Sharpe Thinking with Asymmetrical Preferences

Abstract

As we leave behind the assumption of normality in return distributions, the classical risk-reward Sharpe Ratio becomes a questionable tool for ranking risky projects. In the spirit of Sharpe thinking, a more general risk-reward ratio Phi suitable to compare skewed return distributions with respect to a benchmark, is introduced. This index captures two types of asymmetry information: (1) "good" volatility (above the benchmark) and "bad" volatility (below the benchmark) are differently weighted, (2) asymmetrical preferences to "small" and "large" deviations from the benchmark are modelled. The former goal is achieved by using one-sided volatility measures, and the latter by choosing appropriate order for the one-sided moments involved. The Omega Index (see Cascon et al. (2002) and the Upside Potential Ratio (see Sortino (2000) follow as special cases of the index Phi. Moreover, compatibility of the ranking rule based on ratio Phi with the expected utility framework is proved.

Keywords: Sharpe Index, One-Sided Risk Measures, First Order Risk Aversion

JEL Classification: G0, G1, G2