Table 5.3. Empirical results far tests of the Skarpe-I.inlner version of the CAIM.

Time

/i -value fa -value /;( -value f -value

Fiveyear subperiods

1/65-12/69 2.038

1/70-12/74 2.136

0, 1/75-12/79 1.914

1/80-12/84 1.224

1/85-12/89 1.732

} 1/90-12/94 1.153

j Overall 77.224

hen-year subperiods 1/65-12/74 2.400 1/75-12/84 2.248 1/85-12/94 1.900 Overall 57.690

7 hirty-year period

1/65-12/94 2.159

0.049 0.039 0.066 0.300 0.100 0.344 0.004

0.013 0.020 0.053 0.001

20.867 0.022 21.712 0.017 19.784 0.031 13.378 0.203 18.164 0.052 12.680 0.242 106.586 **

23.883 0.008 22.503 0.013 19.281 0.037 65.667 **

18.432 19.179 17.476 11.818 16.045 11.200 94.151

0.048 0.038 0.064 0.297 0.098 0.342 0.003

22.490 0.013 21.190 0.020 18.157 0.052 61.837 0.001

22.105 0.015 21.397 0.018 27.922 0.002 13.066 0.220 16.915 0.076 12.379 0.260 113.785 **

24.649 0.006

27.192 0.002

16.373 0.089 68.215 **

0.020 21.612 0.017 21.192 0.020 22.176 0.014

Vl.ess than 0.0005. I

results are for ten value-weighted portfolios (N = 10) with stocks assigned to the portfolios based on market value of equity. The CRSP value-weighted index is used as a measure of the market portfolio and a one-month Treasury bill is used as a measure of the riskfree rate. The tests are based on monthly data from January 1905 to December 1994.

been suggested to consider if inferences are sensitive to the use of a proxy in place of the market portfolio.

One approach is advanced in Stambaugh (1982). He examines the sensitivity of tests to the exclusion of assets by considering a number of broader proxies for the market portfolio. He shows that inferences arc-similar whether one uses a stock-based proxy, a slock- and bond-based proxy, or a stock-, bond-, and rcal-csiaic-bascd proxy. This suggests that inferences arc not sensitive to the error in the proxy when viewed as a measure of the market portfolio and thus Rolls concern is not an empirical problem.

Related work considers the possibility ol accounting for the return on human capital. See Mayers (1972), Campbell (I99(ia), ami agannalhan and Wang (1996).

Л second approach to the problem is presented by Kandel and Stambaugh (1987) and Shanken (1987a). Their papers csliinatc an upper hound on the correlation between the market proxy relurn and the true market relurn necessary lo overturn the rejection of the CAIM. The basic finding is that if the correlation between die proxy and the true market exceeds about 0.70, then the rejection of the CAIM with a market proxy would also imply the rejection of die CAIM with the true market portfolio. Thus, as long as we believe there is a high correlation between the true market return and the proxies used, the rejections remain intact.

5.8 Cross-Sectional Regressions

So far in this chapter wc have focused on the mean-variance efficiency of the market portfolio. Another view of the CAPM is that it implies a linear relation between expected returns and market betas which completely explain the cross section of expected returns. These implications can be tested using a cross-sectional regression methodology.

Fama and Mac Beth (1973) first developed the cross-sectional regression approach. The basic idea is, for each cross section, to project the returns on the betas and then aggregate the estimates in the lime dimension. Assuming that the betas are known, the regression model for the (th cross section of /V assets is

Z, = Kn/i + Yu(im 4-/,. (5.8.1)

where X, is the (Лх 1) vector of excess asset returns for time period /, i is an (Лх 1) vector of ones, and /3, is the (Лх I) vector of CAIM betas.

Implementation of the Fama-MacBcth approach involves two steps. First, given 7 periods of data, (5.8.1) is estimated using OI.S for each /,

t = 1.....7, giving the 7 estimates of уш and y\,. Then in the second

step, the time series of pots and yus arc analyzed. Defining ya = £[yo/] and y\ = F[yn], the implications of the Sharpc-I.inlncr CAIM are y = 0 (zero intercept) and y\ > 0 (positive market risk premium). Because the returns are normally distributed and temporally III), the gammas will also be normally distributed and IID. Hence, given time series of yw and y\u

t = 1.....7 , we can test these implications using the usual (-test. Defining

w(yt) as the (-statistic, we have

where

(P>) =

(5.8.2)

5. The Capitol Asset Pricing Model

The distribution of !<(/,) is Student ( with (7 - 1) degrees of freedom and asymptotically is standard normal, (liven the test statistics, inferences can he made in die usual fashion.

The Fama-MacBeih approach is particularly useful because it can easily he modified to atcoinmodaie additional risk measures beyond the СЛРМ beta. Ну adding additional risk measures, we can examine the hypothesis that beta completely describes the cross-sectional variation in expected returns. For example, we can consider if firm size has explanatory power for the cross-section of expected returns where firm size is defined as the logarithm of the market value of equity. Defining <;, as the (Л/х 1) vector wilh elements corresponding to linn size at the beginning of period /. we can augment (5.8.I) lo investigate if firm size has explanatory power not caplulcd by the market beta:

Л = Y i i + Y\i ft , + Yli c + т/с (.r>.s.r>)

Using the p-j/s from (Г>.Н.Г>), we can lest the hypothesis that size does not have any explanatory power beyond beta, that is, y> = 0, by setting / = 2 in (5.8.2)-(5.8.1).

The Fama-Mai Belli methodology, while useful, does have several problems. First, il cannot be directly applied because the market betas are not known. Thus the regressions are conducted using betas estimated from the data, which introduces an errurs-in-variables complication. The errors-in-variables problem can be addressed in two ways. One approach, adopted by Fama and MarBcth, is to minimize the errors-in-variables problem bv grouping ihe slocks into portfolios and increasing the precision of the beta estimates. A second approach, developed by I.itzenberger and Ra-inaswiiiny (197.)) and refined by Shanken (1992b), is to explicitly adjust the standard errors lo correct for the biases introduced by the errors-in-variables. Shanken suggests multiplying <т2 in (5.8.4) by an adjustment factor (1 + (/i , - у )/о~). While this approach eliminates the errors-in-variables bias in the (-statistic in (5.8.2), it does not eliminate the possibility that other variables might enter spuriously in (5.8.5) as a result of the tin-observability of the true betas.

The nnobservabilily of ihe market portfolio is also a potential problem for the cross-sectional regression approach. Roll and Ross (1994) show that if the line market portfolio is efficient, the cross-sectional relation between expected returns and betas can be very sensitive to even small deviations of the market porllolio proxy from the true market portfolio. Thus evidence of the lack of a relation between expected return and beta could be the

result of the fact that empirical work is forced to work with proxies for the market portfolio. Kandel and Stambaugh (1995) show that this extreme sensitivity can potentially be mitigated by using a generalized-least-squares (Gl .S) estimation approach in place of ordinary least squares. However their result depends on knowing the true covariance matrix of returns. The gains from using GIi> with an estimated covariance matrix are as yet uncertain.

5.9 Conclusion

In this chapter we have concentrated on the classical approach to testing the unconditional CAPM. Other lines of research are also of interest. One important topic is the extension of the framework to test conditional versicjns of the CAPM, in which the model holds conditional on state variables that describe the state of the economy. This is useful because the CAPM can hcjld conditionally, period by period, and yet not hold unconditionally. Chapter 8 discusses the circumstances under which the conditional CAPM might held in a dynamic equilibrium setting, and Chapter 12 discusses econometric methods for testing the conditional CAPM.

Another important subject is Bayesian analysis of mean-variance efficiency and the CAPM. Bayesian analysis allows the introduction of prior information and addresses some of the shortcomings of the classical approach such as the stark dichotomy between acceptance and rejection of the model. Harvey and Zhou (1990), Kandel, McCulloch, and Stambaugh (1995), and Shanken (1987c) are examples of work with this perspective;

We have shown that there is some statistical evidence against the CAPM in the past 30 years of US stock-market data. Despite this evidence, the CAPM remainsa widely used tool in finance. There is controversy about how the evidence against the model should be interpreted. Some authors argue that the CAPM should be replaced by multifactor models with several sources of risk; others argue that the evidence against the CAPM is overstated because of mismeasurement of the market portfolio, improper neglect of conditioning information, data-snooping, or sample-selection bias; and yet others claim that no risk-based model can explain the anomalies of stock-market behavior. In the next chapter we explore multifactor asset pricing models and then return to this debate in Section 6.6.

Problems-Chapter 5

5.1 Result 5 states that for a multiple regression of the return on any asset or portfolio Rn on the return of any minimum-variance portfolio Rp (except for the global minimum-variance portfolio) and the relurn of its associated

aie fa = 0Л/ , /J = 1 - P / and = 0. Show this.

5.2 Show that the intercept of the excess-return market model, a, is /его ifthe market portfolio is the tangency portfolio.

5.J Using monthly returns from the 10-ycar period January 1985 to December 1994 for three individual stocks of your choice, a value-weighted market index, and a Treasury bill with one month to maturity, perform the following tests of the Sharpc-Lintncr Capital Asset Pricing Model.

5.3.1 Using the entire 10-year sample, regress excess returns of each stock on the excess (value-weighted) market return, and perform tests with a size of 5% that the intercept is zero. Report the point estimates, /-statistics, and whether or not you reject the CAPM. Perforin regression diagnostics to check your specification.

3.2 For each stock, perform the same test over each of the two cqui-partitioned subsamples and report the point estimates, /-statistics, and whether or not you reject the CAPM in each subperiod. Also include the same diagnostics as above.

5.3.3 Combine all three stocks into a single cqual-wcighlcd portfolio and re-do the tests for the entire sample and for each of the two subsamples, and report the point estimates, (-statistics, and whether or not you reject the CAPM for the whole sample and in each subsample. Include diagnostics.

5.3.4 Jointly test that the intercepts for all three stocks are zero using the / -test statistic Ji in (5.3.23) for the whole sample and for each subsample

5.4 Derive the Gibbons, Ross, and Slianken result in equation (5.5.3).

Multifactor Pricing Models

AT Tl IK KNI) Ol CllAITKR 5 we summarized empirical evidence indicating that the CAPM beta does not completely explain the cross section of expected asset returns. This evidence suggests that one or more additional factors may be required to characterize the behavior of expected returns and naturally leads to consideration of multifactor pricing models. Theoretical arguments also suggest that more than one factor is required, since only under strong assumptions will the CAlM apply period by period. Two main theoretical approaches exist. The Arbitrage Pricing Theory (API) developed by Ross (1976) is based on arbitrage arguments and the Intertemporal Capital Asset Pricing Model (ICAPM) developed by Melton (1973a) is based on equilibrium arguments. In this chapter we will consider (he econometric analysis of inultilactor models.

The chapter proceeds as follows. Section 6.1 briefly discusses the theoretical background of the multifactor approaches. In Section 6.2 we consider estimation and testing of the models with known factors, while in Section 6.3 we develop estimators for risk prcmia and expected returns. Since the factors arc not always provided by theory, we discuss ways to construct them in Section 6.4. Section 6.5 presents empirical results. Because of the lack of specificity of the models, deviations can always be explained by additional factors. This raises an issue of interpreting model violations which we discuss in Section 6.6.

6.1 Theoretical Background

The Arbitrage Pricing Theory (API) was introduced by Ross (1976) as an alternative to the Capital Asset Pricing Model. The APT can be more genera! than the (АРМ in that it allows for multiple risk factors. Also, unlike the CAPM, the APT docs not require the identification of the market portfolio. However, this generality is not without costs. In its most general form

9

ro-beia portfolio ft, - Pn+Pi Iiy+lhRp+tin the regression coefficients