Portfolio Performance Evaluation with Generalized Sharpe Ratios: Beyond the Mean and Variance

Abstract

The main purpose of this paper is to present a theoretically sound portfolio performance measure that takes into account higher moments of the distribution of returns. First, we perform a study of the investor's preferences to higher moments of distribution within expected utility theory and discuss the performance measurement. To illustrate the investor's preferences to higher moments and the computation of a performance measure, we provide an approximation analysis of the optimal capital allocation problem and derive a formula for the Sharpe ratio adjusted for skewness of distribution. This performance measure justifies the notion of the Generalized Sharpe Ratio (GSR) introduced by Hodges (1998) which presumably accounts for all moments of distribution. We present two methods of practical estimation of the GSR: nonparametric and parametric. For the implementation of the parametric method we derive a closed-form solution for the GSR where the higher moments are calibrated to the normal inverse Gaussian distribution. We illustrate how the GSR can mitigate the shortcomings of the Sharpe ratio in resolution of Sharpe ratio paradoxes and reveal the real performance of portfolios with manipulated Sharpe ratios. We also demonstrate the use of this measure in the performance evaluation of hedge funds.

Keywords: Sharpe ratio, skewness, kurtosis, portfolio performance evaluation

JEL Classification: G11