## THE iNternal Rate Of Return

We now come to the most important alternative to NPV, the internal rate of return, universally known as the IRR. As we will see, the IRR is closely related to NPV. With the IRR, we try to find a single rate of return that summarizes the merits of a project. Furthermore, we want this rate to be an "internal" rate in the sense that it depends only on the cash flows of a particular investment, not on rates offered elsewhere.

internal rate of return (IRR)

The discount rate that makes the NPV of an investment zero.

5The AAR is closely related to the return on assets (ROA) discussed in Chapter 3. In practice, the AAR is sometimes computed by first calculating the ROA for each year, and then averaging the results. This produces a number that is similar, but not identical, to the one we computed.

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

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### 288 PART FOUR Capital Budgeting

To illustrate the idea behind the IRR, consider a project that costs \$100 today and pays \$110 in one year. Suppose you were asked, "What is the return on this investment?" What would you say? It seems both natural and obvious to say that the return is 10 percent because, for every dollar we put in, we get \$1.10 back. In fact, as we will see in a moment, 10 percent is the internal rate of return, or IRR, on this investment.

Is this project with its 10 percent IRR a good investment? Once again, it would seem apparent that this is a good investment only if our required return is less than 10 percent. This intuition is also correct and illustrates the IRR rule:

Based on the IRR rule, an investment is acceptable if the IRR exceeds the required return. It should be rejected otherwise.

Imagine that we want to calculate the NPV for our simple investment. At a discount rate of R, the NPV is:

Now, suppose we don't know the discount rate. This presents a problem, but we can still ask how high the discount rate would have to be before this project was deemed unacceptable. We know that we are indifferent between taking and not taking this investment when its NPV is just equal to zero. In other words, this investment is economically a break-even proposition when the NPV is zero because value is neither created nor destroyed. To find the break-even discount rate, we set NPV equal to zero and solve for R:

NPV = 0 = -\$100 + [110/(1 + R)] \$100 = \$110/(1 + R) 1 + R = \$110/100 = 1.1 R = 10%

This 10 percent is what we already have called the return on this investment. What we have now illustrated is that the internal rate of return on an investment (or just "return" for short) is the discount rate that makes the NPV equal to zero. This is an important observation, so it bears repeating:

The IRR on an investment is the required return that results in a zero NPV when it is used as the discount rate.

The fact that the IRR is simply the discount rate that makes the NPV equal to zero is important because it tells us how to calculate the returns on more complicated investments. As we have seen, finding the IRR turns out to be relatively easy for a single-period investment. However, suppose you were now looking at an investment with the cash flows shown in Figure 9.4. As illustrated, this investment costs \$100 and has a cash

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