Omega Compatibility: A Meta-analysis

The Omega performance measure introduced by Keating and Shadwick (An introduction to Omega. AIMA Newsletter, 2002a; J Perform Meas 6(3):59-84, 2002b) is widely used in asset allocation and performance measurement. We contribute to the debates around this measure by focusing on its relation and compatibility with Second-order Stochastic Dominance, introducing two conditions of compatibility: the Non-Strict Dominance Compatibility and the Strict Dominance Compatibility conditions. We show that Omega is compatible with the First-order Stochastic Dominance criterion when using the Non-Strict Dominance Compatibility condition (as already shown), but also in the sense of the Strict Dominance Compatibility condition. We also prove again that Omega is compatible with Second-order Stochastic Dominance when using the Non-Strict Dominance Compatibility condition, but only under some conditions on the threshold used in the computation of the Omega measure, as usual. However, we finally also show that Omega is not compatible (i.e. incompatible) with Second-order Stochastic Dominance criterion when using the Strict Dominance Compatibility condition. We further provide a critical meta-analysis that separates good from approximate statements when comparing the views and results provided in many articles on the topic and point out that the use of Omega in asset selection and optimal asset allocation may entail real computational economics issues and may lead to unreasonable financial decisions. Finally, trying to avoid further disputes, ill-posed optimization procedures, and ultimately incorrect economic decisions in computational financial applications, we recall the main potential drawbacks of Omega that, in our opinion, mainly lies in its incompatibility with the Second-order Stochastic Dominance criterion under the Strict Dominance Compatibility condition.