The conditional higher moment risk measure: second-order asymptotics with FGM contagion

2026-07-17

AI Summary

This study addresses the challenge of accurately estimating tail dependence and extreme quantiles of Conditional Higher-order Moment (CoHM) risk measures in financial or insurance settings characterized by weak contagion effects, where existing methods fall short. By leveraging the Farlie–Gumbel–Morgenstern (FGM) dependence structure and integrating extreme value theory with second-order regular variation theory, this work establishes the first second-order asymptotic expansion for CoHM, overcoming the precision limitations inherent in first-order asymptotic approaches. The proposed method substantially enhances approximation accuracy at extreme confidence levels, as demonstrated through numerical simulations showing markedly reduced errors at high quantiles. Empirical analysis further confirms its superior performance when applied to real-world insurance claims data.

Abstract

This paper investigates second-order asymptotic expansions for the conditional higher moment (CoHM) coherent risk measure under a Farlie-Gumbel-Morgenstern (FGM) dependence structure, capturing a weak contagion between a primary loss risk and a reference risk. Assuming that the primary risk belongs to the Fréchet, Weibull, or Gumbel maximum domain of attraction, we systematically derive second-order asymptotic expansions using extreme value theory and second-order regular variation theory. Compared with existing first-order results, our refined approximations capture higher-order tail behavior and dependence effects more accurately. Numerical simulations confirm that the second-order asymptotics substantially reduce approximation errors, especially at extreme confidence levels. Empirical applications to insurance claim data further illustrate the practical superiority of the second-order approach.

Problem

• conditional higher moment risk measure
• second-order asymptotics
• FGM contagion
• extreme value theory
• tail dependence

Innovation

• second-order asymptotics
• conditional higher moment risk measure
• FGM dependence
• extreme value theory
• tail dependence