# Valuation Measuring and managing the value ofpanies - Koller T.

ISBN 0-471-36190-9

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where rf

E(0

E(rm) -rf

beta

The risk-free rate of return

The expected rate of return on the overall market portfolio

The market risk premium

The systematic risk of the equity

The CAPM is illustrated in Exhibit 10.3. The cost of equity, ks, increases linearly as a function of the measured undiversifiable risk, beta. The beta for the entire market portfolio is 1.0. This means that the average company's equity beta will also be about 1.0. It is very unusual to observe a beta greater than 2.0 or less than 0.3. The market risk premium (the price of risk) is measured as the slope of the CAPM line in Exhibit 10.3, that is, the slope is E (rJ -rr

To carry out the CAPM approach, we need to estimate the three factors that determine the CAPM line: the risk-free rate, the market risk premium, and the systematic risk (beta). The balance of this section describes a recommended approach for estimating each.

Determining the Risk-Free Rate

Hypothetically, the risk-free rate is the return on a security or portfolio of securities that has no default risk and is completely uncorrelated with returns on anything else in the economy. In theory, the best estimate of the risk-free rate would be the return on a zero-beta portfolio, constructed of long and short positions in equities in a way that produces the minimum variance zero-beta portfolio. Because of the cost and complexity of constructing minimum variance zero-beta portfolios, they are not practical for estimating the risk-free rate.

We have three reasonable alternatives that use government securities: the rate for Treasury bills, the rate for 10-year Treasury bonds, and the rate Exhibit 10.3 The Capital Asset Pricing Model

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for 30-year Treasury bonds. We recommend using a 10-year Treasury-bond rate for several reasons:7

• It is a long-term rate that usually comes close to matching the duration of the cash flow of the company being valued. Since the current Treasury-bill rate is a short-term rate, it does not match duration properly. If we were to use short-term rates, the appropriate choice would be the short-term rates that are expected to apply in each future period, not today's short-term interest rate. The 10-year rate is a geometric weighted average estimate of the expected short-term Treasury-bill rates.

• The 10-year rate approximates the duration of the stock market index portfolio—for example, the S&P 500—and its use is therefore consistent with the betas and market risk premiums estimated relative to these market portfolios.8

• The 10-year rate is less susceptible to two problems involved in using a longer-term rate, such as the 30-year Treasury-bond rate. Its price is less sensitive to unexpected changes in inflation, and so has a smaller beta than the 30-year rate. Also, the liquidity premium built into 10-year rates may be slightly lower than that of 30-year bonds. These are technical details, with a minor impact in normal circumstances. But they do argue for using a 10-year bond rate.9

Determining the Market Risk Premium

The market risk premium (the price of risk) is the difference between the expected rate of return on the market portfolio and the risk-free rate, E(rm) - r. The market risk premium is one of the most vexing issues in finance. It can be based on either historical data, assuming that the future will be like the past, or on ex ante estimates that attempt to forecast the future. Both approaches have their proponents and critics.

In early 2000, we were recommending using a 4 1/2 percent to 5 percent historically estimated market risk premium for U.S. companies. Depending on the period chosen and the type of average, historically based estimates of the market risk premium can vary from about 3 percent to almost 8 percent,

7 Theoretically, you should use a distinct WACC for each year's projected cash flows based on the yield curve for risk-free securities. This is rarely done in practice, but should be considered when the cash flows are heavily front-end or back-end loaded or when the yield curve is unusually steep.

8 For an economic rationale of this argument, with supporting evidence, see J. Campbell and L. Viceira, ''Who Should Buy Long-Term Bonds?" Working Paper, National Bureau of Economic Research (November 1998).

9 The market for 30-year U.S. Treasury bonds was in flux in early 2000 because of the federal government's declining borrowing needs.

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as shown on Exhibit 10.4. Following is the series of choices that we make to arrive at our estimate:

• We measure the risk premium over as long a period as possible.

• We use an arithmetic average of rates of return because the CAPM is based on expected returns, which are forward-looking.

• We adjust the historical arithmetic rate of return downward by 1 1/2 percent to 2 percent because the historical rate is biased upward by survivorship bias.

• We calculate the premium over long-term government bond returns to be consistent with the risk-free rate we use to calculate the cost of equity. Historical Period

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