The Expected Return–Beta Relationship
Recall that if the expected return–beta relationship holds with respect to an observable ex ante efficient index, M, the expected rate of return on any security i is
E(ri)5rf1bi3E(rM)2rf4 (13.1) where b i is defined as Cov(ri, rM)/sM2 .
This is the most commonly tested implication of the CAPM. Early simple tests fol- lowed three basic steps: establishing sample data, estimating the SCL (security character- istic line), and estimating the SML (security market line).
Setting Up the Sample Data Determine a sample period of, for example, 60 monthly holding periods (5 years). For each of the 60 holding periods, collect the rates of return on 100 stocks, a market portfolio proxy (e.g., the S&P 500), and 1-month (risk-free) T-bills.
Your data thus consist of
r it 5 Returns on the 100 stocks over the 60-month sample period; i 5 1, . . . , 100, and t 5 1, . . . , 60.
r Mt 5 Returns on the S&P 500 index over the sample period.
r ft 5 Risk-free rate each month.
This constitutes a table of 102 3 60 5 6,120 rates of return.
Estimating the SCL View Equation 13.1 as a security characteristic line (SCL), as in Chapter 8. For each stock, i, you estimate the beta coefficient as the slope of a first-pass regression equation. (The terminology first-pass regression is due to the fact that the esti- mated coefficients will be used as input into a second-pass regression. )
rit2rft5ai1bi(rMt2rft)1eit You will use the following statistics in later analysis:
ri2rf5 Sample averages (over the 60 observations) of the excess return on each of the 100 stocks.
b i 5 Sample estimates of the beta coefficients of each of the 100 stocks.
rM2rf5Sample average of the excess return of the market index.
s 2 ( e i ) 5 Estimates of the variance of the residuals for each of the 100 stocks.
The sample average excess returns on each stock and the market portfolio are taken as estimates of expected excess returns, and the values of b i are estimates of the true beta coefficients for the 100 stocks during the sample period. s 2 ( e i ) estimates the nonsystematic risk of each of the 100 stocks.
CONCEPT CHECK
1
a. How many regression estimates of the SCL do we have from the sample?
b. How many observations are there in each of the regressions?
c. According to the CAPM, what should be the intercept in each of these regressions?
Estimating the SML Now view Equation 13.1 as a security market line (SML) with 100 observations for the stocks in your sample. You can estimate g 0 and g 1 in the following second-pass regression equation with the estimates b i from the first pass as the independent variable:
ri2rf5g01g1bi i51,c, 100 (13.2) Compare Equations 13.1 and 13.2; you should conclude that if the CAPM is valid, then g 0 and g 1 should satisfy
g050 and g15rM2rf
In fact, however, you can go a step further and argue that the key property of the expected return–beta relationship described by the SML is that the expected excess return on securi- ties is determined only by the systematic risk (as measured by beta) and should be inde- pendent of the nonsystematic risk, as measured by the variance of the residuals, s 2 ( e i ), which also were estimated from the first-pass regression. These estimates can be added as a variable in Equation 13.2 of an expanded SML that now looks like this:
ri2rf5g01g1bi1g2s2(ei) (13.3) This second-pass regression equation is estimated with the hypotheses
g050; g15rM2rf; g250
The hypothesis that g 2 5 0 is consistent with the notion that nonsystematic risk should not be “priced,” that is, that there is no risk premium earned for bearing nonsystem- atic risk. More generally, according to the CAPM, the risk premium depends only on beta. Therefore, any additional right-hand-side variable in Equation 13.3 beyond beta should have a coefficient that is insignificantly different from zero in the second-pass regression.
Tests of the CAPM
Early tests of the CAPM performed by John Lintner, 1 and later replicated by Merton Miller and Myron Scholes, 2 used annual data on 631 NYSE stocks for 10 years, 1954 to 1963,
1 John Lintner, “Security Prices, Risk and Maximal Gains from Diversification,” Journal of Finance 20 (December 1965).
2 Merton H. Miller and Myron Scholes, “Rate of Return in Relation to Risk: A Reexamination of Some Recent Findings,” in Studies in the Theory of Capital Markets, ed. Michael C. Jensen (New York: Praeger, 1972).
and produced the following estimates (with returns expressed as decimals rather than percentages):
Coefficient: g 0 5 .127 g 1 5 .042 g 2 5 .310 Standard error: .006 .006 .026 Sample average: rM2rf5.165
These results are inconsistent with the CAPM. First, the estimated SML is “too flat”;
that is, the g 1 coefficient is too small. The slope should equal rM2rf5.165 (16.5% per year), but it is estimated at only .042. The difference, .122, is about 20 times the standard error of the estimate, .006, which means that the measured slope of the SML is less than it should be by a statistically significant margin. At the same time, the intercept of the esti- mated SML, g 0 , which is hypothesized to be zero, in fact equals .127, which is more than 20 times its standard error of.006.
The two-stage procedure employed by these researchers (i.e., first estimate security betas using a time-series regression and then use those betas to test the SML relationship between risk and average return) seems straightforward, and the rejection of the CAPM using this approach is disappointing. However, it turns out that there are several difficulties with this approach. First and foremost, stock returns are extremely volatile, which lessens the precision of any tests of average return. For example, the average standard deviation of annual returns of the stocks in the S&P 500 is about 40%; the average standard deviation of annual returns of the stocks included in these tests is probably even higher.
In addition, there are fundamental concerns about the validity of the tests. First, the market index used in the tests is surely not the “market portfolio” of the CAPM. Second, in light of asset volatility, the security betas from the first-stage regressions are necessarily estimated with substantial sampling error and therefore cannot readily be used as inputs to the second-stage regression. Finally, investors cannot borrow at the risk-free rate, as assumed by the simple version of the CAPM. Let us investigate the implications of these problems in turn.
The Market Index
In what has become known as Roll’s critique, Richard Roll 3 pointed out that:
1. There is a single testable hypothesis associated with the CAPM: The market portfolio is mean-variance efficient.
2. All the other implications of the model, the best-known being the linear relation between expected return and beta, follow from the market portfolio’s efficiency and therefore are not independently testable. There is an “if and only if” relation between the expected return–beta relationship and the efficiency of the market portfolio.
3 Richard Roll, “A Critique of the Asset Pricing Theory’s Tests: Part I: On Past and Potential Testability of the Theory,” Journal of Financial Economics 4 (1977).
CONCEPT CHECK
2
a. What is the implication of the empirical SML being “too flat”?
b. Do high- or low-beta stocks tend to outperform the predictions of the CAPM?
c. What is the implication of the estimate of g 2 ?
3. In any sample of observations of individual returns there will be an infinite number of ex post (i.e., after the fact) mean-variance efficient portfolios using the sample- period returns and covariances (as opposed to the ex ante expected returns and covariances). Sample betas calculated between each such portfolio and individual assets will be exactly linearly related to sample average returns. In other words, if betas are calculated against such portfolios, they will satisfy the SML relation exactly whether or not the true market portfolio is mean-variance efficient in an ex ante sense.
4. The CAPM is not testable unless we know the exact composition of the true market portfolio and use it in the tests. This implies that the theory is not testable unless all individual assets are included in the sample.
5. Using a proxy such as the S&P 500 for the market portfolio is subject to two difficul- ties. First, the proxy itself might be mean-variance efficient even when the true mar- ket portfolio is not. Conversely, the proxy may turn out to be inefficient, but obviously this alone implies nothing about the true market portfolio’s efficiency. Furthermore, most reasonable market proxies will be very highly correlated with each other and with the true market portfolio whether or not they are mean-variance efficient. Such a high degree of correlation will make it seem that the exact composition of the market portfolio is unimportant, whereas the use of different proxies can lead to quite differ- ent conclusions. This problem is referred to as benchmark error, because it refers to the use of an incorrect benchmark (market proxy) portfolio in the tests of the theory.
Roll and Ross 4 and Kandel and Stambaugh 5 expanded Roll’s critique. Essentially, they argued that tests that reject a positive relationship between average return and beta point to inefficiency of the market proxy used in those tests, rather than refuting the theoretical expected return–beta relationship. They demonstrate that even if the CAPM is true, highly diversified portfolios, such as the value- or equally weighted portfolios of all stocks in the sample, may fail to produce a significant average return–beta relationship.
Kandel and Stambaugh considered the properties of the usual two-pass test of the CAPM in an environment in which borrowing is restricted but the zero-beta version of the CAPM holds. In this case, you will recall that the expected return–beta relationship describes the expected returns on a stock, a portfolio E on the efficient frontier, and that portfolio’s zero-beta companion, Z (see Equation 9.12):
E(ri)2E(rZ)5bi3E(rE)2E(rZ)4 (13.4) where b i denotes the beta of security i on efficient portfolio E.
We cannot construct or observe the efficient portfolio E (because we do not know expected returns and covariances of all assets), and so we cannot estimate Equation 13.4 directly. Kandel and Stambaugh asked what would happen if we followed the common procedure of using a market proxy portfolio M in place of E, and used as well the more efficient generalized least squares regression procedure in estimating the second-pass regression for the zero-beta version of the CAPM, that is,
ri2rZ5g01g13(Estimated bi)
4 Richard Roll and Stephen A. Ross, “On the Cross-Sectional Relation between Expected Return and Betas,”
Journal of Finance 50 (1995), pp. 185–224.
5 Schmuel Kandel and Robert F. Stambaugh, “Portfolio Inefficiency and the Cross-Section of Expected Returns,”
Journal of Finance 50 (1995), pp. 185–224; “A Mean-Variance Framework for Tests of Asset Pricing Models,”
Review of Financial Studies 2 (1989), pp. 125–56; “On Correlations and Inferences about Mean-Variance Efficiency,” Journal of Financial Economics 18 (1987), pp. 61–90.
They showed that the estimated values of g 0 and g 1 will be biased by a term proportional to the relative efficiency of the market proxy. If the market index used in the regression is fully efficient, the test will be well specified. But the second-pass regression will provide a poor test of the CAPM if the proxy for the market portfolio is not efficient. Thus, we still cannot test the model in a meaningful way without a reasonably efficient market proxy.
Unfortunately, determining how efficient our market index is relative to the theoretical true market portfolio is difficult, so we cannot tell how good our tests are.
Measurement Error in Beta
Roll’s critique tells us that CAPM tests are handicapped from the outset. But suppose that we could get past Roll’s problem by obtaining data on the returns of the true market portfo- lio. We still would have to deal with the statistical problems caused by measurement error in the estimates of beta from the first-stage regressions.
It is well known in statistics that if the right-hand-side variable of a regression equation is measured with error (in our case, beta is measured with error and is the right-hand-side variable in the second-pass regression), then the slope coefficient of the regression equa- tion will be biased downward and the intercept biased upward. This is consistent with the findings cited above, which found that the estimate of g 0 was higher than predicted by the CAPM and that the estimate of g 1 was lower than predicted.
Indeed, a well-controlled simulation test by Miller and Scholes 6 confirms these argu- ments. In this test a random-number generator simulated rates of return with covariances similar to observed ones. The average returns were made to agree exactly with the CAPM expected return–beta relationship. Miller and Scholes then used these randomly generated rates of return in the tests we have described as if they were observed from a sample of stock returns. The results of this “simulated” test were virtually identical to those reached using real data, despite the fact that the simulated returns were constructed to obey the SML, that is, the true g coefficients were g050, g15rM2rf, and g 2 5 0.
This postmortem of the early test gets us back to square one. We can explain away the disappointing test results, but we have no positive results to support the CAPM-APT implications.
The next wave of tests was designed to overcome the measurement error problem that led to biased estimates of the SML. The innovation in these tests, pioneered by Black, Jensen, and Scholes, 7 was to use portfolios rather than individual securities. Combining securities into portfolios diversifies away most of the firm-specific part of returns, thereby enhancing the precision of the estimates of beta and the expected rate of return of the portfolio of securities. This mitigates the statistical problems that arise from measurement error in the beta estimates.
Obviously, however, combining stocks into portfolios reduces the number of observa- tions left for the second-pass regression. For example, suppose that we group our sample of 100 stocks into five portfolios of 20 stocks each. If the assumption of a single-factor market is reasonably accurate, then the residuals of the 20 stocks in each portfolio will be practically uncorrelated and, hence, the variance of the portfolio residual will be about one-twentieth the residual variance of the average stock. Thus the portfolio beta in the first-pass regression will be estimated with far better accuracy. However, now consider the second-pass regression. With individual securities, we had 100 observations to estimate
6 Miller and Scholes, “Rate of Return in Relation to Risk.”
7 Fischer Black, Michael C. Jensen, and Myron Scholes, “The Capital Asset Pricing Model: Some Empirical Tests,” in Studies in the Theory of Capital Markets, ed. Michael C. Jensen (New York: Praeger, 1972).
the second-pass coefficients. With portfolios of 20 stocks each, we are left with only five observations for the second-pass regression.
To get the best of this trade-off, we need to construct portfolios with the largest possible dispersion of beta coefficients. Other things being equal, a sample yields more accurate regression estimates the more widely spaced are the observations of the independent vari- ables. Consider the first-pass regressions where we estimate the SCL, that is, the relation- ship between the excess return on each stock and the market’s excess return. If we have a sample with a great dispersion of market returns, we have a greater chance of accu- rately estimating the effect of a change in the market return on the return of the stock. In our case, however, we have no control over the range of the market returns. But we can control the range of the independent variable of the second-pass regression, the portfolio betas. Rather than allocate 20 stocks to each portfolio randomly, we can rank portfolios by betas. Portfolio 1 will include the 20 highest-beta stocks and portfolio 5 the 20 lowest- beta stocks. In that case a set of portfolios with small nonsystematic components, e P , and widely spaced betas will yield reasonably powerful tests of the SML.
Fama and MacBeth 8 used this methodology to verify that the observed relationship between average excess returns and beta is indeed linear and that nonsystematic risk does not explain average excess returns. Using 20 portfolios constructed according to the Black, Jensen, and Scholes methodology, Fama and MacBeth expanded the estimation of the SML equation to include the square of the beta coefficient (to test for linearity of the rela- tionship between returns and betas) and the estimated standard deviation of the residual (to test for the explanatory power of nonsystematic risk). For a sequence of many subperiods, they estimated for each subperiod the equation
ri5g01g1bi1g2bi21g3s(ei) (13.5) The term g 2 measures potential nonlinearity of return, and g 3 measures the explanatory power of nonsystematic risk, s ( e i ). According to the CAPM, both g 2 and g 3 should have coefficients of zero in the second-pass regression.
Fama and MacBeth estimated Equation 13.5 for every month of the period January 1935 through June 1968. The results are summarized in Table 13.1 , which shows average coefficients and t -statistics for the overall period as well as for three subperiods. Fama
8 Eugene Fama and James MacBeth, “Risk, Return, and Equilibrium: Empirical Tests,” Journal of Political Economy 81 (March 1973).
Period 1935/6–1968 1935–1945 1946–1955 1956/6–1968
Av. r f 13 2 9 26
Av. g 0 2 r f 8 10 8 5
Av. t ( g 0 2 r f ) 0.20 0.11 0.20 0.10 Av. r M 2 r f 130 195 103 95
Av. g 1 114 118 209 34
Av. t ( g 1 ) 1.85 0.94 2.39 0.34
Av. g 2 226 29 276 0
Av. t ( g 2 ) 20.86 20.14 22.16 0
Av. g 3 516 817 2378 960
Av. t ( g 3 ) 1.11 0.94 20.67 1.11
Av. R -SQR 0.31 0.31 0.32 0.29
Table 13.1
Summary of Fama and MacBeth (1973) study (all rates in basis points per month)
and MacBeth observed that the coefficients on residual standard deviation (nonsystematic risk), denoted by g 3 , fluctuate greatly from month to month and were insignificant, consis- tent with the hypothesis that nonsystematic risk is not rewarded by higher average returns.
Likewise, the coefficients on the square of beta, denoted by g 2 , were insignificant, consis- tent with the hypothesis that the expected return–beta relationship is linear.
With respect to the expected return–beta relationship, however, the picture is mixed.
The estimated SML is too flat, consistent with previous studies, as can be seen from the fact that g 0 2 r f is positive, and that g 1 is, on average, less than r M 2 r f . On the positive side, the difference does not appear to be significant, so that the CAPM is not clearly rejected.
In conclusion, these tests of the CAPM provide mixed evidence on the validity of the theory. We can summarize the results as follows:
1. The results that support the single-factor CAPM and APT are as follows:
a. Expected rates of return are linear and increase with beta, the measure of systematic risk.
b. Expected rates of return are not affected by nonsystematic risk.
2. However, the single-variable expected return–beta relationship predicted by either the risk-free rate or the zero-beta version of the CAPM is not fully consistent with empirical observation.
Thus, although the CAPM seems qualitatively correct in that b mat- ters and s ( e i ) does not, empirical tests do not validate its quantitative predictions.
The EMH and the CAPM
Roll’s critique also provides a positive avenue to view the empirical content of the CAPM and APT. Recall, as Roll pointed out, that the CAPM and the expected return–beta rela- tionship follow directly from the efficiency of the market portfolio. This means that if we can establish that the market portfolio is efficient, we would have no need to further test the expected return–beta relationship.
As demonstrated in Chapter 11 on the efficient market hypothesis, proxies for the mar- ket portfolio such as the S&P 500 and the NYSE index have proven hard to beat by pro- fessional investors. This is perhaps the strongest evidence for the empirical content of the CAPM and APT.
Accounting for Human Capital and Cyclical Variations in Asset Betas
We are reminded of two important deficiencies of the tests of the single-index models:
1. Only a fraction of the value of assets in the United States is traded in capital markets; perhaps the most important nontraded asset is human capital.
2. There is ample evidence that asset betas are cyclical and that accounting for this cyclicality may improve the predictive power of the CAPM.
CONCEPT CHECK
3
According to the CAPM and the data in Table 13.1 , what are the predicted values of g 0 , g 1 , g 2 , and g 3 in the Fama-MacBeth regressions for the period 1946–1955?
CONCEPT CHECK
4
What would you conclude if you performed the Fama and MacBeth tests and found that the coefficients on b 2 and s ( e ) were positive?