**UFBA**

# The Capital Asset Pricing Model: Theory and Evidence

(Parte **1** de 5)

First draft: August 2003 This draft: January 2004

The Capital Asset Pricing Model: Theory and Evidence* 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).

Four decades later, the CAPM is still widely used in applications, such as estimating the cost of capital for firms and evaluating the performance of managed portfolios. It is the centerpiece of

MBA investment courses. Indeed, it is often the only asset pricing model taught in these courses.1

The attraction of the CAPM is that it offers powerful and intuitively pleasing predictions about how to measure risk and the relation between expected return and risk. Unfortunately, the empirical record of the model is poor – poor enough to invalidate the way it is used in applications. The CAPM’s empirical problems may reflect theoretical failings, the result of many simplifying assumptions. But they may also be caused by difficulties in implementing valid tests of the model. For example, the CAPM says that the risk of a stock should be measured relative to a comprehensive “market portfolio” that in principle can include not just traded financial assets, but also consumer durables, real estate, and human capital. Even if we

* Eugene F. Fama eugene.fama@gsb.uchicago.edu is Robert R. McCormick Distinguished Service Professor of

Finance, Graduate School of Business, University of Chicago, Chicago, Illinois. Kenneth R. French kfrench@dartmouth.edu is Carl E. and Catherine M. Heidt Professor of Finance, Tuck School of Business,

Dartmouth College, Hanover, New Hampshire. We gratefully acknowledge the comments of John Cochrane, George Constantinides, Richard Leftwich, Andrei Shleifer, René Stulz, and Timothy Taylor.

1 Although every asset pricing model is a capital asset pricing model, the finance profession reserves the acronym CAPM for the specific model of Sharpe (1964), Lintner (1965), and Black (1972) discussed here. Thus, throughout the paper we refer to the Sharpe – Lintner – Black model as the CAPM.

take a narrow view of the model and limit its purview to traded financial assets, is it legitimate to further limit the market portfolio to U.S. common stocks (a typical choice), or should the market be expanded to include bonds, and other financial assets, perhaps around the world? In the end, we argue that whether the model’s problems reflect weaknesses in the theory or in its empirical implementation, the failure of the CAPM in empirical tests implies that most applications of the model are invalid.

We begin by outlining the logic of the CAPM, focusing on its predictions about risk and expected return. We then review the history of empirical work and what it says about shortcomings of the CAPM that pose challenges to be explained by alternative models.

The Logic of the CAPM The CAPM builds on the model of portfolio choice developed by Harry Markowitz

(1959). In Markowitz’s model, an investor selects a portfolio at time t-1 that produces a stochastic return at t. The model assumes investors are risk averse and, when choosing among portfolios, they care only about the mean and variance of their one-period investment return. As a result, investors choose “mean-variance-efficient” portfolios, in the sense that the portfolios: 1) minimize the variance of portfolio return, given expected return, and 2) maximize expected return, given variance. Thus, the Markowitz approach is often called a “mean-variance model.”

The portfolio model provides an algebraic condition on asset weights in mean-varianceefficient portfolios. The CAPM turns this algebraic statement into a testable prediction about the

prices are to clear the market of all assets |

relation between risk and expected return by identifying a portfolio that must be efficient if asset

Sharpe (1964) and Lintner (1965) add two key assumptions to the Markowitz model to identify a portfolio that must be mean-variance-efficient. The first assumption is complete agreement: given market clearing asset prices at t-1, investors agree on the joint distribution of asset returns from t-1 to t. And this distribution is the true one, 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, which is the same for all investors and does not depend

on the amount borrowed or lent |

Figure 1 describes portfolio opportunities and tells the CAPM story. The horizontal axis shows portfolio risk, measured by the standard deviation of portfolio return; the vertical axis shows expected return. The curve abc, which is called the minimum variance frontier, traces combinations of expected return and risk for portfolios of risky assets that minimize return variance at different levels of expected return. (These portfolios do not include riskfree borrowing and lending.) The tradeoff between risk and expected return for minimum variance portfolios is apparent. For example, an investor who wants a high expected return, perhaps at point a, must accept high volatility. At point T, the investor can have an intermediate expected return with lower volatility. If there is no riskfree borrowing or lending, only portfolios above b along abc are mean-variance-efficient, since these portfolios also maximize expected return, given their return variances.

Adding riskfree borrowing and lending turns the efficient set into a straight line.

Consider a portfolio that invests the proportion x of portfolio funds in a riskfree security and 1-x in some portfolio g. If all funds are invested in the riskfree security – that is, they are loaned at the riskfree rate of interest – the result is the point Rf in Figure 1, a portfolio with zero variance and a riskfree rate of return. Combinations of riskfree lending and positive investment in g plot on the straight line between Rf and g. Points to the right of g on the line represent borrowing at the riskfree rate, with the proceeds from the borrowing used to increase investment in portfolio g.

In short, portfolios that combine riskfree lending or borrowing with some risky portfolio g plot along a straight line from Rf through g in Figure 1. 2 To obtain the mean-variance-efficient portfolios available with riskfree borrowing and lending, one swings a line from Rf in Figure 1 up and to the left as far as possible, to the tangency portfolio T. We can then see that all efficient portfolios are combinations of the riskfree asset (either riskfree borrowing or lending) and a single risky tangency portfolio, T. This key result is Tobin’s (1958) “separation theorem.”

The punch line of the CAPM is now straightforward. With complete agreement about distributions of returns, all investors see the same opportunity set (Figure 1) and they combine the same risky tangency portfolio T with riskfree lending or borrowing. Since all investors hold the same portfolio T of risky assets, it must be the value-weight market portfolio of risky assets. Specifically, each risky asset’s weight in the tangency portfolio, which we now call M (for the

“market”), must be the total market value of all outstanding units of the asset divided by total market value of all risky assets. In addition, the riskfree rate must be set (along with the prices of risky assets) to clear the market for riskfree borrowing and lending.

In short, the CAPM assumptions imply that the market portfolio M must be on the minimum variance frontier if the asset market is to clear. This means that the algebraic relation

2 Formally, the return, expected return, and standard deviation of return on portfolios of the riskfree asset f and a risky portfolio g vary with x, the proportion of portfolio funds invested in f, as,

Rp = xRf + (1-x)Rg,

E(Rp) = xRf + (1-x)E(Rg),

that holds for any minimum variance portfolio must hold for the market portfolio. Specifically, if there are N risky assets,

In this equation, E(Ri) is the expected return on asset i and ßiM, the market beta of asset i, is the covariance of its return with the market return divided by the variance of the market return,

MRR Rb s

The first term on the right-hand side of the minimum variance condition, E(RzM), is the expected return on assets that have market betas equal to zero, which means their returns are uncorrelated with the market return. The second term is a risk premium – the market beta of

minus E(RzM) |

asset i, ßiM, times the premium per unit of beta, which is the expected market return, E(RM), Since the market beta of asset i is also the slope in the regression of its return on the market return, a common (and correct) interpretation of beta is that it measures the sensitivity of the asset’s return to variation in the market return. But there is another interpretation of beta more in line with the spirit of the portfolio model that underlies the CAPM. The risk of the market portfolio, as measured by the variance of its return (the denominator of ßiM), is a weighted average of the covariance risks of the assets in M (the numerators of ßiM for different assets). Thus, ßiM is the covariance risk of asset i in M measured relative to the average covariance risk of assets, which is just the variance of the market return.3 In economic terms, ßiM is proportional to the risk each dollar invested in asset i contributes to the market portfolio.

The last step in the development of the Sharpe – Lintner model is to use the assumption of riskfree borrowing and lending to nail down E(RzM), the expected return on zero-beta assets. A risky asset’s return is uncorrelated with the market return – its beta is zero – when the average

contributes nothing to the variance of the market return |

of the asset’s covariances with the returns on other assets just offsets the variance of the asset’s return (see footnote 3). Such a risky asset is riskless in the market portfolio in the sense that it When there is riskfree borrowing and lending, the expected return on assets that are uncorrelated with the market return, E(RzM), must equal the riskfree rate, Rf. The relation between expected return and beta then becomes the familiar Sharpe – Lintner CAPM equation,

(Sharpe – Lintner CAPM) ()[())],ifMfiMERRERRb=+- | 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 asset’s market beta, ßiM, times the premium per unit of beta risk, E(RM) - Rf. Unrestricted riskfree borrowing and lending is an unrealistic assumption. Fischer Black

(1972) develops a version of the CAPM without riskfree borrowing or lending. He shows that the CAPM’s key result – that the market portfolio is mean-variance-efficient – can be obtained by instead allowing unrestricted short sales of risky assets. In brief, back in Figure 1, if there is no riskfree asset, investors select portfolios from along the mean-variance-efficient frontier from

3 Formally, if xiM is the weight of asset i in the market portfolio, then the variance of the portfolio’s return is:

M M M iM i M iM iMii R Cov R R Cov xR R x CovRRs == a to b. Market clearing prices imply 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. The market portfolio is thus a portfolio of the efficient portfolios chosen by investors. With unrestricted short-selling of risky assets, portfolios made up of efficient portfolios are themselves efficient. Thus, the market portfolio is efficient, which means that the minimum variance condition for M given above holds, and it is the expected return-risk relation of the Black CAPM. The relations between expected return and market beta of the Black and Sharpe – Lintner versions of the CAPM differ only in terms of what each says about E(RzM), the expected return on assets uncorrelated with the market. The Black version says only that E(RzM) must be less than the expected market return, so the premium for beta is positive. In contrast, in the Sharpe –

Lintner version of the model, E(RzM) must be the riskfree interest rate, Rf, and the premium per unit of beta risk is E(RM) - Rf. The assumption that short selling is unrestricted is as unrealistic as unrestricted riskfree borrowing and lending. If there is no riskfree asset and short-sales of risky assets are not allowed, mean-variance investors still choose efficient portfolios – points above b on the abc curve in Figure 1. But when there is no short-selling of risky assets and no riskfree asset, the algebra of portfolio efficiency says that portfolios made up of efficient portfolios are not typically efficient. This means that the market portfolio, which is a portfolio of the efficient portfolios chosen by investors, is not typically efficient. And the CAPM relation between expected return and market beta is lost. This does not rule out predictions about expected return and betas with respect to other efficient portfolios – if theory can specify portfolios that must be efficient if the market is to clear. But so far this has proven impossible.

In short, the familiar CAPM equation relating expected asset returns to their market betas is just an application to the market portfolio of the relation between expected return and portfolio beta that holds in any mean-variance-efficient portfolio. The efficiency of the market portfolio is based on many unrealistic assumptions, including complete agreement and either unrestricted riskfree borrowing and lending or unrestricted short-selling of risky assets. But all interesting models involve unrealistic simplifications, which is why they must be tested against data.

Early Empirical Tests

Tests of the CAPM are based on three implications of the relation between expected return and market beta implied by the model. First, expected returns on all assets are linearly related to their betas, and no other variable has marginal explanatory power. Second, the beta premium is positive, meaning that the expected return on the market portfolio exceeds the expected return on assets whose returns are uncorrelated with the market return. Third, in the Sharpe – Lintner version of the model, assets uncorrelated with the market have expected returns equal to the riskfree interest rate, and the beta premium is the expected market return minus the riskfree rate. Most tests of these predictions use either cross-section or time-series regressions. Both approaches date to early tests of the model.

Tests on Risk Premiums The early cross-section regression tests focus on the Sharpe – Lintner model’s predictions about the intercept and slope in the relation between expected return and market beta. The approach is to regress a cross-section of average asset returns on estimates of asset betas. The model predicts that the intercept in these regressions is the riskfree interest rate, Rf, and the coefficient on beta is the expected return on the market in excess of the riskfree rate, E(RM) - Rf.

(Parte **1** de 5)