Several studies have found that the cross-section of stock returns reflects a risk premium for bearing downside risk; however, existing measures of downside risk have poor power for predicting returns. Therefore, this paper proposes a novel measure of downside risk, the ES-implied beta, to...

In this paper, we provide an axiomatic approach to general premium priciples giving rise to a decomposition into risk, as a generalization of the expected value, and deviation, as a generalization of the variance. We show that, for every premium priciple, there exists a maximal risk measure...