Asymptotic Properties of Generalized Shortfall Risk Measures for Heavy-tailed Risks

TL;DR

This paper analyzes the asymptotic behavior of generalized shortfall risk measures, which are flexible tools for assessing risks under heavy tails, providing theoretical expansions and practical estimation methods.

Contribution

It derives first- and second-order asymptotic expansions for the generalized shortfall risk measure, unifying various risk measures and guiding their estimation at extreme levels.

Findings

Asymptotic expansions unify distortion and utility-based risk measures.

A quantile-based estimator is constructed and studied.

Numerical examples demonstrate the accuracy of the expansions and estimator.

Abstract

We study a general risk measure called the generalized shortfall risk measure, which was first introduced in Mao and Cai (2018). It is proposed under the rank-dependent expected utility framework, or equivalently induced from the cumulative prospect theory. This risk measure can be flexibly designed to capture the decision maker's behavior toward risks and wealth when measuring risk. In this paper, we derive the first- and second-order asymptotic expansions for the generalized shortfall risk measure. Our asymptotic results can be viewed as unifying theory for, among others, distortion risk measures and utility-based shortfall risk measures. They also provide a blueprint for the estimation of these measures at extreme levels, and we illustrate this principle by constructing and studying a quantile-based estimator in a special case. The accuracy of the asymptotic expansions and of the estimator is assessed on several numerical examples.