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