For over thirty years, the mean-variance (MV) capital asset pricing model (CAPM) has formed one of the central paradigms of financial economics. From a theoretical point of view, the Sharpe (1964) - Lintner (1965) model represents an almost perfect blend of elegance and simplicity. Beta is an intuitively appealing measure of risk whether one argues it is an asset's contribution to total societal risk or the part of risk that cannot be diversified away. From an empirical point of view, the model appears to be readily testable. Betas are easily estimated from standard time series regressions. And, a linear risk-return tradeoff seems to be tailor made for empirical testing. But is it? Here, I review the literature which asserts that tests of the CAPM are ambiguous at best.

How might we test the theory? Each of the following five statements has implications for how we might judge whether the CAPM is true or false. First, the market portfolio is MV efficient. Second, there is at least one positively weighted efficient portfolio. Third, in the riskless asset version of the model, the market portfolio is the tangency portfolio, i.e., the point of tangency between a ray emanating from the riskless interest rate and the minimum-variance frontier of risky assets. Fourth, there is a linear relation between the expected returns and market betas of securities. That is, securities plot on the security market line (SML). Fifth, market betas are the only measures of risk needed to explain the cross-section of expected returns.

In the early 1970s, Black, Jensen, and Scholes (1972), Blume and Friend (1973), and Fama and MacBeth (1973) produce the first extensive tests of the model. They focus on the cross-sectional expected return - beta tradeoff and the special prediction of the Sharpe-Lintner version of the model that the returns on "zero-beta" portfolios have expected returns equal to the riskless rate of interest. Their findings are well known. The average return-beta plot is almost linear, but the estimated slope of the SML is too flat and the intercept is too high. The evidence is interpreted as providing grounds for rejection of the Sharpe-Lintner model and as being consistent with Black's (1972) zero-beta version. Yet, barely twenty years later, Fama and French (1992) find no cross-sectional relationship between average returns and beta. Rather, they report that size and book-to-market equity combine to capture the cross-sectional variation in average stock returns. Suddenly, the CAPM is dead. Or is it?

The most serious doubts regarding the validity of the tests focus on logical rather than statistical considerations. Roll's (1977) critique is the best known. He asserts: "(a) No correct and unambiguous test of the theory has appeared in the literature, and (b) there is practically no possibility that such a test can be accomplished in the future" (p.129). He argues that the theory is equivalent to the assertion that the market portfolio is MV efficient. When portfolios that include only a subset of assets are used as proxies for the true market portfolio, the CAPM is not being tested.

Cheng and Grauer (1980) identify further ambiguities associated with the tests. We note that tests of the CAPM are tests of a joint hypothesis: prices are determined by the CAPM and return distributions and betas are constant over time. Unfortunately, the joint hypothesis leads to a truly remarkable conclusion: relative prices never change. But, by relaxing the assumption of constant betas and focusing on the Invariance Law of Prices, which asserts that there is an exact linear relation between the values of any three assets, we are able to test the CAPM without having to identify the true market portfolio. Perhaps not surprisingly, our alternative test of the Invariance Law provides little support for the CAPM.

Turnbull and Winter (1982) and Sweeney (1982) identify a fundamental inconsistency in the joint hypothesis shared in all previous tests of the CAPM.

Furthermore, they argue that relaxing the stationarity assumption will alleviate the problem. Cheng and Grauer (1982) argue that the problem is even more fundamental than Turnbull and Winter and Sweeney thought. The CAPM is a single-period model. We show that attempting to embed it in a multi-period setting implies that ex post returns are not drawn from the ex ante return distributions envisioned by investors (Rosenberg and Ohlson (1976) express a similar concern). Moreover, we show that this deeper problem cannot be corrected by assuming nonstationary return distributions. While finance may be blessed with the cleanest and most plentiful data in all of economics, it is small comfort to those attempting to test the CAPM if these returns do not reflect investors' ex ante beliefs.

In the intervening years, we have seen major statistical advances. The multivariate tests of Gibbons, Ross, and Shanken (1989) and Jobson and Korkie (1982) represent a statistical tour de force. The test can be interpreted either as a multivariate test of the securities' deviations from the SML or as the difference in the squared Sharpe ratios of the proxy and tangency portfolios. The test, however, requires the use of a proxy for the market portfolio. Moreover, it assumes stationary return distributions. This is particularly troubling when we estimate the tangency portfolio in periods where the riskless interest rate changes as much as it has in the last 30 years (Best and Grauer (1992) document the extreme sensitivity of MV portfolio weights to small perturbations in the means). Tests of conditional forms of the CAPM are tests of joint hypotheses that replace assumptions of return stationarity with specific assumptions about how means, variances, covariances or betas evolve through time. But, for the most part, these tests still require the use of a proxy portfolio and do not address Cheng and Grauer's concern that in a CAPM world realized returns are not drawn from the ex ante distributions envisioned by investors.

Critics of the tests were more or less silent during this period. That is, until Fama and French (1992) report that there is no relationship between average returns and beta. Then, Roll and Ross (1994) and Kandel and Stambaugh (1995) highlight the danger of focusing exclusively on mean-beta space. Roll and Ross demonstrate that a market proxy can be almost MV efficient even though the slope from an ordinary least squares (OLS) regression of population expected returns on population betas is zero. Conversely, Kandel and Stambaugh show that there can be a near perfect OLS fit between means and betas calculated relative to a proxy that is grossly inefficient. More importantly, they show that, in a generalized least squares (GLS) regression of mean returns on betas, the slope and R-square are determined uniquely by the mean-variance location of the market index relative to the minimum-variance boundary. However, neither Roll and Ross nor Kandel and Stambaugh verify whether the minimum-variance frontier contains a positively weighted portfolio. Thus, we cannot be sure whether their results hold if the CAPM is true, i.e., when the positively weighted market portfolio is MV efficient. Furthermore, any reasonable proxy portfolio should contain positive weights. Roll and Ross are able to construct one example where a proxy portfolio contains positive weights, but the proxies in Kandel and Stambaugh's paper do not contain positive weights.

In Grauer (1999), I examine scenarios where the MV CAPM is true and where it is false. All the implications of the model mentioned in the second paragraph of this note hold exactly when the CAPM is true. However, in some cases, the positions of the market and tangency portfolios differ dramatically when the CAPM is false. I do not employ a proxy for the market in most cases, but when I do, with one exception, I employ a positively weighted proxy. I then investigate whether the coefficients from OLS and GLS regressions of population expected excess returns on population betas, and expected excess returns on betas and size, allow us to distinguish between the true and false scenarios. I show two main results. First, when the CAPM is true, coefficients of OLS and GLS regressions employing proxy portfolios that are almost efficient can incorrectly indicate that the model is false. Second, and perhaps more importantly, when the CAPM is false, coefficients of OLS, GLS, or both OLS and GLS regressions employing market portfolio betas can incorrectly indicate that the model is true, even when the market is grossly inefficient.

The lack of any clear-cut agreement among the different implications of the CAPM when it is false is particularly disturbing. It does not bode well for those seeking to design an unambiguous test of the model.

Finally, Grauer and Janmaat (1999) examine the effects of grouping. We argue that the econometric benefits of grouping -- reducing measurement error in cross-sectional tests and reducing the dimension of the covariance matrix in multivariate tests -- must be balanced against the costs. We consider two scenarios in a world where there is no measurement error. In the first scenario, the CAPM holds exactly. In the second, it is false. Then, we identify four costs or unintended consequences of grouping.

First, we show that the most basic CAPM relationships may not hold with grouped data. Second, we demonstrate that grouping can cause fundamental problems with the cross-sectional regression methodology. Third, we present examples where, with ungrouped data, the CAPM is dead wrong. The tangency portfolio is driven to near minus infinity in expected return-standard deviation space in one case and there is no relation between expected returns and betas in another. Yet, with grouped data, the model is absolutely correct. Fourth, we show that grouping can cause the slope of the cross-sectional regression of expected returns on betas to be flatter than it really is, exacerbating the very problem it was meant to alleviate.

So where does this leave us? Clearly, some believe early empirical evidence is consistent with the CAPM. And, some believe recent evidence leads to the conclusion that the CAPM is dead. But others await a meaningful test.