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 then show how the resulting granular beta functions may be used in the estimation of new "risk premium functions." Implementing the proposed new methods with a large cross-section of U.S. equity returns, we find significant 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 also provide new insights into where in the "factor space" the compensation for exposures to systematic risks is mostly earned