Comparative risk aversion vs. threshold choice in the Omega ratio

We study conditions under which the threshold parameter in the Omega ratio represents risk aversion in the sense of monotonicity of risk premia. To this end, we derive asymptotic expansions for risk premia associated with taking a small additional risk on top of a background risk. These risk premia have the expected monotonicity behavior if, roughly speaking, the variance of the additional risk decreases with the background risk and if the density of the background risk is log-concave. When these conditions are violated, the threshold in the Omega ratio does not represent risk aversion in general. Finally, we compare our sufficient conditions for the Omega ratio to those that are needed to guarantee monotonicity of risk premia with an expected utility criterion under background risk. We argue that the conditions that are needed for the Omega threshold to represent risk aversion are comparable to those that are needed for expected utility with exponential utility functions.

Automated summary

This study examines whether the threshold parameter in the Omega ratio represents risk aversion by analyzing the asymptotic expansions for risk premia. The results suggest that the threshold in the Omega ratio represents risk aversion when the variance of the additional risk decreases with the background risk and when the density of the background risk is log-concave. Comparisons with expected utility criteria indicate that the conditions for the Omega threshold to represent risk aversion are comparable to those for exponential utility functions.