One-sided performance measures under Gram-Charlier distributions

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 moment (LPM) measures can be expressed as linear functions on both skewness and excess kurtosis. Under this framework, we study the behavior of portfolio rankings with performance measures based on partial moments, that is, both Farinelli-Tibiletti (FT) 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 than the GC distribution.