The asset-pricing model of Sharpe (1964), Lintner (1965), and Black (1972) has long shaped the way academics and practitioners think about average returns and risk. The central prediction of the model is that the market portfolio of invested wealth is mean-variance efficient in the sense of Markowitz (1959). The efficiency of the market portfolio implies that (a) expected returns on securities are a positive linear function of their market *β*s (the slope in the regression of a security's return on the market's return), and (b) market *β*s suffice to describe the cross-section of expected returns.

There are several empirical contradictions of the Sharpe-Lintner-Black (SLB) model. The most prominent is the size effect of Banz (1981). He finds that market equity, ME (a stock's price times shares outstanding), adds to the explanation of the cross-section of average returns provided by market *β*s. Average returns on small (low ME) stocks are too high given their *β* estimates, and average returns on large stocks are too low.

Another contradiction of the SLB model is the positive relation between leverage and average return documented by Bhandari (1988). It is plausible that leverage is associated with risk and expected return, but in the SLB model, leverage risk should be captured by market *β*. Bhandari finds, however, that leverage helps explain the cross-section of average stock returns in tests that include size (ME) as well as *β*.

Stattman (1980) and Rosenberg, Reid, and Lanstein (1985) find that average returns on U.S. stocks are positively related to the ratio of a firm's book value of common equity, BE, to its market value, ME. Chan, Hamao, and Lakonishok (1991) find that book-to-market equity,

Finally, Basu (1983) shows that earnings-price ratios *β*. Ball (1978) argues that

Ball's proxy argument for *β*, size,

Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) find that, as predicted by the SLB model, there is a positive simple relation between average stock returns and *β* during the pre-1969 period. Like Reinganum (1981) and Lakonishok and Shapiro (1986), we find that the relation between *β* and average return disappears during the more recent 1963–1990 period, even when *β* is used alone to explain average returns. The appendix shows that the simple relation between *β* and average return is also weak in the 50-year 1941–1990 period. In short, our tests do not support the most basic prediction of the SLB model, that average stock returns are positively related to market *β*s.

Unlike the simple relation between *β* and average return, the univariate relations between average return and size, leverage, *β* does not seem to help explain the cross-section of average stock returns, and (b) the combination of size and book-to-market equity seems to absorb the roles of leverage and

If assets are priced rationally, our results suggest that stock risks are multidimensional. One dimension of risk is proxied by size, ME. Another dimension of risk is proxied by

It is possible that the risk captured by

Whatever the underlying economic causes, our main result is straightforward. Two easily measured variables, size (ME) and book-to-market equity

In the next section we discuss the data and our approach to estimating *β*. Section II examines the relations between average return and *β* and between average return and size. Section III examines the roles of