Managing Extreme Events for Insurance and Finance

Abstract

Insuring losses from natural perils such as hurricanes and earthquakes has traditionally been challenging to the insurance sector, partly because of their heavy-tailed nature. Due to this characteristic of the losses, the valuation of risk management methods is met with a high degree of uncertainty and critical computational costs. By using techniques involving regular variation and Extreme Value Theory (EVT), in this thesis, we propose convenient analytical approximations to key quantities of two risk management mechanisms: catastrophe risk pooling and catastrophe bonds (CAT bonds). Through the derivation of these approximations, we aim at compensating computational limitations arising from the analysis of extreme events as well as providing useful insights for practical implementations. The first focus of this thesis is on optimizing a structure of catastrophe risk pool so that participants can attain a Pareto optimal diversification benefit from joining the pool. Determining the practical optimal pool entails solving a high-dimensional optimization problem, for which analytical solutions are typically unavailable and numerical methods can be computationally intensive and potentially unreliable. To address this challenge, we evaluate the diversification benefit in the limit and use the result to derive an asymptotically optimal pool as an approximation to the practical optimal pool, and a canonical pool as a benchmark. Through numerical studies, we show that the proposed pools provide accurate and reliable approximations to the practical optimal pool. We also conduct an empirical analysis using data from the U.S. National Flood Insurance Program (NFIP) to illustrate the relatively straightforward implementation of this framework. The second focus of this thesis is on constructing a catastrophe risk pool by considering three objectives: optimal diversification benefits for participants, social equity through equal benefits, and premiums which are mindful of both the pool’s solvency and participants’ benefits. Each design requirement can be expressed as an optimization problem with respect to the participants’ losses covered by the pool and the loadings they pay on top of the fair charge. We investigate a benchmark flexible pooling scheme and a truncated pooling scheme. For each scheme, optimal solutions for each problem are derived and compared using insights from EVT. We then combine the results and propose under each pooling scheme a balanced pooling design which can potentially balance the three objectives. We observe that the balanced pools under the two pooling structures asymptotically coincide in the limit case, and a weaker tail dependence between participants’ losses would improve participants’ benefits within the balanced pools. Theoretical results are verified through a numerical study, and the implementation of the balanced pool under the truncated scheme is illustrated through an empirical analysis of the U.S. NFIP data. The third focus of this thesis is on the question of pricing a CAT bond - a financial product which allows catastrophe risk to be transferred from the insurance sector to the bond market. Despite the rapid development of the CAT bond market, the market pricing of this product proves to be challenging due to the nature of catastrophe losses, as well as since its evaluation requires using numerical methods which could be computationally demanding. By using insights from EVT and a product pricing measure previously proposed in the literature, we derive two tractable approximations to the expected maturity payment of a zero-coupon CAT bond. This value can then be discounted by an independent risk-free rate process to obtain the bond price. Four simulations are then carried out to showcase the flexibility of the framework in accommodating various loss models, especially under changing climate, as well as to verify its coherence with the dependence-free property of the expected maturity payment. The results show that our approximations are accurate, flexible, and provide a considerable reduction in computational costs. Through the analysis on these three research questions, we aim at enhancing the implementation of risk pooling and CAT bonds in practice through a rigorous application of EVT techniques, thus enriching the risk management toolbox for extreme event risk in finance and insurance.