CAPM and Time-Varying Beta: The Cross-Section of Expected Returns

Abstract

The failure of the static-beta CAPM to explain the cross-section of returns on portfolios sorted on firm size, book-to-market ratio, momentum, and even portfolios sorted on past CAPM betas, is well documented. In this paper we show that the model's performance dramatically improves when portfolio betas are allowed to be time-varying functions of (lagged) business cycle variables. We use an approach based on Hansen and Richard (1987) to construct a candidate stochastic discount factor (SDF), using the excess return on the market portfolio as the single factor, scaled by a time-varying coe±cient. The result is a model in which the conditional factor risk premium is a non-linear function of the business cycle variables. We assess the performance of our model by computing the R2 of the cross-sectional regression of realized on model-implied expected returns, as for example in Jagannathan and Wang (1996). While this is not a formal test of the model's ability to price the assets correctly, it does provide an informative summary statistic that allows us to compare the performance of our scaled model with that of the static version, and also to compare our findings to those of other similar studies.

In the post-1980 period, where the static CAPM is known to perform particularly poorly, our scaled model explains around 60% of the cross-sectional variation in returns on beta and book-to-market portfolios, and 87% for momentum portfolios. Moreover, the model captures 70% of the value premium (the return spread between the highest and lowest book-to-market decile portfolios), and 75% of the momentum premium (the spread between the past 'winner' and 'loser' portfolios). Our results thus confirm the crucial importance of time-varying risk premiums in explaining the cross-section of average returns on these sets of portfolios. Moreover, the conditional market risk premium and hence also the betas implied by our model exhibits considerable non-linearity in the business cycle instruments.

Keywords: Capital Asset Pricing Model, Time-Varying Risk Premium

JEL Classification: C12, G11, G12