are five sign changes, there are either five IRRs, three IRRs, or one IRR. If there are two sign changes, there are either two IRRs or no IRRs.

Ross et al.: Fundamentals I IV. Capital Budgeting I 9. Net Present Value and I I © The McGraw-Hill of Corporate Finance, Sixth Other Investment Criteria Companies, 2002

Edition, Alternate Edition

294 PART FOUR Capital Budgeting mutually exclusive investment decisions

A situation in which taking one investment prevents the taking of another.

Mutually Exclusive Investments Even if there is a single IRR, another problem can arise concerning mutually exclusive investment decisions. If two investments, X and Y, are mutually exclusive, then taking one of them means that we cannot take the other. Two projects that are not mutually exclusive are said to be independent. For example, if we own one corner lot, then we can build a gas station or an apartment building, but not both. These are mutually exclusive alternatives.

Thus far, we have asked whether or not a given investment is worth undertaking. There is a related question, however, that comes up very often: Given two or more mutually exclusive investments, which one is the best? The answer is simple enough: the best one is the one with the largest NPV. Can we also say that the best one has the highest return? As we show, the answer is no.

To illustrate the problem with the IRR rule and mutually exclusive investments, consider the following cash flows from two mutually exclusive investments:

Year

Investment A

Investment B

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