One-Sided Performance Measures Under Gram-Charlier Distributions

Abstract

We derive closed-form expressions for risk measures based on partial moments by assuming the Gram-Charlier (GC) density for stock returns. As a result, the lower partial moments (LPM) can be expressed as linear functions on both skewness and excess kurtosis. Under this framework, we study the behaviour of portfolio rankings with performance measures based on partial moments, that is, both Farinelli-Tibiletti and Kappa ratios. Contrary to previous results, significant differences are found in ranking portfolios between the Sharpe ratio and the FT family. We also obtain closed-form expressions for LPMs under the semi non-parametric (SNP) distribution which allows higher flexibility (in terms of third- and fourth- order moments) than the GC distribution.