Risk, Return, and Equilibrium: Empirical Tests TOR Eugene F Fama. James D. Mac Beth The Journal of political Economy, Volume 81, Issue 3(May -Jun, 1973),607-636 Stable url: http://links.jstor.org/sici?sici=0022-3808%28197305%2f197306%2981%03a3%03c607%3arraeett3e2.0.c0%03b2-c Your use of the JSTOR archive indicates your acceptance of jSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JStoR archive only for your personal, non-commercial use Each copy of any part of a jSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission The Journal of political Economy is published by The University of Chicago Press. Please contact the publisher for further permissions regarding the use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/ucpress.html The Journal of Political economy o1973 The University of Chicago Press jStOR and the jStOR logo are trademarks of JSTOR, and are Registered in the U.S. Patent and Trademark Office For more information on STOR contact jstor- info @umich. edu @2002 JSTOR http://www」]stor.org Wed nov1321:33:052002
Risk, Return, and Equilibrium Empirical Tests Eugene F. Fama and James D. Mac Beth University of chicago This paper tests the relationship between average return and risk for New York Stock Exchange common stocks. The theoretical basis of the tests is the"two-parameter" portfolio model and models of market equilibrium derived from the two-parameter portfolio model. We can not reject the hypothesis of these models that the pricing of common stocks reflects the attempts of risk-averse investors to hold portfolios that are"efficient""in terms of expected value and dispersion of return Moreover, the observed "fair game"properties of the coefficients and esiduals of the risk-return regressions are consistent with an"efficient capital market-that is, a market where prices of securities fully refect available information . Theoretical background In the two-parameter portfolio model of Tobin(1958), Markowitz(1959) and Fama(1965b), the capital market is assumed to be perfect in the sense that investors are price takers and there are neither transactions costs nor information costs. Distributions of one-period percentage returns on all assets and port folic ssumed to be normal or to conform to ome other two-parameter member of the symmetric stable class. Investors are assumed to be risk averse and to behave as if they choose among portfolios on the basis of maximum expected utility. A perfect capital market, investor risk aversion, and two-parameter return distributions imply the important "efficient set theorem": The optimal portfolio for any investor must be efficient in the sense that no other portfolio with the same or higher expected return has lower dispersion of return received for publication September 2, 197 Foundation. The ents of p Gonedes, M. Jensen, M. Miller, R. Office re gratefully acknowledged. A special note of hanks is d I Althoug rameter is arbitrary, the standard deviation
JOURNAL OF POLITICAL ECONOMY In the portfolio model the investor looks at individual assets only in erms of their contributions to the expected value and dispersion, or risk of his portfolio return. With normal return distributions the risk of port- folio p is measured by the standard deviation, G(R,), of its return, R, and the risk of an asset for an investor who holds p is the contribution of the asset to o(R,). If x is the proportion of portfolio funds invested in asset i, Gy=cov(R,, R,)is the covariance between the returns on assets i and j, and N is the number of assets, then cov(Ri, Rp) Cip g(R RP) Thus, the contribution of asset i to o(Rp)-that is, the risk of asset i in the portfolio p-is proportional to x10/((2)=cov(R,R)/(R Note that since the weights xip vary from portfolio to portfolio, the risk of an asset is different for different portfolios For an individual investor the relationship between the risk of an asset ind its expected return is implied by the fact that the investors optimal portfolio is efficient. Thus, if he chooses the portfolio m, the fact that m is efficient means that the weights xim, i=1, 2, .., N, maximize ex portfolio return E(Rm) E(R1) subject to the constraints is common when return distributions are assumed to be normal. whereas an inter fractile range is usually suggested when returns are generated from some other symmetric stable distribution It is well known that the mean-standard deviation version of the two-parameter portfolio model can be derived from the assumption that investors have quadratic case, the empirical evidence of Fama (1965a) (1970),Roll(1970),K, Millet (1971), and Officer (1971)provides support for the "distribution"approach to the model. For a discussion of the issues and a detailed treatment of the two-parameter model, see Fama and Miller (1972, chaps. 6-8) 'e also concentrate on the special case of the two-parameter model obtained witl the assumption of normally distributed returns. As shown in Fama(1971)or Fama metric stable model are the same as those of the normal model tIldes (-)are used to denote random variables. And the one-period percentage eturn is most often referred to just as the return
RISK, RETURN, AND EQUILIBRIUM o(Rp)=o(Rm)and methods can then be used to show that the weights xim must be chosen in such a way that for any asset i in m E(Ri-E(Rm)=s, o(Rm 0(Rm) where Sm is the rate of change of E(R,) with respect to a change in o(Rp)at the point on the efficient set corresponding to portfolio m.If there are nonnegativity constraints on the weights (that is, if short selling is prohibited), then (1)only holds for assets i such that xim >0 Although equation (1)is just a condition on the weights xim that is re- quired for portfolio efficiency, it can be interpreted as the relationship be tween the risk of asset i in portfolio m and the expected return on the asset. The equation says that the difference between the expected return on the asset and the expected return on the portfolio is proportional to the differ ence between the risk of the asset and the risk of the portfolio. The pro portionality factor is Smt, the slope of the efficient set at the point corres- ponding to the portfolio m. and the risk of the asset is its contribution to total portfolio risk, o(Rm) IL. Testable Implications Suppose now that we posit a market of risk-averse investors who make portfolio decisions period by period according to the two-parameter model. 3 We are concerned with determining what this implies for observable properties of security and portfolio returns. We consider two categories of implications. First, there are conditions on expected returns that are im ied by the fact that two-parameter world investors hold efficient portfolios. Second, there are conditions on the behavior of returns through time that are implied by the assumption of model th the capital market is perfect or frictionless in the sense that there are neither transactions costs nor information costs A. Expected Returns The implications of the two-parameter model for expected returns derive irom the efficiency condition or expected return-risk relationship of equa- tion (1). First, it is convenient to rewrite(1)as 8 A multiperiod version of the two-parameter model is in Fama (1970a)or F and Miller (1972, chap. 8)
JOURNAL OF POLITICAL ECONOMY E(RS= JE(Rm)-Sm o(Rm)]+Sm O(Rm)B (Ra, Rn cov(Ri, Rm)/G(Rm) (Rm) 0(Rn) The parameter B, can be interpreted as the risk of asset i m, measured relative to o(Rm), the total risk of The E(R0)=E(Rm)一Sm0(Rm) is the expected return on a security whose return is uncorrelated with Rmthat is, a zero-P security. Since B=0 implies that a security con- tributes nothing to o(Rm), it is appropriate to say that it is riskless in this portfolio. It is well to note from (3), however, that since xim oi=tim o(Ri) is just one of the N terms in Bi, Bi=0 does not imply that security i has zero variance of return From (4), it follows that R (R so that(2)can be rewritten E(R,)=E(R)+E(Rm)-E(R,)1B. In words, the expected return on security i is E(Ro), the expected return on a security that is riskless in the portfolio m, plus a risk premium that is Bi times the difference between E(rm) and E(ro) Equation (6)has three testable implications: (C1) The relationshi between the expected return on a security and its risk in any efficient port folio m is linear.(C2)B: is a complete of the risk of security i in he efficient portfolio m: no other measure of the risk of i appears in(6) (C3)In a market of risk-averse investors, higher risk should be associated with higher expected return; that is, E(Rm)-E(R)>0 The importance of condition C3 is obvious. The importance of C1 and C2 should become clear as the discussion proceeds. At this point suffice it to say that if Cl and C2 do not hold, market returns do not reflect th attempts of investors to hold efficient portfolios: Some assets are syste- matically underpriced or overpriced relative to what is implied by the expected return-risk or efficiency equation (6) B. Market Equilibrium and the efficiency of the Market Portfoli To test conditions C1-C3 we must identify some efficient portfolio m This in turn requires specification of the characteristic of market equi-
RISK, RETURN, AND EQUILIBRIUM 6 librium when investors make portfolio decisions according to the two- parameter model Assume again that the capital market is perfect. In addition, suppose that from the information available without cost all investors derive the same and correct assessment of the distribution of the future value asset or portfolio- an assumption usually called"homogeneous ex tions. Finally, assume that short selling of all assets is allowed Black (1972)has shown that in a market equilibrium, the so-called market portfolio, defined by the weights total market value of all units of im≡ total market value of all assets is always efficient Since it contains all assets in positive amounts, the market portfolio is a convenient reference point for testing the expected return-risk conditions C1-C3 of the two-parameter model. And the homogeneous-expectation assumption implies a correspondence between ex ante assessments of return distributions and distributions of ex post returns that is also re quired for meaningful tests of these three hypotheses C. A Stochastic Model for returns Equation(6)is in terms of expected returns. But its implications must be ested with data on period-by-period security and portfolio returns. We wish to choose a model of period-by-period returns that allows us to use observed average returns to test the expected-return conditions Cl-C but one that is nevertheless as general as possible We suggest the follor ing stochastic generalization of (6) Ra=Yot+Y1B1+y2B2+73:51+ The subscript t refers to period t, so that Ru is the one-period percent ge return on security i from t-1 to t. Equation(7)allows Yot and y1t to vary stochastically from period to period. The hypothesis of condition C3 is that the expected value of the risk premium Yit, which is the slope IE(Rmt)-E(Rot)] in (6), is positive-that is, E(t)=E(Rmt) E(Rot)> The variable P2 is included in (7)to test linearity. The hypothesis of condition CI is E(Yet)=0, although Yet is also allowed to vary stochast- cally from period to period. Similar statements apply to the term involving of the risk of security i that not deterministically related to Bi. The hypothesis of condition C2 is E(Yar)=0, but Yat can vary stochastically through time The disturbance mit is assumed to have zero mean and to be independent of all other variables in(7). If all portfolio return distributions are to be
bI JOURNAL OF POLITICAL ECONOMY normal (or symmetric stable), then the variables mat, Yot, Y1t, Yet and Yar must have a multivariate normal (or symmetric stable)distribution D. Capital Market Efficiency: The Behavior of Returns through Time C1-C3 are conditions on expected returns and risk that are implied by the two-parameter model. But the model, and especially the underlyin assumption of a perfect market, implies a capital market that is efficient in the sense that prices at every point in time fully refect available informa tion. This use of the word efficient is, of course not to be confused with portfolio efficiency. The terminology, if a bit unfortunate, is at least standard Market efficiency in combination with condition C1 requires that scrutin of the time series of the stochastic nonlinearity coefficient Yet does not lead to nonzero estimates of expected future values of Y2t. Formally, Yet must be a fair game. In practical terms, although nonlinearities are ob- served ex post, because Y2 is a fair game, it is always appropriate for the investor to act ex ante under the presumption that the two-parameter model, as summarized by (6), is valid. That is, in his portfolio decisions he always assumes that there is a linear relationship between the risk of a security and its expected return. Likewise, market efficiency in the two parameter model requires that the non-B risk coefficient 13t and the time series of return disturbances mit are fair games. And the fair-game hypo difference between the risk premium for period t and its expected value o time series o of Yu-[E(Rmt)-E(Rot1,th In the terminology of Fama(1970b), these are "weak-form"proposi tions about capital market efficiency for a market where expected returns are generated by the two-parameter model. The propositions are weak since they are only concerned with whether prices fully reflect any information in the time series of past returns. "Strong-form"tests would be concerned with the speed-of-adjustment of prices to all available information E. Market Equilibrium with Riskless Borrowing and Lending We have as yet presented no hypothesis about for in(7). In the general two-parameter model, given E(Y )=E( r)=E(u)=0, then, from (6), E(Yor)is just E(Ror), the expected return on any zero-B security And market efficiency requires that Yot-E(Ror)be a fair game But if we add to the model as presented thus far the assumption that there is unrestricted riskless borrowing and lending at the known rate R, then one has the market setting of the original two-parameter "capital asset pricing model"of Sharpe(1964)and Lintner(1965 ). In this world, since B,=0, E(You)=Ri. And market efficiency requires that Yot-Rrtbe a fair game
RISK, RETURN, AND EQUILIBRIUM It is well to emphasize that to refute the proposition that E(You)=R, only to refute a specific two-parameter model of market equilibrium ur view is that tests of conditions C1-C3 are more fundamental, We egard CI-C3 as the general expected return implications of the two- parameter model in the sense that they are the implications of the fact that in the two-parameter portfolio model investors hold efficient portfolios, and they are consistent with any two-parameter model of market equi librium in which the market portfolio is efficient F. The Hypotheses To summarize, given the stochastic generalization of(2)and(6)that is provided by (7), the testable implications of the two-parameter model CI (linearity )-E(Yet)=0 C2(no systematic effects of non-B risk)-E(Yar)=0 C3 (positive expected return-risk tradeoff)-E(1) E(Rt) E(Ro)>0 Sharpe-Lintner(S-L) Hypothesis-E(You=Rye Finally, capital market efficiency in a two-parameter world requires ME(market efficiency )-the, stochastic coefficients Yat, at-e ia TE(R mt)-E(Ro)L, Yo-E(Rut), and the disturbances ma are fa I. Previous Wor The earliest tests of the two-parameter model were done by Douglas (1969), whose results seem to refute condition C2. In annual and quarterl return data, there seem to be measures of risk, in addition to b, that con tribute systematically to observed average returns. These results, if vali are inconsistent with the hypothesis that investors attempt to hold efficient portfolios. Assuming that the market portfolio is efficient, premiums are paid for risks that do not contribute to the risk of an efficient portfolio Miller and Scholes (1972)take issue both with Douglas's statistical techniques and with his use of annual and quarterly data. Using different methods and simulations, they show that Douglass negative results could be expected even if condition C2 holds. Condition C2 is tested below with extensive monthly data, and this avoids almost all of the problems dis- mes,then E(Y r)=e(y3)=o. Th the expected return conditions separate, however, be phasizes the economic basis of the various hypotheses. A comprehensive survey of empirical and theoretical work on the two-parameter
6I4 JOURNAL OF POLITICAL ECONOMY Much of the available empirical work on the two-parameter model is oncerned with testing the s-L hypothesis that E(Yor )=R. The tests of Friend and Blume(1970) and those of Black, Jensen, and Scholes(1972) indicate that, at least in the period since 1940, on average Yot is system atically greater than Ry. The results below support this conclusion In the empirical literature to date, the importance of the linearity condi tion CI has been largely overlooked, Assuming that the market portfolio m is efficient, if E(Yet) in (7)is positive, the prices of high-P securities are on average too low--their expected returns are too high-relative to those of low-B securities, while the reverse holds if E(Yer) is negative. In short, if the process of price formation in the capital market reflects the attempts of investors to hold efficient portfolios, then the linear relation ship of (6) between expected return and risk must hold Finally, the previous empirical work on the two-parameter model has not been concerned with tests of market efficiency. ar IV. Methodology The data for this study are monthly percentage returns (including dends and capital gains, with the appropriate adjustments for capital changes such as splits and stock dividends for all common stocks traded on the New York Stock Exchange during the period January 1926 through June 1968. The data are from the Center for Research in Security Prices f the university of Chicago 4. General Approach Testing the two-parameter model immediately presents an unavoidable "errors-in-the-variables''problem: The efficiency condition or expected return-risk equation (6) is in terms of true values of the relative risk measure B, but in empirical tests estimates, Bi, must be used. In this paper cov (ri, R where COV(R, R,) and 6(Rm) are estimates of cov(Ri, Rm) and o"(Rm) obtained from monthly returns, and where the proxy chosen for Rmt is Fisher's arithmetic Index, an equally weighted average of the returns ll stocks listed on the New York Stock Exchange in month t. The properties of this index are analyzed in Fisher (1966) Blume (1970)shows that for any portfolio P, defined by the weights Cov(Rp, Rm) COV(Ri, Rm) 谷2(Rn xp合2(Rm)
RISK, RETURN, AND EQUILIBRIUM 6I If the errors in the Ba are substantially less than perfectly positively cor- lated,the bs of portfolios can be much more precise estima β s than theβ 's for individual securities To reduce the loss of information in the risk-return tests caused by using portfolios rather than individual securities, a wide range of values of portfolio B 's is obtained by forming portfolios on the basis of ranked values of B: for individual securities. But such a procedure, naively exe- cuted could result in a serious regression phenomenon. In a cross section of B high observed B, tend to be above the corresponding true Be and low observed Bi tend to be below the true B Forming portfolios on the bas ranked B, thus causes bunching of positive and negative sampling errors within portfolios. The result is that a large portfolio B, would tend to over state the true Bp, while a low B, would tend to be an underestimate The regression phenomenon can be avoided to a large extent by forming portfolios from ranked B2 computed from data for one time period but then using a subsequent period to obtain the B for these portfolios that are used to test the two-parameter model. With fresh data, within a portfolio errors in the individual securit ty B: are to a large extent random across securities, so that in a portfolio Bp the effects of the regression phenomenon are, it is hoped, minimized B. Details The specifics of the approach are as follows. let n be the total number of securities to be allocated to portfolios and let int(N /20)be the largest steger equal to or less than N/20. Using the first 4 years(1926-29)of monthly return data, 20 portfolios are formed on the basis of ranked B 2 for individual securities. The middle 18 portfolios each has int(N/20) securities. If N is even, the first and last portfolios each has int(N/20)+ 1 IN-20 int(N /20)] securities. The last(highest B)portfolio gets an additional security if N is odd The following 5 years(1930-34)of data are then used to recompute the B, and these are averaged across securities within portfolios to obtain o initial portfolio Bpt for the risk-return tests. The subscript t is added to indicate that each month t of the following four years(1935-38)these are recomputed as simple averages of individual security B, thus ad justing the portfolio month by month to allow for delisting of securi ties. The component B2 for securities are themselves updated yearly--that The errors-in-ti problem and the by Blume (1970).T by Friend and blur and Black, Jensen, ar menon that then by black ind Scholes (1972), who offer a solution to
