Assessment, Mitigation, and Backtesting of Extreme Risks in Insurance and Finance

Abstract

In this thesis, we focus on the quantitative assessment, mitigation, and backtesting of extreme risks in insurance and finance. We adopt tools from extreme value theory (EVT), dependence modeling, and statistical testing to address fundamental challenges in managing low-probability but high-impact events. The research contributes to three key aspects of extreme risk management: systemic risk assessment, catastrophe risk mitigation, and risk model evaluation.

We begin with the assessment of systemic risk under extreme scenarios. Systemic events are characterized by strong dependence among individual entities and heavy-tailed risk behavior. In Chapter 3, we develop an asymptotic framework for systemic risk measures, covering both Value-at-Risk-based and expectile-based risk measures. Second-order asymptotic approximations are derived to improve the accuracy of risk quantification beyond conventional first-order results. Special attention is given to expectile-based systemic risk measures, which provide a more conservative assessment of systemic risk.

The limitation of diversification for heavy-tailed risks has been well documented in the literature, especially for extremely heavy-tailed risks with infinite first moment. However, in practical insurance markets, catastrophic risks often exhibit extremely heavy tails but are also subject to truncation due to limited liability or policy design. In Chapter 4, we investigate the effectiveness of catastrophe risk pooling under these realistic constraints. Building on EVT, we characterize conditions under which diversification benefits remain achievable. The analysis incorporates tail heaviness, loss scaling, liability structure, and risk-sharing rules, providing theoretical foundations and practical guidance for catastrophe risk management.

In Chapter 5, we turn to the problem of risk model evaluation through backtesting. Conventional backtesting methods often rely on strict model assumptions and may fail under model misspecification, structural change or dependence uncertainty. Building on existing model-free testing approaches using e-values and e-processes, we extend these ideas to develop a backtesting framework for identifiable and elicitable risk measures, including Value-at-Risk, Expected Shortfall, and expectiles. The proposed framework delivers valid statistical inference for both standard and comparative backtests, and supports robust risk assessment across a wide range of risk levels, not limited to extreme events.

Throughout the thesis, theoretical results are complemented by extensive simulation studies and real-world applications. These findings contribute to advancing the theoretical foundations and practical methodologies for extreme risk management in modern insurance and financial systems.