Can stochastic discount factor models explain the cross-section of equity returns?

We propose a multivariate test of the capital asset pricing model (C-CAPM) of the cross-sectional variation in equity returns in which we compare cross-sectional variation in equity returns to the cross-sectional variation in their conditional covariance with stochastic discount factors. We use a multivariate generalized heteroskedasticity in mean model to estimate 25 portfolios that are formed on size and the book-to-market ratio. Each portfolio is allowed to have its own no-arbitrage condition. We find that although the conditional covariances of returns with consumption exhibit negative variation across size, they do not vary across the book-to-market ratio. Thus, C-CAPM can capture the size effect, but not the value effect. The fit is, however, improved by allowing the coefficients on the consumption covariances to be different. The value effect appears to be associated with the book-to-market ratio as well as size. On its own the book-to-market ratio does not generate additional information about average returns to C-CAPM. A possible explanation for these findings is that both small and low book-to-market ratio firms are expected to have higher rates of growth.