## Ex Ante Estimates of the Market Risk Premium

An alternative to the historically estimated market risk premium is an ex ante estimate, one based on the current value of the share market relative to projections of earnings or cash flows. One approach estimates the expected rate of return on the market portfolio, E(rm), by adding the analysts' consensus estimate of

10 E. Fama and K. French, ''Dividend Yields and Expected Stock Returns," Journal of Financial Economics (October 1988), pp. 3-26; A. Lo and C. MacKinlay, "Stock Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test," Review of Financial Studies (1988), pp. 41-66; J. Poterba and L. Summers, "Mean Reversion in Stock Prices," Journal of Financial Economics (October 1988), pp. 27-60.

11 S. Brown, W. Goetzmann, and S. Ross, "Survivorship Bias," Journal of Finance (July 1995), pp. 853- 873.

12 P. Jorion and W. Goetzmann, "Global Stock Markets in the Twentieth Century," Working Paper (New Haven,

CT: Yale School of Management, 1999).

growth in the dividend of the S&P 500 Index, g, to the dividend yield for the index, Div/S:

The risk-free rate is then subtracted from the expected return on the market to obtain the forecast of the market risk premium. Analysts have shown limited skill in forecasting price changes in the S&P 500. In addition, the formula that provides the basis for this approach implicitly assumes perpetual growth at a constant rate, g. This is a particularly stringent assumption.

There is a growing literature that uses either the above-mentioned dividend discount model, or an equity free cash flow model and observed stock prices, to impute the equity discount rate (and the market risk premium) as an internal rate of return. A typical example is the work of Gebhardt, Lee, and Swaminathan.13 They estimate the cost of equity at the individual company level using a residual income model to value the equity of a company (this is the same approach in principle as the economic profit model described in Chapter 8). Assume a flat term structure and that the firm uses clean surplus accounting (implying that all gains and losses affecting the book value of equity are also run through the income statement). Then the value of the stock, S,, is equal to the book value of equity, B,, plus the discounted expected economic profit (defined as the spread between the return on equity, ROE, and the cost of equity, r, multiplied by the book value of equity). An ex ante estimate of the cost of equity for a company is estimated through the following process: plug in the current stock price and the book value of equity; use I/B/E/S estimates of expected earnings per share to estimate net income as (ROE x Bw-1) for the next three years; implicitly forecast economic profit to year 12 by assuming that the company return on equity fades to equal the industry median return on equity by year 12; assume that economic profit during the terminal value period is a perpetuity, and then solve the above equation for the cost of equity.

Both the Gebhardt, Lee, and Swaminathan paper and a paper by Claus and Thomas find similar results.14 The ex-ante market risk premium

13 W. Gebhardt, C. Lee, and B. Swaminathan, ''Toward an Ex-Ante Cost of Capital," Working Paper (Ithaca, NY: Cornell University, 1999).

14 J. Claus and J. Thomas, The Equity Risk Premium Is Much Lower than You Think It Is: Empirical Estimates from a New Approach, Working Paper (New York: Columbia University, 1998).

(estimated by subtracting the yield on 10-year U.S. Treasury bonds from the cost of equity) is in the two to three percent range while the historical risk premium is about 5 percent. We prefer not to use the ex ante approach because it always fits the data. Even if one makes an error forecasting future cash flows, the ex ante approach will produce an internal rate of return that is consistent with the observed stock price.

Exhibit 10.9 shows the valuation of 31 companies in August 1999. It duplicates the results of Exhibit 5.5, which showed a 92 percent r-squared between the market value and the DCF estimate of the value of these companies—except for one thing. In Exhibit 10.9, the free cash flows are discounted at a WACC that assumes a market risk premium of 3 percent (based on published ex ante estimates of the market risk premium). The results show that the r-squared falls to 79 percent, the slope of the line is significantly below unity, and so the DCF model vastly overvalues companies. Clearly, the 3-percent market risk premium assumption worsened the results. The higher market risk premium that we used in Exhibit 5.5 better fits the actual market values than the ex ante premium.

A number of investment banks have begun publishing estimates of the market risk premium, several using ex ante approaches. In early 2000, most of these estimates were 3 1/2 percent to 5 percent.

Estimating the Systematic Risk (Beta)

For listed companies, using published estimates of beta is the easiest approach. BARRA publishes betas for more than 10,000 companies around the world, but we recommend checking several reliable sources because beta estimates vary considerably. You should also compare the beta with the industry average beta. If the betas

Exhibit 10.9 Lower Risk Premium Distorts Value

from several sources vary by more than .2 or the beta for a company is more than .3 from the industry average, consider using the industry average. An industry average beta is typically more stable and reliable than an individual company beta because measurement errors tend to cancel out. When constructing the industry average, unlever the betas to estimate the average. The average unlevered beta can then be relevered using the company's capital structure.

For unlisted companies and business units, you should generally use industry averages as well. See Chapter 14 for a more detailed discussion.

Is Beta Dead? Criticism of the CAPM

In June 1992, Eugene Fama and Ken French of the University of Chicago published a paper in The Journal of Finance that received a great deal of attention because they concluded:

In short, our tests do not support the most basic prediction of the SLB model [The Sharpe-Lintner-Black Capital Asset Pricing Model] that average stock returns are positively related to market betas.15

At that time, theirs was the most recent in a long line of empirical studies that questioned the usefulness of measured betas in explaining the risk premium (above the riskless rate) on equities. Banz (1981) and Reinganum (1981) found a prominent size effect that added to the explanation of cross-sectional returns, in addition to beta. Basu (1983) found a seasonal (January) effect. Bhandari (1988) demonstrated that the degree of financial leverage was important. And Stattman (1980) as well as Rosenberg, Reid, and Lanstein (1985) found that average returns are positively related to the firm's equity book-to-market ratio.16

If beta is not dead, then surely it's wounded. Fama and French concluded that equity returns are inversely related to the size of a company measured by the value of its equity capitalization, and positively related to the ratio of the book value of the company's equity to its market value.

15 E. Fama and K. French, ''The Cross-Section of Expected Stock Returns," Journal of Finance (June 1992), pp. 427- 465.

16 R. Blanz, "The Relationship between Return and the Market Value of Common Stocks," Journal of Financial Economics (March 1981), pp. 3-18; M. Reinganum, "Misspecification of Capital Asset Pricing: Empirical Anomalies Based on Earnings Yields and Market Values," Journal of Financial Economics (March 1981), pp. 1946; S. Basu, "The Relationship between Earnings Yield, Market Value and Return for NYSE Common Stocks: Further Evidence," Journal of Financial Economics (June 1983), pp. 129-156; L. Bhandari, "Debt/Equity Ratio and Expected Common Stock Returns: Empirical Evidence," Journal of Finance (April 1988), pp. 507-528; D. Stattman, "Book Values and Stock Returns," The Chicago MBA: A Journal of Selected Papers (1980), pp. 25-45; and B. Rosenberg, K. Reid, and R. Lanstein, "Persuasive Evidence of Market Inefficiency," Journal of Portfolio Management (1985), pp. 9-17.

When these variables were taken into account, beta added nothing to the ability to explain the returns on equity.

A practical implication is that one might estimate the required return on the equity of a company by looking up its size and market-to-book ratio in a table of average risk premia over the risk free rate. This might give better results than using estimates of beta and the CAPM. Another possible implication, and one that Fama and French hint at, is the need to use a multifactor approach like the arbitrage pricing model, which is discussed in the next section.

There have been some papers that add to the debate and give practitioners some reasons to stick with the CAPM. In particular, the work of Kothari, Shanken, and Sloan (1995) is worth reading.17 They have five major conclusions. First, and most important:

Our examination of the cross-section of expected returns reveals economically and statistically significant compensation (about 6 to 9 percent per annum) for beta risk . . .

Second, they point out that the Fama and French statistical tests were of sufficiently low power that they could not reject a non-trivial (beta-related) risk premium of 6 percent over the post-1940 period. Third, if annual returns are used to estimate beta (to avoid seasonality in returns) there is a statistically significant linear relationship between beta and returns over the 1941 to 1990 period. Fourth, size is also significantly related to returns; however, the incremental economic contribution is not large (less than 1 percent). Finally, the relationship between return and the book/market ratio is the result of survivorship bias in the Compustat database, and is not economically significant.

To explain this last, important conclusion, let's go back to the history of the Compustat database. In 1978 it was expanded from 2,700 to 6,000 companies by adding small companies that had survived until 1978. By definition their rates of return were higher than their failed peers and that were not included in the database. Consequently, the returns on small Compustat companies are biased upward. The Center for Research and Securities Prices returns database do not have this bias because it includes all companies ever listed on the exchange and does not remove them should they fail. Kothari, Shanken, and Sloan found that the returns for the set of companies that were on CRSP but not on Compustat were, in fact, lower than the set of companies that were on both data sets.

What's the bottom line? It takes a better theory to kill an existing theory, and we have not seen the better theory yet. Therefore, we continue to

17 S. Kothari, J. Shanken, and R. Sloan, ''Another Look at the Cross-Section of Expected Returns," Journal of Finance (December 1995).

use the CAPM (and sometimes the arbitrage pricing model), being wary of all of the problems with estimating it.

### The Arbitrage Pricing Model

The APM can be thought of as a multifactor analogue to the CAPM. The CAPM explains securities returns as a function of one factor, which is called the market index, and is usually measured as the rate of return on a well-diversified portfolio. The APM cost of equity is defined as follows:

where E(Fk) The expected rate of return on a portfolio that mimics the kth factor and is = independent of all others betak = The sensitivity of the stock return to the kth factor

Instead of one measure of systematic risk, the APM includes many. Each beta measures the sensitivity of a company's stock return to a separate underlying factor in the economy. Empirical work has suggested that five fundamental factors are changes in:

• The industrial production index, a measure of how well the economy is doing in terms of actual physical output.

• The short-term real interest rate, measured by the difference between the yield on Treasury bills and the Consumer Price Index.

• Short-term inflation, measured by unexpected changes in the Consumer Price Index.

• Long-term inflation, measured as the difference between the yield to maturity on long- and short-term U.S. government bonds.

• Default risk, measured by the difference between the yield to maturity on Aaa- and Baa-rated long-term corporate bonds.

Empirical evidence also confirms that the APM explains expected returns better than the single-factor CAPM (for example, see Chen 1983; Chen, Ross, and Roll 1986; or Berry, Burmeister, and McElroy 1988).18 In addition, the APM can add insight into the type of risk that is relevant. This is illustrated in Exhibit 10.10. The axes are two of the fundamental factors, the industrial production index and short-term inflation. The diagonal dotted lines

18 N. Chen, R. Roll, and S. Ross, "Economic Forces and the Stock Market," Journal of Business (July 1986), pp. 383-403; and M. Berry, E. Burmeister, and M. McElroy, "Sorting Out Risks Using Known APT Factors," Financial Analysts Journal, vol. 44 (March/April 1988), pp. 29-42.

Beta; (short-term inflation)

Beta; (short-term inflation)

represent constant returns with different combinations of risk. Any portfolio at the origin (point F) has no exposure to either factor, and therefore earns the riskless rate, rf.

For a portfolio at point G, exposure to the systematic risk of unexpected inflation has increased but is offset by decreased risk relative to the industrial production index. The net result is that point G earns the riskless rate, just like point F, but is exposed to a different bundle of risks. A similar story can be told about points A, M, and B. All earn the same expected return as the CAPM market portfolio, E(rm), but have varying exposures to the risk of unexpected inflation and changes in the industrial production index.

Exhibit 10.11 shows the difference in risk premiums as calculated by the APM and the CAPM for five industries. Oil and money center banks are riskier in every dimension. Forest products are less risky, and electric utilities have much less default risk. A larger risk premium means that the industry is more sensitive to a given type of risk than would be predicted by the CAPM. Banks and other financial institutions are more sensitive to unexpected changes in long-term inflation, and the market charges a risk premium—that is, it requires a higher cost of equity.

Exhibit 10.12 shows the net effect of using the CAPM versus the APM to estimate the cost of equity for nine industries. The importance of these differences for valuation of an all-equity perpetual stream of cash flows is reflected in the last column. The 4.7 percent higher APM cost-of-equity estimate in the oil industry means that equity cash flows discounted using the CAPM would be overvalued by 25 percent. Cost-of-equity estimates using the APM are significantly lower for forest products and electric utilities and significantly higher for money center banks and for oil companies with more than 50 percent of their assets in oil reserves.

Exhibit 10.11 Differences in Risk Premiums between APM and CAPM

Exhibit 10.11 Differences in Risk Premiums between APM and CAPM

Exhibit 10.12 Comparison of CAPM and APM Cost of Equity Estimates

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