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the expected risk premium on the size factor is 3.1 percent, and the expected risk premium on the book-to-market factor is 4.4 percent. (These were the realized premia from 1928–2000.)

*Return on small-firm stocks less return on large-firm stocks.

†Return on high book-to-market-ratio stocks less return on low book-to-market-ratio stocks.

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C A P I T A L

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LONG BEFORE THE development of modern theories linking risk and expected return, smart financial managers adjusted for risk in capital budgeting. They realized intuitively that, other things being equal, risky projects are less desirable than safe ones. Therefore, financial managers demanded a higher rate of return from risky projects, or they based their decisions on conservative estimates of the cash flows.

Various rules of thumb are often used to make these risk adjustments. For example, many companies estimate the rate of return required by investors in their securities and then use this company cost of capital to discount the cash flows on new projects. Our first task in this chapter is to explain when the company cost of capital can, and cannot, be used to discount a project’s cash flows. We shall see that it is the right hurdle rate for those projects that have the same risk as the firm’s existing business; however, if a project is more risky than the firm as a whole, the cost of capital needs to be adjusted upward and the project’s cash flows discounted at this higher rate. Conversely, a lower discount rate is needed for projects that are safer than the firm as a whole.

The capital asset pricing model is widely used to estimate the return that investors require.1 It states

Expected return r rf 1beta2 1rm rf 2

We used this formula in the last chapter to figure out the return that investors expected from a sample of common stocks but we did not explain how to estimate beta. It turns out that we can gain some insight into beta by looking at how the stock price has responded in the past to market fluctuations. Beta is difficult to measure accurately for an individual firm: Greater accuracy can be achieved by looking at an average of similar companies. We will also look at what features make some investments riskier than others. If you know why Exxon Mobil has less risk than, say, Dell Computer, you will be in a better position to judge the relative risks of different capital investment opportunities.

Some companies are financed entirely by common stock. In these cases the company cost of capital and the expected return on the stock are the same thing. However, most firms finance themselves partly by debt and the return that they earn on their investments must be sufficient to satisfy both the stockholders and the debtholders. We will show you how to calculate the company cost of capital when the firm has more than one type of security outstanding.

There is still another complication: Project betas can shift over time. Some projects are safer in youth than in old age; others are riskier. In this case, what do we mean by the project beta? There may be a separate beta for each year of the project’s life. To put it another way, can we jump from the capital asset pricing model, which looks one period into the future, to the discounted-cash-flow formula for valuing long-lived assets? Most of the time it is safe to do so, but you should be able to recognize and deal with the exceptions.

We will use the capital asset pricing model, or CAPM, throughout this chapter. But don’t infer that it is therefore the last word on risk and return. The principles and procedures covered in this chapter work just as well with other models such as arbitrage pricing theory (APT).

1In a survey of financial practice, Graham and Harvey found that 74 percent of firms always, or almost always, used the capital asset pricing model to estimate the cost of capital. See J. Graham and C. Harvey, “The Theory and Practice of Corporate Finance: Evidence from the Field,” Journal of Financial Economics 60 (May/June 2001), pp. 187–244.

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9.1 COMPANY AND PROJECT COSTS OF CAPITAL

The company cost of capital is defined as the expected return on a portfolio of all the company’s existing securities. It is used to discount the cash flows on projects that have similar risk to that of the firm as a whole. For example, in Table 8.2 we estimated that investors require a return of 9.2 percent from Pfizer common stock. If Pfizer is contemplating an expansion of the firm’s existing business, it would make sense to discount the forecasted cash flows at 9.2 percent.2

The company cost of capital is not the correct discount rate if the new projects are more or less risky than the firm’s existing business. Each project should in principle be evaluated at its own opportunity cost of capital. This is a clear implication of the value-additivity principle introduced in Chapter 7. For a firm composed of assets A and B, the firm value is

Firm value PV1AB 2 PV1A 2 PV1B 2sum of separate asset values

Here PV(A) and PV(B) are valued just as if they were mini-firms in which stockholders could invest directly. Investors would value A by discounting its forecasted cash flows at a rate reflecting the risk of A. They would value B by discounting at a rate reflecting the risk of B. The two discount rates will, in general, be different. If the present value of an asset depended on the identity of the company that bought it, present values would not add up. Remember, a good project is a good project is a good project.

If the firm considers investing in a third project C, it should also value C as if C were a mini-firm. That is, the firm should discount the cash flows of C at the expected rate of return that investors would demand to make a separate investment in C. The true cost of capital depends on the use to which that capital is put.

This means that Pfizer should accept any project that more than compensates for the project’s beta. In other words, Pfizer should accept any project lying above the upward-sloping line that links expected return to risk in Figure 9.1. If the project has a high risk, Pfizer needs a higher prospective return than if the project has a low risk. Now contrast this with the company cost of capital rule, which is to accept any project regardless of its risk as long as it offers a higher return than the company’s cost of capital. In terms of Figure 9.1, the rule tells Pfizer to accept any project above the horizontal cost of capital line, that is, any project offering a return of more than 9.2 percent.

It is clearly silly to suggest that Pfizer should demand the same rate of return from a very safe project as from a very risky one. If Pfizer used the company cost of capital rule, it would reject many good low-risk projects and accept many poor high-risk projects. It is also silly to suggest that just because another company has a low company cost of capital, it is justified in accepting projects that Pfizer would reject.

The notion that each company has some individual discount rate or cost of capital is widespread, but far from universal. Many firms require different returns

2Debt accounted for only about 0.3 percent of the total market value of Pfizer’s securities. Thus, its cost of capital is effectively identical to the rate of return investors expect on its common stock. The complications caused by debt are discussed later in this chapter.

from different categories of investment. For example, discount rates might be set as follows:

Perfect Pitch and the Cost of Capital

The true cost of capital depends on project risk, not on the company undertaking the project. So why is so much time spent estimating the company cost of capital?

There are two reasons. First, many (maybe, most) projects can be treated as average risk, that is, no more or less risky than the average of the company’s other assets. For these projects the company cost of capital is the right discount rate. Second, the company cost of capital is a useful starting point for setting discount rates for unusually risky or safe projects. It is easier to add to, or subtract from, the company cost of capital than to estimate each project’s cost of capital from scratch.

There is a good musical analogy here.3 Most of us, lacking perfect pitch, need a well-defined reference point, like middle C, before we can sing on key. But anyone who can carry a tune gets relative pitches right. Businesspeople have good intuition about relative risks, at least in industries they are used to, but not about absolute risk or required rates of return. Therefore, they set a companywide cost of capital as a benchmark. This is not the right hurdle rate for everything the company does, but adjustments can be made for more or less risky ventures.

3The analogy is borrowed from S. C. Myers and L. S. Borucki, “Discounted Cash Flow Estimates of the Cost of Equity Capital—A Case Study,” Financial Markets, Institutions, and Investments 3 (August 1994), p. 18.

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F I G U R E 9 . 2

We have used past returns to estimate the betas of three stocks for the periods August 1988 to January 1995 (lefthand diagrams) and February 1995 to July 2001 (right-hand diagrams). Beta is the slope of the fitted line. Notice that in both periods Dell had the highest beta and Exxon Mobil the lowest. Standard errors are in parentheses below the betas. The standard error shows the range of possible error in the beta estimate. We also report the proportion of total risk that is due to market movements (R 2).

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beta was well below 1 in the previous period, while Dell’s estimated beta was well above 1. Nevertheless, there is always a large margin for error when estimating the beta for individual stocks.

Fortunately, the estimation errors tend to cancel out when you estimate betas of portfolios.6 That is why financial managers often turn to industry betas. For example, Table 9.1 shows estimates of beta and the standard errors of these estimates for the common stocks of four large railroad companies. Most of the standard errors are above .2, large enough to preclude a precise estimate of any particular utility’s beta. However, the table also shows the estimated beta for a portfolio of all four railroad stocks. Notice that the estimated industry beta is more reliable. This shows up in the lower standard error.

The Expected Return on Union Pacific Corporation’s Common Stock

Suppose that in mid-2001 you had been asked to estimate the company cost of capital of Union Pacific Corporation. Table 9.1 provides two clues about the true beta of Union Pacific’s stock: the direct estimate of .40 and the average estimate for the industry of .50. We will use the industry average of .50.7

In mid-2001 the risk-free rate of interest rf was about 3.5 percent. Therefore, if you had used 8 percent for the risk premium on the market, you would have concluded that the expected return on Union Pacific’s stock was about 7.5 percent:8

6If the observations are independent, the standard error of the estimated mean beta declines in proportion to the square root of the number of stocks in the portfolio.

7Comparing the beta of Union Pacific with those of the other railroads would be misleading if Union Pacific had a materially higher or lower debt ratio. Fortunately, its debt ratio was about average for the sample in Table 9.1.

8This is really a discount rate for near-term cash flows, since it rests on a risk-free rate measured by the yield on Treasury bills with maturities less than one year. Is this, you may ask, the right discount rate for cash flows from an asset with, say, a 10or 20-year expected life?

Well, now that you mention it, possibly not. In 2001 longer-term Treasury bonds yielded about

5.8percent, that is, about 2.3 percent above the Treasury bill rate.

The risk-free rate could be defined as a long-term Treasury bond yield. If you do this, however,

you should subtract the risk premium of Treasury bonds over bills, which we gave as 1.8 percent in Table 7.1. This gives a rough-and-ready estimate of the expected yield on short-term Treasury bills over the life of the bond:

Expected average T-bill rate T-bond yield premium of bonds over bills

.058 .019 .039, or 3.9%

The expected average future Treasury bill rate should be used in the CAPM if a discount rate is needed for an extended stream of cash flows. In 2001 this “long-term rf” was a bit higher than the Treasury bill rate.

Expected stock return rf 1rm rf 2

3.5 .518.0 2 7.5%

We have focused on using the capital asset pricing model to estimate the expected returns on Union Pacific’s common stock. But it would be useful to get a check on this figure. For example, in Chapter 4 we used the constant-growth DCF formula to estimate the expected rate of return for a sample of utility stocks.9 You could also use DCF models with varying future growth rates, or perhaps arbitrage pricing theory (APT). We showed in Section 8.4 how APT can be used to estimate expected returns.

9.3 CAPITAL STRUCTURE AND THE COMPANY COST OF CAPITAL

In the last section, we used the capital asset pricing model to estimate the return that investors require from Union Pacific’s common stock. Is this figure Union Pacific’s company cost of capital? Not if Union Pacific has other securities outstanding. The company cost of capital also needs to reflect the returns demanded by the owners of these securities.

We will return shortly to the problem of Union Pacific’s cost of capital, but first we need to look at the relationship between the cost of capital and the mix of debt and equity used to finance the company. Think again of what the company cost of capital is and what it is used for. We define it as the opportunity cost of capital for the firm’s existing assets; we use it to value new assets that have the same risk as the old ones.

If you owned a portfolio of all the firm’s securities—100 percent of the debt and 100 percent of the equity—you would own the firm’s assets lock, stock, and barrel. You wouldn’t share the cash flows with anyone; every dollar of cash the firm paid out would be paid to you. You can think of the company cost of capital as the expected return on this hypothetical portfolio. To calculate it, you just take a weighted average of the expected returns on the debt and the equity:

For example, suppose that the firm’s market-value balance sheet is as follows:

Note that the values of debt and equity add up to the firm value (D E V ) and that the firm value equals the asset value. (These figures are market values, not book values: The market value of the firm’s equity is often substantially different from its book value.)

9The United States Surface Transportation Board uses the constant-growth model to estimate the cost of equity capital for railroad companies. We will review its findings in Chapter 19.