Moreover,E(RM),the expected return on assets whose returns are uncorrelated with RM,is the riskfree rate,Re and(4a)becomes the familiar Sharpe-Lintner CAPM risk-return relation, (5) E(R)=R+[E(R)-R)]BM i=1.....N. In words,the expected return on any asset i is the riskfree interest rate,Rr,plus a risk premium which is the beta risk of asset i in M BiM,times the price per unit of beta risk,E(RM)-R(the market risk premium).And BiM is the covariance risk of i in M,cov(Ri,RM),measured relative to the overall risk of the M,s(RM),which is itself a weighted average of the covariance risks of all assets(see equations(1b) and(2)).Finally,note from(4b)that BiM is also the slope in the regression of Ri on RM.This leads to its commonly accepted interpretation as the sensitivity of the asset's return to variation in the market return. Unrestricted riskfree borrowing and lending is an unrealistic assumption.The CAPM risk-return relation(4a)can hold in its absence,but the cost is high.Unrestricted short sales of risky assets must be allowed.In this case,we get Fischer Black's(1972)version of the CAPM.Specifically,without riskfree borrowing or lending,investors choose efficient portfolios from the risky set(points above b on the abc curve in Figure D).Market clearing requires that when one weights the efficient portfolios chosen by investors by the ir(positive)shares of aggregate invested wealth,the resulting portfolio is the market portfolio M.But when unrestricted short-selling of risky assets is allowed,portfolios of positively weighted efficient portfolios are efficient.Thus,market equilibrium again requires that M is efficient, which means assets must be priced so that(4a)holds. Unfortunately,the efficiency of the market portfolio does require either unrestricted riskfree borrowing and lending or unrestricted short selling of risky assets.If there is no riskfree asset and short- sales of risky assets are not allowed,Markowitz'investors still choose efficient portfolios,but portfolios made up of efficient portfolios are not typically efficient.This means the market portfolio almost surely is not efficient,so the CAPM risk-return relation(4a)does not hold.This does not rule out predictions about the relation between expected return and risk if theory can specify the portfolios that must be efficient if the market is to clear.But so far this has proven impossible 5
5 Moreover, E(RzM), the expected return on assets whose returns are uncorrelated with RM, is the riskfree rate, Rf , and (4a) becomes the familiar Sharpe-Lintner CAPM risk-return relation, (5) ( ) [ ( ) )] , E R R i = f + - E R R M f biM i=1,…,N. In words, the expected return on any asset i is the riskfree interest rate, Rf , plus a risk premium which is the beta risk of asset i in M, ßiM, times the price per unit of beta risk, E(RM) – Rf (the market risk premium). And ßiM is the covariance risk of i in M, cov(Ri , RM), measured relative to the overall risk of the M, s 2 (RM), which is itself a weighted average of the covariance risks of all assets (see equations (1b) and (2)). Finally, note from (4b) that ßiM is also the slope in the regression of Ri on RM. This leads to its commonly accepted interpretation as the sensitivity of the asset’s return to variation in the market return. Unrestricted riskfree borrowing and lending is an unrealistic assumption. The CAPM risk-return relation (4a) can hold in its absence, but the cost is high. Unrestricted short sales of risky assets must be allowed. In this case, we get Fischer Black’s (1972) version of the CAPM. Specifically, without riskfree borrowing or lending, investors choose efficient portfolios from the risky set (points above b on the abc curve in Figure 1). Market clearing requires that when one weights the efficient portfolios chosen by investors by their (positive) shares of aggregate invested wealth, the resulting portfolio is the market portfolio M. But when unrestricted short-selling of risky assets is allowed, portfolios of positively weighted efficient portfolios are efficient. Thus, market equilibrium again requires that M is efficient, which means assets must be priced so that (4a) holds. Unfortunately, the efficiency of the market portfolio does require either unrestricted riskfree borrowing and lending or unrestricted short selling of risky assets. If there is no riskfree asset and shortsales of risky assets are not allowed, Markowitz’ investors still choose efficient portfolios, but portfolios made up of efficient portfolios are not typically efficient. This means the market portfolio almost surely is not efficient, so the CAPM risk-return relation (4a) does not hold. This does not rule out predictions about the relation between expected return and risk if theory can specify the portfolios that must be efficient if the market is to clear. But so far this has proven impossible
In short,the central testable implication of the CAPM is that assets must be priced so that the market portfolio M is mean-variance efficient,which implies that the risk-return relation(4a)holds for all assets.This result requires the availability of either unrestricted riskfree borrowing and lending (the Sharpe-Lintner CAPM)or unrestricted short-selling of risky securities(the Black version of the model). II.Early Tests Tests of the CAPM are based on three implications of(4a)and(5).If the market portfolio is efficient, (C1)The expected returns on all assets are linearly related to their market betas,and no other variable has marginal explanatory power; (C2)The risk premium,E(RM)-E(RM)is positive; (C3)In the Sharpe-Lintner version of the model,E(RM)is equal to the riskfree rate,Re Two approaches,cross-section and time-series regressions,are common in tests of(C1)to (C3). Both date to the early tests of the model. Testing (C2)and (C3)-The early cross-section tests focus on (C2)and (C3),and use a approach suggested by (5):Regress average security returns on estimates of their market betas,and test whether the slope is positive and the intercept equals the average riskfree interest rate.Two problems in these tests quickly became apparent.First,there are common sources of variation in the regression residuals(for example,industry effects in average returns)that produce downward bias in OLS estimates of the standard errors of the cross-section regression slopes.Second,estimates of beta for individual securities are imprecise,creating a measurement error problem when they are used to explain average returns Following Blume (1970),Friend and Blume(1970)and Black,Jensen,and Scholes(1972)use a grouping approach to the beta measurement error problem,which becomes the norm in later tests. Expected returns and betas for portfolios are weighted averages of expected asset returns and betas, (6) E(R,)=∑x,E(R, A-g2-Σ G(Ru) 6
6 In short, the central testable implication of the CAPM is that assets must be priced so that the market portfolio M is mean-variance efficient, which implies that the risk-return relation (4a) holds for all assets. This result requires the availability of either unrestricted riskfree borrowing and lending (the Sharpe-Lintner CAPM) or unrestricted short-selling of risky securities (the Black version of the model). II. Early Tests Tests of the CAPM are based on three implications of (4a) and (5). If the market portfolio is efficient, (C1) The expected returns on all assets are linearly related to their market betas, and no other variable has marginal explanatory power; (C2) The risk premium, E(RM) – E(RzM) is positive; (C3) In the Sharpe-Lintner version of the model, E(RzM) is equal to the riskfree rate, Rf . Two approaches, cross-section and time-series regressions, are common in tests of (C1) to (C3). Both date to the early tests of the model. Testing (C2) and (C3) – The early cross-section tests focus on (C2) and (C3), and use an approach suggested by (5): Regress average security returns on estimates of their market betas, and test whether the slope is positive and the intercept equals the average riskfree interest rate. Two problems in these tests quickly became apparent. First, there are common sources of variation in the regression residuals (for example, industry effects in average returns) that produce downward bias in OLS estimates of the standard errors of the cross-section regression slopes. Second, estimates of beta for individual securities are imprecise, creating a measurement error problem when they are used to explain average returns. Following Blume (1970), Friend and Blume (1970) and Black, Jensen, and Scholes (1972) use a grouping approach to the beta measurement error problem, which becomes the norm in later tests. Expected returns and betas for portfolios are weighted averages of expected asset returns and betas, (6) 1 ( ) ( ) N p ip i i E R xER = =å , 2 1 cov( , ) , ( ) p M N pM i ip iM M R R x R b b s = = = å
where xp,i=1,...,N,are the weights for assets in portfolio p.Since expected returns and market betas combine in the same way,if the CAPM explains security returns it also explains portfolio returns.And since beta estimates for diversified portfolios are more precise than estimates for securities,the beta measurement error problem in cross-section regressions of average returns on betas can be reduced by using portfolios.To mitigate the shrinkage in the range of betas(and the loss of statistical power)caused by grouping,Friend and Blume(1970)and Black,Jensen,and Scholes(1972)form portfolios based on ordered beta estimates for securities,an approach that becomes standard. Fama and MacBeth(1973)provide a solution to the inference problem caused by correlation of the residuals in cross-section regressions that also becomes standard.Rather than a single regression of average returns on betas,they estimate monthly cross-section regressions, (7) Rp=Ya +YubpM +Epr p=1,,P,t=1,t, where P is the number of portfolios in the cross-section regression for month t,bMt is the beta estimate for portfolio p,and t is the number of monthly cross-section regressions. Fama (1976,ch.9)shows that the slope in(7)is the return for month t on a zero investment portfolio (sum of the weights equal to 0.0)of the left hand side (LHS)returns that has an estimated market beta,BoM,equal to 1.0.If the market portfolio is efficient,(4a)implies that the expected return on zero investment portfolios that have BoM equal to 1.0 is the expected market premium,E(RM)-E(RM). Inferences about the expected market premium can thus be based on the mean of the monthly estimates of and its standard error.Likewise,?o is the return on a standard portfolio(sum of the weights equal to 1.0)of the LHS returns whose estimated BoM equals zero.The mean of the month-by-month intercepts, ?ot,can be used to test the prediction of the Sharpe-Lintner CAPM that the expected return on portfolios with BoM equal to zero is the average riskfree rate.The advantage of this approach is that the month-by- month variation in the regression coefficients,which determines the standard errors of the means, captures all estimation error implied by the covariance matrix of the cross-section regression residuals.In effect,the difficult problem of estimating the covariance matrix is avoided by repeated sampling. 7
7 where xip, i=1,…,N, are the weights for assets in portfolio p. Since expected returns and market betas combine in the same way, if the CAPM explains security returns it also explains portfolio returns. And since beta estimates for diversified portfolios are more precise than estimates for securities, the beta measurement error problem in cross-section regressions of average returns on betas can be reduced by using portfolios. To mitigate the shrinkage in the range of betas (and the loss of statistical power) caused by grouping, Friend and Blume (1970) and Black, Jensen, and Scholes (1972) form portfolios based on ordered beta estimates for securities, an approach that becomes standard. Fama and MacBeth (1973) provide a solution to the inference problem caused by correlation of the residuals in cross-section regressions that also becomes standard. Rather than a single regression of average returns on betas, they estimate monthly cross-section regressions, (7) R b pt t0 1t pMt pt = g + + g e , p = 1,…, P, t = 1,…, t, where P is the number of portfolios in the cross-section regression for month t, bpMt is the beta estimate for portfolio p, and t is the number of monthly cross-section regressions. Fama (1976, ch.9) shows that the slope ?1t in (7) is the return for month t on a zero investment portfolio (sum of the weights equal to 0.0) of the left hand side (LHS) returns that has an estimated market beta, ßpM, equal to 1.0. If the market portfolio is efficient, (4a) implies that the expected return on zero investment portfolios that have ßpM equal to 1.0 is the expected market premium, E(RM) – E(RzM). Inferences about the expected market premium can thus be based on the mean of the monthly estimates of ?1t and its standard error. Likewise, ?0t is the return on a standard portfolio (sum of the weights equal to 1.0) of the LHS returns whose estimated ßpM equals zero. The mean of the month-by-month intercepts, ?0t, can be used to test the prediction of the Sharpe-Lintner CAPM that the expected return on portfolios with ßpM equal to zero is the average riskfree rate. The advantage of this approach is that the month-bymonth variation in the regression coefficients, which determines the standard errors of the means, captures all estimation error implied by the covariance matrix of the cross-section regression residuals. In effect, the difficult problem of estimating the covariance matrix is avoided by repeated sampling
The second approach to testing the CAPM,time-series regressions,has its roots in Jensen(1968) and is first applied by Friend and Blume(1970)and Black,Jensen,and Scholes(1972).Jensen(1968) notes that if the Sharpe-Lintner risk-return relation(5)holds,the intercept in the time-series regression of the "excess"return on asset i on the excess market return, (8) Ri-R=a+Biy(RM-Rn)+i, is zero for all assets i.Estimates of the intercept in (8)can thus be used to test the prediction of the Sharpe-Lintner CAPM that an asset's average excess return(the average value of Rit-Ra)is completely expla ined by its realized CAPM risk premium(its estimated beta times the average value of RM-Ra). The early cross-section regression tests (Douglas (1968),Black,Jensen and Scholes (1972), Miller and Scholes(1972),Blume and Friend (1973),Fama and MacBeth(1973))reject prediction(C3) of the Sharpe-Lintner version of the CAPM.Specifically,the average value of in estimates of(7)is greater than the average riskfree rate(typically proxied as the return on a one-month Treasury bill),and the average value of is less than the observed average market return in excess of the bill rate.These results persist in more recent cross-section regression tests(for example,Fama and French(1992)).And they are confirmed in time-series regression tests (Friend and Blume(1970),Black,Jensen,and Scholes (1972),Stambaugh(1982)).Specifically,the intercept estimates in(8)are positive for low Bm portfolios and negative for high BM portfolios. When average return is plotted against beta,however,he relation seems to be linear.This suggests that the Black model(4a),which predicts only that the beta premium is positive,describes the data better than the Sharpe-Lintner model(5).Indeed Black's(1972)model is directly motivated by the early evidence that the relation between average return and beta is flatter than predicted by the Sharpe- Lintner model Testing (Cl)-If the market portfolio is efficient,condition(C1)holds:Market betas suffice to explain differences in expected returns across securities and portfolios.This prediction plays a prominent role in tests of the CAPM,and in the early work,the weapon of choice is cross-section regressions.In the 8
8 The second approach to testing the CAPM, time-series regressions, has its roots in Jensen (1968) and is first applied by Friend and Blume (1970) and Black, Jensen, and Scholes (1972). Jensen (1968) notes that if the Sharpe-Lintner risk-return relation (5) holds, the intercept in the time-series regression of the “excess” return on asset i on the excess market return, (8) ( ) Rit - Rft i =a + b e iM R R Mt - + ft it , is zero for all assets i. Estimates of the intercept in (8) can thus be used to test the prediction of the Sharpe-Lintner CAPM that an asset’s average excess return (the average value of Rit – Rft) is completely expla ined by its realized CAPM risk premium (its estimated beta times the average value of RMt – Rft). The early cross-section regression tests (Douglas (1968), Black, Jensen and Scholes (1972), Miller and Scholes (1972), Blume and Friend (1973), Fama and MacBeth (1973)) reject prediction (C3) of the Sharpe-Lintner version of the CAPM. Specifically, the average value of ?0t in estimates of (7) is greater than the average riskfree rate (typically proxied as the return on a one-month Treasury bill), and the average value of ?1t is less than the observed average market return in excess of the bill rate. These results persist in more recent cross-section regression tests (for example , Fama and French (1992)). And they are confirmed in time-series regression tests (Friend and Blume (1970), Black, Jensen, and Scholes (1972), Stambaugh (1982)). Specifically, the intercept estimates in (8) are positive for low ßiM portfolios and negative for high ßiM portfolios. When average return is plotted against beta, however, the relation seems to be linear. This suggests that the Black model (4a), which predicts only that the beta premium is positive, describes the data better than the Sharpe-Lintner model (5). Indeed Black’s (1972) model is directly motivated by the early evidence that the relation between average return and beta is flatter than predicted by the SharpeLintner model. Testing (C1) – If the market portfolio is efficient, condition (C1) holds: Market betas suffice to explain differences in expected returns across securities and portfolios. This prediction plays a prominent role in tests of the CAPM, and in the early work, the weapon of choice is cross-section regressions. In the