Amos Tuck school of business at Dartmouth College Working Paper No 03-26 Center for Research in Security Prices(CRSP) University of Chicago Working Paper No 550 The CaPM: Theory and evidence Eugene f fama University of chicago Kenneth french artmouth College: MIT, NBER August 2003 This paper can be downloaded without charge from the Social Science Research Network Electronic Paper Collection at http:/ssrn.com/abstract=440920
Amos Tuck School of Business at Dartmouth College Working Paper No. 03-26 Center for Research in Security Prices (CRSP) University of Chicago Working Paper No. 550 The CAPM: Theory and Evidence Eugene F. Fama University of Chicago Kenneth R. French Dartmouth College; MIT; NBER August 2003 This paper can be downloaded without charge from the Social Science Research Network Electronic Paper Collection at: http:/ssrn.com/abstract=440920
First draft: August 2003 Not for Comments solicited The CAPM: Theory and Evidence by Eugene f Fama and Kenneth french The capital asset pricing model( CAPM) of William Sharpe(1964)and John Lintner (1965) marks the birth of asset pricing theory(resulting in a Nobel Prize for Sharpe in 1990). Before their breakthrough, there were no asset pricing models built from first principles about the nature of tastes and investment opportunities and with clear testable predictions about risk and return. Four decades later, the CAPM is still widely used in applications, such as estimating the cost of equity capital for firms and evaluating the performance of managed portfolios. And it is the centerpiece, indeed often the only asset pricing model taught in MBA level investment courses The attraction of the CAPm is its powerfully simple logic and intuitively pleasing predictions about how to measure risk and about the relation between expected return and risk. Unfortunately perhaps because of its simplicity, the empirical record of the model is poor- poor enough to invalidate the way it is used in applications. The model's empirical problems may reflect true failings. (It is, after all, just a model. But they may also be due to shortcomings of the empirical tests, most notably, poor proxies for the market portfolio of invested wealth, which plays a central role in the models predictions We argue, however, that if the market proxy problem invalidates tests of the model, it also invalidates most applications, which typically borrow the market proxies used in empirical tests For perspective on the CAPm's predictions about risk and expected return, we begin with a brief summary of its logic. We then review the history of empirical work on the model and what it says about shortcomings of the CAPM that pose challenges to be explained by more complicated modes Graduate School of Business, University of Chicago(Fama), and Tuck School of Business, Dartmouth College rench
First draft: August 2003 Not for quotation Comments solicited The CAPM: Theory and Evidence by Eugene F. Fama and Kenneth R. French* The capital asset pricing model (CAPM) of William Sharpe (1964) and John Lintner (1965) marks the birth of asset pricing theory (resulting in a Nobel Prize for Sharpe in 1990). Before their breakthrough, there were no asset pricing models built from first principles about the nature of tastes and investment opportunities and with clear testable predictions about risk and return. Four decades later, the CAPM is still widely used in applications, such as estimating the cost of equity capital for firms and evaluating the performance of managed portfolios. And it is the centerpiece, indeed often the only asset pricing model taught in MBA level investment courses. The attraction of the CAPM is its powerfully simple logic and intuitively pleasing predictions about how to measure risk and about the relation between expected return and risk. Unfortunately, perhaps because of its simplicity, the empirical record of the model is poor – poor enough to invalidate the way it is used in applications. The model’s empirical problems may reflect true failings. (It is, after all, just a model.) But they may also be due to shortcomings of the empirical tests, most notably, poor proxies for the market portfolio of invested wealth, which plays a central role in the model’s predictions. We argue, however, that if the market proxy problem invalidates tests of the model, it also invalidates most applications, which typically borrow the market proxies used in empirical tests. For perspective on the CAPM’s predictions about risk and expected return, we begin with a brief summary of its logic. We then review the history of empirical work on the model and what it says about shortcomings of the CAPM that pose challenges to be explained by more complicated models. * Graduate School of Business, University of Chicago (Fama), and Tuck School of Business, Dartmouth College (French)
. The CAPm The CAPM builds on Harry Markowitz'(1952, 1959)mean-variance portfolio model. In Markowitz' model, an investor selects a portfolio at time t-l that produces a random return Rpt at t. The model assumes that investors are risk averse and, when choosing among portfolios, they care only about the mean and variance of their one-period investment return. The models main result follows from these assumptions. Specifically, the portfolios relevant for choice by investors are mean-variance efficient, which means (i) they minimize portfolio return variance, s"(Rpt), given expected return, E(Rpt), and (ii) they maximize expected return given variance The way assets combine to produce efficient portfolios provides the template for the relation between expected return and risk in the CAPM. Suppose there are n risky assets available to investors It is easy to show that the portfolio e that minimizes return variance, subject to delivering expected return E(Re), allocates proportions of invested wealth, xe 2 is =1.0), to portfolio assets so as to produce a linear relation between the expected return on any asset i and its beta risk in portfolio e (1a) E(R)=E(R)+[E(R2)-BR2)B,i=1,N (1b) B Cov(R,R)∑xCo(R,R a2(R2) x2∑2xCov(R,R) In these equations, Cov denotes a covariance, E(rze)is the expected return on assets whose returns are uncorrelated with the return on e(they have Cov(Ri, re)=0), and the subscript t that should appear on all returns is fo To interpret(la)and (lb), note first that in the portfolio model, expected returns on assets and covariances between asset returns are parameters supplied by the investor. Equations(la)and(1b)then say that given these inputs, finding the portfolio that minimizes return variance subject to having expected return E(Re) implies choosing asset weights(xie, i=l,., N) that produce beta risks(Be, F1,.,N) that cause(la)to be satisfied for each asset
2 I. The CAPM The CAPM builds on Harry Markowitz’ (1952, 1959) mean-variance portfolio model. In Markowitz’ model, an investor selects a portfolio at time t-1 that produces a random return Rpt at t. The model assumes that investors are risk averse and, when choosing among portfolios, they care only about the mean and variance of their one-period investment return. The model’s main result follows from these assumptions. Specifically, the portfolios relevant for choice by investors are mean-variance efficient, which means (i) they minimize portfolio return variance, s 2 (Rpt), given expected return, E(Rpt), and (ii) they maximize expected return given variance. The way assets combine to produce efficient portfolios provides the template for the relation between expected return and risk in the CAPM. Suppose there are N risky assets available to investors. It is easy to show that the portfolio e that minimizes return variance, subject to delivering expected return E(Re), allocates proportions of invested wealth, xie 1 ( 1.0) N i ie x = å = , to portfolio assets so as to produce a linear relation between the expected return on any asset i and its beta risk in portfolio e, (1a) ( ) ( ) [ ( ) ( )] , E Ri =+- E Rze E Re E Rze bie i=1,…,N, (1b) 1 2 1 1 ( , ) (,) . ( ) ( , ) N je i j i e j ie N N e ie je i j i j CovRR x CovRR R x x CovRR b s = = = = = å å å In these equations, Cov denotes a covariance, E(Rze) is the expected return on assets whose returns are uncorrelated with the return on e (they have Cov(Ri , Re) = 0), and the subscript t that should appear on all returns is, for simplicity, dropped. To interpret (1a) and (1b), note first that in the portfolio model, expected returns on assets and covariances between asset returns are parameters supplied by the investor. Equations (1a) and (1b) then say that given these inputs, finding the portfolio that minimizes return variance subject to having expected return E(Re) implies choosing asset weights (xie, i=1,…,N) that produce beta risks (ßie, i=1,…,N) that cause (1a) to be satisfied for each asset
The beta risk of asset i has an intuitive interpretation. In Markowitz 'model, a portfolios risk is the variance of its return, so the risk of portfolio e is s"(Re). The portfolio return variance is the sum of the weighted covariances of each asset's return with the portfolio return, d(R)=∑xCov(RR) Thus, Bie= Cov(Ri, Re)s(Re) can be interpreted as the covariance risk of asset i in portfolio e, measured relative to the risk of the portfolio, which is just an average of the covariance risks of all assets Equation(la) is the result of algebra, the condition on asset weights that produces the minimum variance portfolio with expected return equal to E(Re). The CAPM turns it into a restriction on market clearing prices and expected returns by identifying a portfolio that must be efficient if asset prices are to clear the market of all securities. Applied to such a portfolio, equation(la)becomes a relation between expected return and risk that must hold in a market equilibrium Sharpe and Lintner add two key assumptions to the Markowitz model to identify a portfolio that must be efficient if the market is to clear. The first is complete agreement: Given market clearing prices at t-1, investors agree on the joint distribution of asset returns from t-l to t. And it is the true distribution, that is the distribution from which the returns we use to test the model are drawn The second assumption is that there is borrowing and lending at a riskfree rate, R which is the same for all investors and does not depend on the amount borrowed or lent. Such unrestricted riskfree borrowing and lending implies a strong form of Tobins(1958)separation theorem. Figure 1, which describes portfolio opportunities in the (E(R), s(R)plane, tells the story. The curve abc traces combinations of E(R)and s(R) for portfolios that minimize return variance at different levels of expected return, but ignoring riskfree borrowing and lending. In this restricted set, only portfolios above b alon abc are efficient(they also maximize expected return, given their return variances) Adding riskfree borrowing and lending simplifies the efficient set Consider a portfolio that invests the proportion x of portfolio funds in a riskfree security and l-x in some portfolio g (3a) Re=xR+(l-xRE
3 The beta risk of asset i has an intuitive interpretation. In Markowitz’ model, a portfolio’s risk is the variance of its return, so the risk of portfolio e is s 2 (Re). The portfolio return variance is the sum of the weighted covariances of each asset’s return with the portfolio return, (2) 2 ( ) ( , ). Re ie i e s =åx Cov R R Thus, ßie = Cov(Ri , Re)/s 2 (Re) can be interpreted as the covariance risk of asset i in portfolio e, measured relative to the risk of the portfolio, which is just an average of the covariance risks of all assets. Equation (1a) is the result of algebra, the condition on asset weights that produces the minimum variance portfolio with expected return equal to E(Re). The CAPM turns it into a restriction on market clearing prices and expected returns by identifying a portfolio that must be efficient if asset prices are to clear the market of all securities. Applied to such a portfolio, equation (1a) becomes a relation between expected return and risk that must hold in a market equilibrium. Sharpe and Lintner add two key assumptions to the Markowitz model to identify a portfolio that must be efficient if the market is to cle ar. The first is complete agreement: Given market clearing prices at t-1, investors agree on the joint distribution of asset returns from t-1 to t. And it is the true distribution, that is, the distribution from which the returns we use to test the model are drawn. The second assumption is that there is borrowing and lending at a riskfree rate , Rf , which is the same for all investors and does not depend on the amount borrowed or lent. Such unrestricted riskfree borrowing and lending implies a strong form of Tobin’s (1958) separation theorem. Figure 1, which describes portfolio opportunities in the (E(R), s(R)) plane, tells the story. The curve abc traces combinations of E(R) and s(R) for portfolios that minimize return variance at different levels of expected return, but ignoring riskfree borrowing and lending. In this restricted set, only portfolios above b along abc are efficient (they also maximize expected return, given their return variances). Adding riskfree borrowing and lending simplifies the efficient set. Consider a portfolio that invests the proportion x of portfolio funds in a riskfree security and 1-x in some portfolio g, (3a) Rp = xRf + (1-x)Rg, x = 1.0
The expected return and the standard deviation of the return on p are E(Rp)=xR+(l-XE(Rg) S(RD=|1-x S(Rg) These equations imply that the portfolios obtained by varying x in(3a) plot along a straight line n Figure 1. The line starts at Rr(x=1.0, all funds are invested in the riskfree asset), runs to the point g(x 0.0, all funds are invested in g and continues on for portfolios that involve borrowing at the riskfree rate(x<0.0, with the proceeds from the borrowing used to increase the investment in g). It is then easy to see that to obtain the efficient portfolios available with riskfree borrowing and lending, one simply swings a line from R in Figure I up and to the left, to the tangency portfolio t, which is as far as one can go without passing into infeasible territory The key result is that with unrestricted riskfree borrowing and lending, all efficient portfolios are combinations of the single risky tangency portfolio T with either lending at the riskfree rate(points below T along the line from Rf or riskfree borrowing(points above t along the line from ra. This is Tobin's (958)separation theorem The CAPM's punch line is now straightforward. With complete agreement about distributions of returns, all investors combine the same tangency portfolio T with riskfree borrowing or lending. Since all investors hold the same portfolio of risky assets, the market for risky assets does not clear at time tl unless each asset is priced so its weight in T is its total market value at t-l divided by the total value of all isky assets. But this is just the assets weight, XM, in the market portfolio of invested wealth, M. Thus the critical tangency portfolio must be the market portfolio. In addition, the riskfree rate must be set along with the prices of risky assets) to clear the market for riskfree borrowing and lending Since the tangency portfolio is the market portfolio, the market portfolio M is efficient and (la) and(lb) hold for M E(R)=E(RM)+[(R)E(RM)IBM, i=1,... N B=COV(R,RM)
4 The expected return and the standard deviation of the return on p are, (3b) E(Rp) = xRf + (1-x)E(Rg), (3c) s(Rp) = |1-x| s(Rg). These equations imply that the portfolios obtained by varying x in (3a) plot along a straight line in Figure 1. The line starts at Rf (x = 1.0, all funds are invested in the riskfree asset), runs to the point g (x = 0.0, all funds are invested in g) and continues on for portfolios that involve borrowing at the riskfree rate (x < 0.0, with the proceeds from the borrowing used to increase the investment in g). It is then easy to see that to obtain the efficient portfolios available with riskfree borrowing and lending, one simply swings a line from Rf in Figure 1 up and to the left, to the tangency portfolio T, which is as far as one can go without passing into infeasible territory. The key result is that with unrestricted riskfree borrowing and lending, all efficient portfolios are combinations of the single risky tangency portfolio T with either lending at the riskfree rate (points below T along the line from Rf) or riskfree borrowing (points above T along the line from Rf). This is Tobin’s (1958) separation theorem. The CAPM’s punch line is now straightforward. With complete agreement about distributions of returns, all investors combine the same tangency portfolio T with riskfree borrowing or lending. Since all investors hold the same portfolio of risky assets, the market for risky assets does not clear at time t-1 unless each asset is priced so its weight in T is its total market value at t-1 divided by the total value of all risky assets. But this is just the asset’s weight, xiM, in the market portfolio of invested wealth, M. Thus the critical tangency portfolio must be the market portfolio. In addition, the riskfree rate must be set (along with the prices of risky assets) to clear the market for riskfree borrowing and lending. Since the tangency portfolio is the market portfolio, the market portfolio M is efficient and (1a) and (1b) hold for M, (4a) ( ) ( ) [ ()( )] , E Ri =+- E RzM M E R E RzM biM i=1,…,N, (4b) 2 cov( , ) . ( ) i M iM M R R R b s =
Moreover, E(RzM), the expected return on assets whose returns are uncorrelated with Rm, is the riskfree rate, R, and(4a)becomes the familiar Sharpe -Lintner CAPM risk-return relation E(R )=R+[E(RM)-RBiar, FlN In words, the expected return on any asset i is the riskfree interest rate, R, plus a risk premium which the beta risk of asset i in M BiM, times the price per unit of beta risk, E(RM)-Re(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(Ib) 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 surel 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 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) IL. 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(r, M)is equal to the riskfree rate, Re Two approaches, cross-section and time-series regressions, are common in tests of ( cl)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 pproach suggested by (5): Regress average security returns on estimates of their market betas, and test whether the slope is positive and the intercept equal 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) B=COV(Rp,Ru) (R)
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, FI, 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 correhtion 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 Rpr=ya+y1,bpM+E, P where P is the number of portfolios in the cross-section regression for month t bMt is the beta estimate or portfolio p, and t is the number of monthly cross-section regressions Fama (1976, ch 9)shows that the slope ?t 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, BpM, equal to 1.0. If the market portfolio is efficient, (4a)implies that the expected return on zero investment portfolios that have BpM 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 ?I and its standard error. Likewise, ?ot is the return on a standard portfolio (sum of the weights equal to 1.0)of the LHs returns whose estimated BpM equal zero. The mean of the month-by-month intercepts, or, can be used to test the prediction of the Sharpe - Lintner CAPm that the expected return on portfolios with BpM 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 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, RI-RA=a+BiM(Rm-R1)+Eir is zero for all assets 1. 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-Ro) is completel expla ined by its realized CAPM risk premium(its estimated beta times the average value of RMt-Ro) 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 ?a 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 %t 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 Blacks(1972)model is directly motivated by the early evidence that the relation between average return and beta is flatter than predicted by the Sharp Lintner model Testing(Cl)-If the market portfolio is efficient, condition(Cl)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 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
Fama-MacBeth framework, one simply adds pre-determined explanatory variables Zipa-1,j=2,,J, to the eriod-by-period cross-section regression(7), Rn=0+bh…+∑/2?n p=1,,P,t=1,,T Generalizing the interpretation of the one-variable cross-section regression(7), the Ols intercept in(9) is the return on a standard portfolio(sum of the weights equal to 1. 0)of the LhS portfolio returns that has zero weighted average values of each of the other explanatory variables. And each regression slope is the return on a zero-investment portfolio(sum of the weights equal to 0.0)of the LHS returns that has a weighted average value of 1.0 for its explanatory variable and weighted average values of zero for other explanatory variables. (See Fama(1976, ch. 9. ) The average values of the period-by-period cross- section regression coefficients in(9)thus provide focused tests of the CAPm predictions(Cl)to( C3) If market betas suffice to explain expected returns(condition(CI), the time-series means of the slopes?jt on the Z variables in(9)should not be reliably different from zero. For example, in Fama and MacBeth(1973)the Z variables are squared market betas( to test the prediction of (4a)that the relation between expected return and beta is linear) and residual variances from regressions of returns on the market return( to test the prediction of (4a) that market beta is the only measure of risk needed to explain expected returns). The tests suggest that these Z variables do not add to the explanation of expected returns provided by beta. Since the tests on ?it suggest that the average market premium is positive, the results of Fama and MacBeth(1973) are consistent with the hypothesis that the ir market proxy(an equal weight portfolio of NYSE stocks) is efficient In the cross-section regression approach of (9), the alternative hypothesis is specific; a particular set of Z variables chosen by the researcher provides the alternative to the CaPm prediction(Cl) that market betas suffice to explain expected returns. Because the alternative hypothesis is specific, t-tests on the average slopes for the Z variables provide tests of (Cl)(though strictly speaking, a joint test on the average slopes for all the Z variables is more appropriate). The trick in this approach is to choose Z variables likely to expose any problems of the CAPM
9 Fama-MacBeth framework, one simply adds pre-determined explanatory variables Zjp,t-1, j = 2,…, J, to the period-by-period cross-section regression (7), (9) 0 1 , 1 2 J pt t t pMt jt j p t pt j R g g b Z g e - = = + + + å , p = 1,…, P, t = 1,…, T. Generalizing the interpretation of the one-variable cross-section regression (7), the OLS intercept in (9) is the return on a standard portfolio (sum of the weights equal to 1.0) of the LHS portfolio returns that has zero weighted average values of each of the other explanatory variables. And each regression slope is the return on a zero-investment portfolio (sum of the weights equal to 0.0) of the LHS returns that has a weighted average value of 1.0 for its explanatory variable and weighted average values of zero for other explanatory variables. (See Fama (1976, ch. 9.)) The average values of the period-by-period crosssection regression coefficients in (9) thus provide focused tests of the CAPM predictions (C1) to (C3). If market betas suffice to explain expected returns (condition (C1)), the time-series means of the slopes ?jt on the Z variables in (9) should not be reliably different from zero. For example, in Fama and MacBeth (1973) the Z variables are squared market betas (to test the prediction of (4a) that the relation between expected return and beta is linear) and residual variances from regressions of returns on the market return (to test the prediction of (4a) that market beta is the only measure of risk needed to explain expected returns). The tests suggest that these Z variables do not add to the explanation of expected returns provided by beta. Since the tests on ?1t suggest that the average market premium is positive, the results of Fama and MacBeth (1973) are consistent with the hypothesis that the ir market proxy (an equalweight portfolio of NYSE stocks) is efficient. In the cross-section regression approach of (9), the alternative hypothesis is specific; a particular set of Z variables chosen by the researcher provides the alternative to the CAPM prediction (C1) that market betas suffice to explain expected returns. Because the alternative hypothesis is specific, t-tests on the average slopes for the Z variables provide tests of (C1) (though strictly speaking, a joint test on the average slopes for all the Z variables is more appropriate). The trick in this approach is to choose Z variables likely to expose any problems of the CAPM
