Asymptotics of Systemic Risk in a Renewal Model with Multiple Business Lines and Heterogeneous Claims

TL;DR

This paper develops asymptotic formulas for systemic risk measures in a multi-line insurance model with heterogeneous claims, using a Lévy process framework and validating results through Monte Carlo simulations.

Contribution

It introduces a multi-dimensional Lévy process-based renewal risk model with asymptotic formulas for systemic risk measures under pairwise asymptotic independence of claims.

Findings

Asymptotic tail probabilities derived for aggregate claims and total loss.

Asymptotics of systemic expected shortfall and marginal expected shortfall obtained.

Monte Carlo simulations confirm the accuracy of the asymptotic formulas.

Abstract

Systemic risk is receiving increasing attention in the insurance industry. In this paper, we propose a multi-dimensional L\'{e}vy process-based renewal risk model with heterogeneous insurance claims, where every dimension indicates a business line of an insurer. We use the systemic expected shortfall (SES) and marginal expected shortfall (MES) defined with a Value-at-Risk (VaR) target level as the measurement of systemic risk. Assuming that all the claim sizes are pairwise asymptotically independent (PAI), we derive asymptotic formulas for the tail probabilities of discounted aggregate claims and the total loss, which hold uniformly for all time horizons. We further obtain the asymptotics of the above systemic risk measures. The main technical issues involve the treatment of uniform convergence in the dynamic time setting. Finally, we perform a detailed Monte Carlo study to validate our asymptotics and analyze the impact and sensitivity of key parameters in the asymptotic expressions both analytically and numerically.