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The cost of equity is the rate of return investors require on an equity investment in a firm. The risk and return models described in chapter 4 need a riskless rate and a risk premium (in the CAPM) or premiums (in the APM and multi-factor models), which we estimated in the last chapter. They also need measures of a firm's exposure to market risk in the form of betas. These inputs are used to arrive at an expected return on an equity investment using the CAPM.

Expected Return = Riskless rate + Beta (Risk Premium) This expected return to equity investors includes compensation for the market risk in the investment and is the cost of equity. In this section, we will concentrate on the estimation of the beta of a firm. While much of our discussion is directed at the CAPM, it can be extended to apply to the arbitrage pricing and multi factor models, as well.

Betas

In the CAPM, the beta of an investment is the risk that the investment adds to a market portfolio. In the APM and Multi-factor model, the betas of the investment relative to each factor have to be measured. There are three approaches available for estimating these parameters. The first is to use historical data on market prices for individual investments. The second is to estimate the betas from the fundamental characteristics of the investment. The third is to use accounting data. We describe all three approaches in this section.

The conventional approach for estimating the beta of an investment is a regression of the historical returns on the investment against the historical returns on a market index. For firms that have been publicly traded for a length of time, it is relatively straightforward to estimate returns that an investor would have made on investing in stock in intervals (such as a week or a month) over that period. In theory, these stock returns on the assets should be related to returns on a market portfolio, i.e. a portfolio that includes all traded assets, to estimate the betas of the assets. In practice, we tend to use a stock index, such as the S&P 500, as a proxy for the market portfolio, and we estimate betas for stocks against the index.

Regression Estimates of Betas

The standard procedure for estimating betas is to regress1 stock returns (Rj) against market returns (Rm) -

R} = a + bRm where a = Intercept from the regression

The slope of the regression corresponds to the beta of the stock and measures the riskiness of the stock.

The intercept of the regression provides a simple measure of performance of the investment during the period of the regression, when returns are measured against the expected returns from the capital asset pricing model. To see why, consider the following rearrangement of the capital asset pricing model: Rj = Rf + t(Rm -Rf )= Rf (1 -p)+pRm

Compare this formulation of the return on an investment to the return equation from the regression:

Thus, a comparison of the intercept (a) to Rf (1-b) should provide a measure of the stock's performance, at least relative to the capital asset pricing model.2 In summary, then:

If a > Rf (1-b) Stock did better than expected during regression period.

1 The appendix to this chapter provides a brief overview of ordinary least squares regressions.

2 The regression is calculated using returns in excess of the riskless rate for both the stock and the market. In this case, the intercept of the regression should be zero if the actual returns equal the expected returns from the CAPM, greater than zero if the stock does better than expected and less than zero if it does worse than expected.

a = Rf (1-b) Stock did as well as expected during regression period. a < Rf (1-b) Stock did worse than expected during regression period. The difference between a and Rf (1-b) is called Jensen's alpha3 and provides a measure of whether the investment in question earned a return greater than or less than its required return, given both market performance and risk. For instance, a firm that earned 15% during a period, when firms with similar betas earned 12%, will have earned an excess return of 3%; its intercept will also exceed Rf (1-b) by 3%.

The third statistic that emerges from the regression is the R squared (R2) of the regression. While the statistical explanation of the R squared is that it provides a measure of the goodness of fit of the regression, the economic rationale is that it provides an estimate of the proportion of the risk of a firm that can be attributed to market risk; the balance (1 - R2) can then be attributed to firm-specific risk.

The final statistic worth noting is the standard error of the beta estimate. The slope of the regression, like any statistical estimate, may be different from the true value; and the standard error reveals just how much error there could be in the estimate. The standard error can also be used to arrive at confidence intervals for the "true" beta value from the slope estimate.

Illustration 8.1: Estimating a Regression Beta for Boeing

Boeing is a dominant firm in both the aerospace and defense businesses and has been traded on the NYSE for decades. In assessing risk parameters for Boeing, we compute the returns on the stock and the market index as follows.

(1) The returns to a stockholder in Boeing are computed month by month from January 1996 to December 2000. These returns include both dividends and price appreciation are defined as follows.

3 The terminology is confusing, since the intercept of the regression is sometimes also called the alpha and is sometimes compared to zero as a measure of risk-adjusted performance. The intercept can be compared to zero only if the regression is run with excess returns for both the stock and the index; the riskless rate has to be subtracted from the raw return in each month for both.

PriceBnemp, -PriceBnejngj1 + Dividends: , ,

Boeing,J Price

ReturnBoeing,j = Returns to a stockholder in Boeing in month j

PriceB0eing,j = Price of Boeing stock at the end of month j

PriceBoeing,j_i = Price of Boeing stock at the end of month j-1 (the previous month) Dividendsj = Dividends on Boeing stock in month j Dividends are added to the returns of the month in which stockholders are entitled to the dividend.4

(2) The returns on the S&P 500 market index are computed for each month of the period, using the level of the index at the end of each month and the monthly dividend on the stocks in the index.

Index, -Index,, + Dividends, Market Return, = -j---1

where Market Returnj = returns of the index in month j Indexj = the level of the index at the end of month j

Indexj-1 = the level of the index at the end of month j-1 (the previous month) Dividendj = the dividends paid on the index in month j While the S&P 500 and the NYSE Composite are the most widely used indices for U.S. stocks, they are, at best, imperfect proxies for the market portfolio in the CAPM, which is supposed to include all assets.

Figure 8.1 graphs the monthly returns on Boeing against the monthly returns on the S&P 500 index from January 1996 to December 2000.

4 The stock has to be bought by the day called the ex-dividend day for investors to be entitled to dividends. The returns in a month include dividends if the ex-dividend day is in that month.

» ♦ ♦ |

Beta is slope^of this line ^ Each point represents a month of data. - * ♦ t « |

Returns on Boeing

Returns on Boeing

The regression statistics for Boeing are as follows:

(a) Slope of the regression = 0.56. This is Boeing's beta, based on monthly returns from 1996 to 2000. Using a different time period for the regression or different return intervals (weekly or daily) for the same period can result in a different beta.

(b) Intercept of the regression = 0.54%. This is a measure of Boeing's performance, when it is compared with Rf(1-b). The monthly riskless rate (since the returns used in the regression are monthly returns) between 1996 and 2000 averaged 0.4%, resulting in the following estimate for the performance:

Rf (1-b) = 0.4% (1-0.56) = 0.18% Intercept - Rf (1-p) = 0.54% - 0.18% = 0.36% This analysis suggests that Boeing performed 0.36% better than expected, when expectations are based on the CAPM and on a monthly basis between January 1996 and December 2000. This results in an annualized excess return of approximately 4.41%. Annualized Excess Return = (1 + Monthly Excess Return)12 - 1

Note, however, that this does not imply that Boeing would be a good investment in the future. The performance measure also does not provide a breakdown of how much of this excess return can be attributed to the performance of the entire sector (aerospace and defense) and how much is specific to the firm. To make that breakdown, we would need to compute the excess over the same period for other firms in the aerospace and defense industry and compare them with Boeing's excess return. The difference would be then attributable to firm-specific actions. In this case, for instance, the average annualized excess return on other aerospace/defense firms between 1996 and 2000 was -0.85%, suggesting that the firm-specific component of performance for Boeing is actually 5.26%. (Firm-specific Jensen's alpha = 4.41% - (-0.85%))

(c) R squared of the regression = 9.43%. This statistic suggests that 9.43% of the risk (variance) in Boeing comes from market sources and that the balance of 90.57% of the risk comes from firm-specific components. The latter risk should be diversifiable and therefore will not be rewarded with a higher expected return. Boeing's R squared is higher than the median R squared of companies listed on the New York Stock Exchange, which was approximately 19% in 2000.

(d) Standard Error of Beta Estimate = 0.23. This statistic implies that the true beta for Boeing could range from 0.33 to 0.79 (subtracting and adding one standard error to beta estimate of 0.56) with 67% confidence and from 0.10 to 1.02 (subtracting and adding two standard error to beta estimate of 0.56) with 95% confidence. While these ranges may seem large, they are not unusual for most U.S. companies. This suggests that we should consider estimates of betas from regressions with caution.

Most of us who use betas obtain them from an estimation service; Merrill Lynch, Barra, Value Line, Standard and Poor's, Morningstar and Bloomberg are some of the well known services. All these services begin with the regression beta described above and adjust them to reflect what they feel are better estimates of future risk. Although many of these services do not reveal their estimation procedures, Bloomberg is an exception.

Figure 8.2 is the beta calculation page from Bloomberg for Boeing, using the same period as our regression (January 1996 to December 2000):

Figure 8.2: Bloomberg Beta Estimate for Boeing

Figure 8.2 is the beta calculation page from Bloomberg for Boeing, using the same period as our regression (January 1996 to December 2000):

Figure 8.2: Bloomberg Beta Estimate for Boeing

While the time period used is identical to the one used in our earlier regression, there are subtle differences between this regression and the earlier one in Figure 8.1. First, Bloomberg uses price appreciation in the stock and the market index in estimating betas and ignores dividends5. The fact that dividends are ignored does not make much of a difference for a company like Boeing, but it could make a difference for a company that either pays no dividends or pays significantly higher dividends than the market. This explains the mild differences in the intercept (0.50% versus 0.54%) and the beta (0.57 versus 0.56).

Second, Bloomberg also computes what it calls an adjusted beta, which is estimated as follows.

5 This is done purely for computational convenience.

These weights (0.67 and 0.33) do not vary across stocks and this process pushes all estimated betas toward one. Most services employ similar procedures to adjust betas towards one. In doing so, they are drawing on empirical evidence that suggests that the betas for most companies, over time, tend to move towards the average beta, which is one. This may be explained by the fact that firms get more diversified in their product mix and client base as they get larger.

There are three decisions we must make in setting up the regression described above. The first concerns the length of the estimation period. Most estimates of betas, including those by Value Line and Standard and Poor's, use five years of data, while Bloomberg uses two years of data. The trade-off is simple: A longer estimation period provides more data, but the firm itself might have changed in its risk characteristics over the time period. Boeing, during the period of our analysis, acquired both Rockwell and McDonnell Douglas changing its business mix and its basic risk characteristics.

The second estimation issue relates to the return interval. Returns on stocks are available on an annual, monthly, weekly, daily and even on an intra-day basis. Using daily or intra-day returns will increase the number of observations in the regression, but it exposes the estimation process to a significant bias in beta estimates related to non-

trading.6 For instance, the betas estimated for small firms, which are more likely to suffer from non-trading, are biased downwards when daily returns are used. Using weekly or monthly returns can reduce the non-trading bias significantly.7 In this case, using weekly returns for 2 years yields a beta estimate for Boeing of only 0.88, while the monthly beta estimate is 0.96. The latter is a much more reliable estimate of the firm's beta.

The third estimation issue relates to the choice of a market index to be used in the regression. The standard practice used by most beta estimation services is to estimate the betas of a company relative to the index of the market in which its stock trades. Thus, the

6 The non-trading bias arises because the returns in non-trading periods are zero (even though the market may have moved up or down significantly in those periods). Using these non-trading period returns in the regression will reduce the correlation between stock returns and market returns and, ultimately, the beta of the stock.

7 The bias can also be reduced using statistical techniques suggested by Dimson and Scholes-Williams.

betas of German stocks are estimated relative to the Frankfurt DAX, British stocks relative to the FTSE, Japanese stocks relative to the Nikkei and U.S. stocks relative to the NYSE composite or the S&P 500. While this practice may yield an estimate that is a reasonable measure of risk for the domestic investor, it may not be the best approach for an international or cross-border investor, who would be better served with a beta estimated relative to an international index. For instance, Boeing's beta between 1993 and 1998 estimated relative to the Morgan Stanley Capital Index, an index that is composed of stocks from different global markets, yields a beta of 0.82.

To the extent that different services use different estimation periods, different market indices and different beta adjustments, they will often provide different beta estimates for the same firm at the same point in time. While these beta differences are troubling, note that the beta estimates delivered by each of these services comes with a standard error and it is very likely that all the betas reported for a firm fall within the range of standard errors from the regressions.

Historical Beta Estimate for Companies in Smaller (or Emerging) Markets

The process for estimating betas in markets with fewer stocks listed on them is no different from the process described above, but the estimation choices on return intervals, the market index and the return period can make a much bigger difference in the estimate.

• When liquidity is limited, as it often is in many stocks in emerging markets, the betas estimated using short return intervals tend to be much more biased. In fact, using daily or even weekly returns in these markets will tend to yield betas that are not good measures of the true market risk of the company.

• In many emerging markets, both the companies being analyzed and the market itself change significantly over short periods of time. Using five years of returns, as we did for Boeing, for a regression may yield a beta for a company (and market) that bears little resemblance to the company (and market) as it exists today.

• Finally, the indices that measure market returns in many smaller markets tend to be dominated by a few large companies. For instance, the Bovespa ( the Brazilian index) was dominated for several years by Telebras, which represented almost half the index. Nor is this just a problem with emerging markets. The DAX, the equity index for Germany, is dominated by Allianz, Deutsche Bank. Siemens and Daimler. When an index is dominated by one or a few companies, the betas estimated against that index are unlikely to be true measures of market risk. In fact, the betas are likely to be close to one for the large companies that dominate the index and wildly variable for all other companies.

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