Lower Partial Moments under Gram Charlier Distribution: Performance Measures and Efficient Frontiers

We derive closed-form expressions for the performance measures(PMs)based on the lower partial moments (LPMs), such as the Farinelli-Tibiletti and Kappa measures, with Gram-Charlier (GC) density for returns. It is verified that the LPMs can be obtained as a linear function on both higher moments, skewness and excess kurtosis. We also show that these PMs influence differently to the Sharpe ratio in ranking portfolios due to the effects of the higher moments. We also obtain the efficient frontiers (EFs) based on the mean-LPM framework. We find important differences between portfolios from different EFs regarding their stock compositions, portfolio skewness and excess kurtosis levels. Finally, we also obtain closed-form expressions for PMs under a more flexible density like the SNP density which nests the GC density.