Risk, Return, and Equilibrium: Empirical Tests
Summary (5 min read)
11. Testable Implications
- Suppose now that the authors posit a market of risk-averse investors who make portfolio decisions period by period according to the two-parameter model.".
- IVe are concerned with determining what this implies for observable properties of security and portfolio returns.
- First, there are conditions on expected returns that are implied by the fact that in a 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 the two-parameter model that 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 from the efficiency condition or expected return-risk relationship of equation ( l ) .
- The relationship between the expected return on a security and its risk in any efficient portfolio nz is linear.
- The importance of C1 and C2 should become clear as the discussion proceeds.
- At this point suffice it to say that if C1 and C2 do not hold, market returns do not reflect the attempts of investors to hold efficient portfolios:.
- Some assets are systematically underpriced or overpriced relative to what is implied by the expected return-risk or efficiency equation ( 6 ) .
B . &favket Equilibriu~~z and the E!ficie~cy of the Market Portfolio
- T o test conditions CI-C3 the authors must identify some efficient portfolio In.
- This in turn requires specification of the characteristic of market equi-librium when investors make portfolio decisions according to the twoparameter model.
- Assume again that the capital market is perfect.
C. A Stochastic Model for Returns
- But its implications must be tested with data on period-by-period security and portfolio returns.
- The authors wish to choose a model of period-by-period returns that allows us to use observed average returns to test the expected-return conditions C1-C3, but one that is nevertheless as general as possible.
- The hypothesis of condition C1 is E(y,,) = 0,although 72t is also allowed to vary stochastically from period to period.
- The hypothesis of condition C2 is E(C;+t) = 0, but can vary stochastically through time.
- If all portfolio return distributions are to be normal (or symmetric stable), then the variables Tit, Gt,vlt,S;2t and 73t must have a multivariate normal (or symmetric stable) distribution.
D. Capital Market Eficiency:
- The Behavior of Returns through Timc C1-C3 are conditions on expected returns and risk that are implied by the two-parameter model.
- But the model, and especially the underlying assumption of a perfect market, implies a capital market that is efficient in the sense that prices at every point in time fully reflect available information.
- This use of the word efficient is, of course, not to be confused with portfolio efficiency.
- Likewise, market efficiency in the twoparameter model requires that the non-/3 risk coefficient yit and the time series of return disturbances q,, are fair games.
- The propositions are weak since they are only concerned with whether prices fully reflect any information in the time series of past returns.
E . Market Equilibrium with Riskless Bon.ozr:ing and Lending
- And market efficiency requires that -E(R,,t) be a fair game.
- But if the authors add to the model as presented thus far the assumption that there is unrestricted riskless borrowing and lending at the known rate Rft, then one has the market setting of the original two-parameter "capital asset pricing model" of Sharpe (1963) and Lintner (1965) .
- And market efficiency requires that Tot-Rjt be a fair game.
- I t is well to emphasize that to refute the proposition that E(it;,t) = Rrt is only to refute a specific two-parameter model of market equilibrium.
- The authors view is that tests of conditions C1-C3 are more fundamental.
IV. Methoclology
- The data for this study are monthly percentage returns (including dividends and capital gains, with the appropriate adjustments for capital changes such as splits and stock dividends) for all common stocks traded on the S e w York Stock Exchange during the period January 1926 through June 1968.
- The data are from the Center for Research in Security Prices of the Cniversity of Chicago.
'4. Genesal 11ppsoach
- Testing the two-parameter model immediately presents an unavoidable "errors-in-the-variables" problem:.
- But such a procedure, nai'vely executed could result i t a serious regression phenomenon.
- Let S be the total number of securities to be allocated to portfolios and let int(hT/20) be the largest integer equal to or less than X 20 Using the first 4 years of monthly return data, 20 portfolios are formed on the basis of ranked p, for individual securities.
- Given a market dominated by risk averters, this model would predict that a security's expected return is related to its total return dispersion rather than just to the contribution of the security to the dispersion in the return on an efficient portfolio.'.
- The nine different portfolio formation periods (all except the first 7 years in length), initial 5-year estimation periods, and testing periods (all but the last 4 years in length) are shown in table 1 .
C. Some Observations on the Approach
- Table 2 shows the values of the 20 portfolios 6,t-l and their standard errors s(P, t-l) for four of the nine 5-year estimation periods.
- Also shown are: r(R,, R,,,)', the coefficient of determination between Rilt and R,,,t; s ( R , ) , the sample standard deviation of R,: and s(2,), the standard deviation of the portfolio residuals from the market model of ( 8) , not to be confused nith F,tpl(2,), the average for individual securities, nhich is also shown.
- But it is important to emphasize that since these ratios are generally less than 3 3 , interdependence among the Zt of different securities does not destroy the value of using portfolios to reduce the dispersion of the errors in estimated p's.
- Finally, all the tests of the two-parameter model are predictive in the A sense that the explanatory variables P, t-l and S, t-l(Q,) in ( 10) are computed from data for a period prior to the month of the returns, the R,(, on which the regression is run.
- Examining the extent to which it is helpful in describing actual return data-the model was initially developed by Jfarkowitz ( 1959) as a normative theory-that is, as a model to help people make better decisions.the authors.
V. Results
- The major tests of the implications of the two-parameter model are in table 3 .
- Finally, t-statistics for testing the hypothesis that 9, = 0 are presented.
- In interpreting these t-statistics one should keep in mind the evidence of Fama ( 1 9 6 5 ~) and Blume (1970) which suggests that distributions of common stock returns are "thick-tailed" relative to the normal distribution and probably conform better to nonnormal symmetric stable distributions than to the normal.
- But it is important to note that, with the exception of condition C3 (positive expected return-risk tradeoff), upward-biased probability levels lead to biases toward rejkction of the hypotheses of the two-parameter model.
A. Tests of the Major Hypotheses of the Two-Parameter Model
- Consider first condition C2 of the two-parameter model, which says that no measure of risk, in addition to (3, systematically affects expected returns.
- That is, the authors are not happy with the model unless there is on average a positive tradeoff between risk and return.
- Except for the period 1956-60, the values of t ( 9 , ) are also systematically positive in the subperiods, but not so systematically large.
- But a t least with the sample of the overall period t ( q l ) achieves values supportive of the conclusion that on average there is a statistically observable positive relationship between return and risk.
B . Thr, Bchavior oj thr Markct
- Some perspective on the behavior of the market during different periods and on the interpretation of the coefficients and in the risk-return regressions can be obtained from table 1.
- For the various periods of table 3, table 3 shows the sample means (and with some exceptions), the standard " T h e sizri;il correlations o i 9, and Q:l about means that are assumed to be zc,ro provide a test of the fair game property of an efficient market.
- Likewise, p,,(Q,,, -R,, ) provides a test of market efficiency with respect to the beha\-ior o i Q,,, thr0ut.h time, given the validity of thc Sharpc-Lintner hypothesis iabout which the authors have as yet said nothing).
- But, a t least for and q:,,,computinr the serial correlations about sample means produces essentially the _same results.
- Finally, it is well to note that in terms of the implications of the serial correl:~tions for making good portfolio decisions-and thus ior judging whether market efficiency is a workal~le representation of reality-the iact that the 'serial corl-elations are low in terms of explanatory power is more important than whether or not they are lo\v in terms of statistical significance.
TABLE 4
- The t-statistics on sample means are computed in the same way as those in table 3 .
- I n fact, for the one-variable regressions of panel A, hloreover, the least-squares estimate can always be interpreted as the return for month t on a zero$ portfolio, where the weights given to each !'There is some degree of approximation in ( 1 2 ) .
- The averages over P of R,, and fir, arc R,,,, and 1.0, respectivcly, only if every security in the market is in some portfolio.
- CVith their methodology (see table 1 ) this is never true.
- But the degree of approximation turns out to he small:.
b27 TABLE 4 (Continued)
- STATISTIC* of the 20 portfolios to form this zero$ portfolio are the least-squares weights that are applied to the RPtin computing (Iot.lo.
- In addition to providing important information about the precision of the coefficient estimates used to test the two-parameter model, the answer to this question can be used to test hypotheses about the stochastic process generating returns.
D. Tests of the S-L Hypothesis
- In the Sharpe-Lintner two-parameter model of market equilibrium one has.
- Except for the most recent period 1961-6/68, the values of -RI in panel A are all positive and generally greater than 0.4 percent per month.
- The tests of the S-L hlpothesis in panel of table 3 are conceptually the same as those of Black, Jensen, and Scholes If one makes the assumptions of the Gauss-Markok Theorem on the underlling disturbances of the models of panels B-D of table 3, the regression intercepts for these models can likewise he interpreted as returns on minimum-variance zero-@ portfolios.
- As indicated above, however, at the moment the most efficient tests of the S-L hypothesis are provided by the one-variable model of panel A, table 3, and the results for that model support the neqative conclusions of others.
- And using the mean of the Qot as an estimate of ~( 2 , ~~) does not work as well in this case as it does for the market efficiency tests on ylt.
VI. Conclusions
- I n sum their results support the important testable implications of the twoparameter model.
- Given that the market portfolio is efficient-or, more specifically, given that their proxy for the market portfolio is at least approximately efficient-the authors cannot reject the hypothesis that average returns on hTew York Stock Exchange common stocks reflect the attempts of riskaverse investors to hold efficient portfolios.
- Specifically, on average there seems to be a positive tradeoff between return and risk, with risk measured from the portfolio viewpoint.
- I n addition, although there are "stochastic nonlinearities" from period to period, the authors cannot reject the hypothesis that on average their effects are zero and unpredictably different from zero from one period to the next.
- The authors also cannot reject the hypothesis of the two-parameter model that no measure of risk, in addition to portfolio risk, systematically affects average returns.
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