On the Foundation of Performance Measures Under Asymmetric Returns

Abstract

This chapter examines two performance measures advocated for asymmetric return distributions-the Sortino ratio-originally introduced by Sortino and Price-and a measure based on power utility. It investigates the role of the Maximum Principle in this context, and assesses the conditions under which the measures satisfy it. A sensible measure of an economic or financial relationship customarily follows as the consequence of an equilibrium in a model that is consistent with known characteristics of observable data. This model does not rest on assumptions about the distributional properties of the market portfolio, but requires the existence of a representative agent; this chapter gives results which prove that, under certain conditions, such a representative agent exists. Practitioners today already use semi-variance as a risk measure and mean-semi variance optimization methods have been used sporadically since their introduction. This model does not rest on assumptions about the distributional properties of the market portfolio, but requires the existence of a representative agent.