On Consistency of the Omega Ratio with Stochastic Dominance Rules

Abstract

Omega ratios have been introduced in Keating and Shadwick (2002) as a performance measure to compare the performance of different investment opportunities. It does not have some of the drawbacks of the famous Sharpe ratio. In particular, it is consistent with first order stochastic dominance. Omega ratios also have an interesting relation to expectiles, which found increasing interest recently as risk measures. There is some confusion in the literature about consistency with respect to second order stochastic dominance. In this paper we clarify this and extend it to a consistency result with respect to stochastic dominance of order 1 + γ recently introduced in Muller, Scarsini, Tsetlin, and Winkler (2017) and generalizing the classical concepts of stochastic dominance of first and second order. Several examples illustrate the usefulness of this result. Finally, some consistency results for even more general stochastic dominance rules are shown, including the concept of ϵ-almost stochastic dominance introduced by Leshno and Levy (2002).