A Moment Expansion of Downside Risk Measures

Abstract

A decomposition of a sub-class of spectral risk measures is introduced in terms of L-moments accounting for geometric characteristics of the return distribution similar to the ones described by the ordinary moments. The decomposition characterizes completely the spectral risk measures with square-integrable risk aversion functions and can be regarded as a link between higher-order moment risk and downside risk measures. Coherent approximations based on only a few L-moments can be successfully constructed for continuous risk aversion functions and can be applied to problems in portfolio theory to analyze the incremental impact of higher order moments on optimal allocations.

Keywords: spectral risk measures, L-moments, coherent risk measures, mean-variance analysis, Gini mean difference

JEL Classification: C13, C61, G11