C
In terms of mechanics, we used smoothed historical growth rates in earnings and dividends as our projected growth rates and a twostage dividend discount model.
Looking at these numbers, we would draw the following conclusions.
• The implied equity premium has seldom been as high as the historical risk premium. Even in 1978, when the implied equity premium peaked, the estimate of 6.50% is well below what many practitioners use as the risk premium in their risk and return models. In fact, the average implied equity risk premium has been between about 4% over the last 40 years.
• The implied equity premium did increase during the seventies, as inflation increased. This does have interesting implications for risk premium estimation. Instead of assuming that the risk premium is a constant and unaffected by the level of inflation and interest rates, which is what we do with historical risk premiums, it may be more realistic to increase the risk premium as expected inflation and interest rates increase.
histimpl.xls: This data set on the web shows the inputs used to calculate the premium in each year for the U.S. market.
Year
^ implprem.xls: This spreadsheet allows you to estimate the implied equity premium in a market.
^ 4.4: Implied and Historical Premiums
Assume that the implied premium in the market is 3%, and that you are using a historical premium of 7.5%. If you valued stocks using this historical premium, you are likely a. to find more under valued stocks than over valued ones b. to find more over valued stocks than under valued ones c. to find about as many undervalued as overvalued stocks
III. Risk Parameters
The final set of inputs we need to put risk and return models into practice are the risk parameters for individual assets and projects. In the CAPM, the beta of the asset has to be estimated relative to the market portfolio. In the APM and Multifactor model, the betas of the asset relative to each factor have to be measured. There are three approaches available for estimating these parameters; one is to use historical data on market prices for individual assets; the second is to estimate the betas from fundamentals and the third is to use accounting data. We will use all three approaches in this section.
A. Historical Market Betas
This is the conventional approach for estimating betas used by most services and analysts. For firms that have been publicly traded for a length of time, it is relatively straightforward to estimate returns that a investor would have made on the assets in intervals (such as a week or a month) over that period. These returns can then be related to a proxy for the market portfolio to get a beta in the capital asset pricing model, or to multiple macro economic factors to get betas in the multi factor models, or put through a factor analysis to yield betas for the arbitrage pricing model.
Standard Procedures for Estimating CAPM Parameters  Betas and Alphas
The standard procedure for estimating betas is to regress21 stock returns (Rj) against market returns (Rm) Rj = a + b Rm where a = Intercept from the regression b = Slope of the regression = Covariance (Rj, Rm) / a2m 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 during the
Jensen's Alpha: This is the difference between the actual returns on an asset and the return you would have expected it to make during a past period, given what the market did, and the asset's beta.
period of the regression, relative to the capital asset pricing model. Rj = Rf + p (Rm  Rf)
Rj = a + b Rm Regression Equation
Thus, a comparison of the intercept (a) to Rf (1p) should provide a measure of the stock's performance, at least relative to the capital asset pricing model.22 If a > Rf (1p) Stock did better than expected during regression period a = Rf (1p) Stock did as well as expected during regression period a < Rf (1p) Stock did worse than expected during regression period The difference between a and Rf (1p) is called Jensen's alpha, and provides a measure of whether the asset in question under or out performed the market, after adjusting for risk, during the period of the regression.
The third statistic that emerges from the regression is the R squared (R2) of the regression.
R Squared: The R squared measures the proportion of the variability of a dependent variable that is explained by an independent variable or variables.
21 The appendix to this chapter provides a brief overview of ordinary least squares regressions.
22 The regression can be run using returns in excess of the riskfree rate, for both the stock and the market. In that 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.
While the statistical explanation of the R squared is that it provides a measure of the goodness of fit of the regression, the financial rationale for the R squared is that it provides an estimate of the proportion of the risk (variance) of a firm that can be attributed to market risk; the balance (1  R2) can then be attributed to firmspecific risk.
The final statistic worth noting is the standard error of the beta estimate. The slope of the regression, like any statistical estimate, is made with noise, and the standard error reveals just how noisy the estimate is. The standard error can also be used to arrive at confidence intervals for the "true" beta value from the slope estimate.
Estimation Issues
There are three decisions the analyst must make in setting up the regression described above. The first concerns the length of the estimation period. The tradeoff is simple: A longer estimation period provides more data, but the firm itself might have changed in its risk characteristics over the time period. Disney and Deutsche Bank have changed substantially in terms of both business mix and financial leverage over the last few years and any regression that we run using historical data will be affected by these changes.
The second estimation issue relates to the return interval. Returns on stocks are available on an annual, monthly, weekly, daily and even on an intraday basis. Using daily or intraday 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 nontrading.23 For instance, the betas estimated for small firms, which are more likely to suffer from nontrading, are biased downwards when daily returns are used. Using weekly or monthly returns can reduce the nontrading bias significantly.24
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 betas of German stocks are estimated relative to the Frankfurt DAX, British stocks
23 The nontrading bias arises because the returns in nontrading periods is zero (even though the market may have moved up or down significantly in those periods). Using these nontrading period returns in the regression will reduce the correlation between stock returns and market returns and the beta of the stock.
24 The bias can also be reduced using statistical techniques suggested by Dimson and ScholesWilliams.
relative to the FTSE, Japanese stocks relative to the Nikkei, and U.S. stocks relative to the S&P 500. While this practice may yield an estimate that is a reasonable measure of risk for the parochial investor, it may not be the best approach for an international or crossborder investor, who would be better served with a beta estimated relative to an international index.
Illustration 4.1: Estimating CAPM risk parameters for Disney
In assessing risk parameters for Disney, the returns on the stock and the market index are computed as follows 
(1) The returns to a stockholder in Dsiney are computed month by month from January 1999 to December 2003. These returns include both dividends and price appreciation and are defined as follows 
Stock Returnintel, j =( PriceIntel, j  Price^, H+Dividendsj) / Price^^ where Stock ReturnIntel,j = Returns to a stockholder in Disney in month j PriceIntel_ j = Price of Disney stock at the end of month j Dividendsj = Dividends on Disney stock in month j Dividends are added to the returns of the month in which the stock went exdividend.25 If there was a stock split26 during the month, the returns have to take into account the split factor, since stock prices will be affected.27
(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 yield on stocks in the index. 
Market Returnintel_ j =( Indexj  Index jx + Dividends,) / IndexjX
25 The exdividend day is the day by which the stock has to be bought for an investor to be entitled to the dividends on the stock.
26 A split changes the number of shares outstanding in a company without affecting any of its fundamentals. Thus, in a threefortwo split, there will be 50% more shares outstanding after the split. Since the overall value of equity has not changed, the stock price will drop by an equivalent amount (1 100/150 = 33.33%)
27 While there were no stock splits in the time period of the regression, Disney did have a 3 for 1 stock split in July 1998. The stock price dropped significantly, and if not factored in will result in very negative returns in that month. Splits can be accounted for as follows 
Returnintel, j =( Factorj * PriceIntel, j  PriceIntel, J1+ Factor * Dividends^ / PriceIntelj1 The factor would be set to 3 for July 1998 and the ending price would be multiplied by 3, as would the dividends per share, if they were paid after the split.
where IndeXj is the level of the index at the end of month j and Dividendj is 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 4.3 graphs monthly returns on Disney against returns on the S&P 500 index from January 1999 to December 2003.
• y^» ••• • ♦ ♦  
* 20.00% 30.00%  
S&P 500 The regression statistics for Disney are as follows:28 (a) Slope of the regression = 1.01. This is Disney's beta, based on returns from 1999 to 2003. 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 = 0.0467%. This is a measure of Disney's performance, when it is compared with Rf (1p).29 The monthly riskfree rate (since the returns used in 28 The regression statistics are computed in the conventional way. The appendix explains the process in more detail. 29 In practice, the intercept of the regression is often called the alpha and compared to zero. Thus, a positive intercept is viewed as a sign that the stock did better than expected and a negative intercept as a sign that the stock did worse than expected. In truth, this can be done only if the regression is run in terms of excess returns, i.e., returns over and above the riskfree rate in each month for both the stock and the market index. the regression are monthly returns) between 1999 and 2003 averaged 0.313%, resulting in the following estimate for the performance: Rf (1p) = 0.313% (11.01) = .0032% Intercept  Rf (1P) = 0.0467%  (0.0032%) = 0.05% This analysis suggests that Disney's stock performed 0.05% better than expected, when expectations are based on the CAPM, on a monthly basis between January 1999 and December 2003. This results in an annualized excess return of approximately 1.81%. Annualized Excess Return = (1 + Monthly Excess Return)12  1 = (10.0005)12 1 = .0060 or 0.60% By this measure of performance, Disney did slightly better than expected during the period of the regression, given its beta and the market's performance over the period. Note, however, that this does not imply that Disney would be a good investment looking forward. It also does not provide a breakdown of how much of this excess return can be attributed to 'industrywide' effects, and how much is specific to the firm. To make that breakdown, the excess returns would have to be computed over the same period for other firms in the entertainment industry and compared with Disney's excess return. The difference would be then attributable to firmspecific actions. In this case, for instance, the average annualized excess return on other entertainment firms between 1999 and 2003 was 1.33%. This would imply that Disney stock underperformed it's peer group by 0.73% between 1999 and 2003, after adjusting for risk. (Firmspecific Jensen's alpha = 0.60%  1.33% =  0.73%) (c) R squared of the regression = 29%. This statistic suggests that 29% of the risk (variance) in Disney comes from market sources (interest rate risk, inflation risk etc.), and that the balance of 71% of the risk comes from firmspecific components. The latter risk should be diversifiable, and therefore unrewarded. Disney's R squared is slightly higher than the median R squared of companies listed on the New York Stock Exchange, which was approximately 21% in 2003. (d) Standard Error of Beta Estimate = 0.20. This statistic implies that the true beta for Disney could range from 0.81 to 1.21 (subtracting adding one standard error to beta estimate of 1.01) with 67% confidence and from 0.61 to 1.41 (subtracting adding two standard error to beta estimate of 1.01) with 95% confidence. While these ranges may seem large, they are not unusual for most U.S. companies. This suggests that we should consider regression estimates of betas from regressions with caution. ^ 4.5: The Relevance of Rsquared to an Investor Assume that, having done the regression analysis, both Disney and Amgen, a biotechnology company, have betas of 1.01. Disney, however, has an Rsquared of 31%, while Amgen has an Rsquared of only 15%. If you had to pick between these investments, which one would you choose? a. Disney, because it's higher Rsquared suggests that it is less risky b. Amgen, because it's lower Rsquared suggests a greater potential for high returns c. I would be indifferent, because they both have the same beta Would your answer be any different if you were running a welldiversified fund? In Practice: Using a Service beta Most analysts 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 regression betas and make what they feel are necessary changes to make them better estimates for the future. While most of these services do not reveal the internal details of this estimation, Bloomberg is an honorable exception. The following is the beta calculation page from Bloomberg for Disney, using the same period as our regression (January 1999 to December 2003): 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 4.1. First, Bloomberg uses price appreciation in the stock and the market index in estimating betas and ignores dividends.30 This does not make much of a difference for a Disney, but it could make a difference for a company that either pays no dividends or pays significantly higher dividends than the market. Second, Bloomberg also computes what they call an adjusted beta, which is estimated as follows: Adjusted Beta = Raw Beta (0.67) + 1 (0.33) These weights do not vary across stocks, and this process pushes all estimated betas towards 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. 30 This is why the intercept in the Bloomberg print out (0.03%) is slightly different from the intercept estimated earlier in the chapter (0.05%). The beta and Rsquared are identical. In general, betas reported by different services for the same firm can be very different because they use different time periods (some use 2 years and others 5 years), different return intervals (daily, weekly or monthly), different market indices and different postregression estimates. While these beta differences may be troubling, the beta estimates delivered by each of these services comes with a standard error, and it is very likely that all of the betas reported for a firm fall within the range of the standard errors from the regressions. Illustration 4.2: Estimating Historical Betas for Aracruz and Deutsche Bank Aracruz is a Brazilian company and we can regress returns on the stock against a Brazilian index to obtain risk parameters. The stock also had an ADR listed on the U.S. exchanges and we can regress returns on the ADR against a U.S. index to obtain parameters. Figure 4.4 presents both graphs for the January 1999 December 2003 time period: Figure 4.4: Estimating Aracruz's Beta: Choice of Indices Aracruz ADR vs S&P 500 Aracruz ADR vs S&P 500 
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