Granular Betas and Risk Premium Functions

We propose new refined measures of the local covariation between the return on an asset and a risk factor. Our proposed “granular betas” generalize the notion of up- and down-side betas to multi-factor functional measures of covariation. We show how the resulting granular beta functions may be used in the estimation of new “risk premium functions.” Implementing the proposed methods with a large cross-section of U.S. equity returns, we find evidence against the traditional (non-granular) CAPM, the Fama–French three and five-factor models, and the Fama–French-Carhart model in favor of the new granular versions of these models. Our empirical results in turn provide new insights into where in the factor-space the compensation for exposures to systematic risks is mostly earned.