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

Haifan Hu; Bingzhen Geng; Jiajun Liu; Shijie Wang

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

Risk Management 2026-07-18 v1 Probability

Authors: Haifan Hu , Bingzhen Geng , Jiajun Liu , Shijie Wang

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 Fr\'echet, 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.

Keywords

conditional higher moment risk measure second-order asymptotics FGM dependence structure extreme value theory regular variation coherent risk measure

Cite

@article{arxiv.2607.16601,
  title  = {The conditional higher moment risk measure: second-order asymptotics with FGM contagion},
  author = {Haifan Hu and Bingzhen Geng and Jiajun Liu and Shijie Wang},
  journal= {arXiv preprint arXiv:2607.16601},
  year   = {2026}
}