Optimal Catastrophe Risk Pooling
Abstract
Catastrophe risk has long been recognized to pose a serious threat to the insurance sector. Although natural disasters such as flooding, hurricane or severe drought are rare events, they generally lead to devastating damages that traditional insurance schemes may not be able to efficiently cover. Catastrophe risk pooling is an effective way to diversify the losses from such risks. In this paper, we improve the catastrophe risk pool by Pareto-optimally allocating the diversification benefits among participants. Finding the practical Pareto-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. We propose evaluating the diversification benefits at the limit case and using it to approximate the optimal pool by deriving an asymptotic optimal pool. Simulation studies are undertaken to explore the implications of the results and an empirical analysis from the U.S. National Flood Insurance Program is also carried out to illustrate how this framework can be applied in practice.
Summary
This paper addresses the problem of optimal catastrophe risk pooling, where the goal is to efficiently allocate diversification benefits among participants. The authors focus on situations where traditional insurance schemes struggle due to the severity of losses from natural disasters. They propose a method to improve catastrophe risk pools by Pareto-optimally allocating diversification benefits. The challenge lies in the high-dimensional optimization problem involved in finding a practical Pareto-optimal pool, which lacks analytical solutions and can be computationally intensive. The authors' approach involves evaluating diversification benefits at the limit case as the Value-at-Risk (VaR) level approaches 1, then using this to approximate the optimal pool by deriving an asymptotic optimal pool. They consider two models: one where losses are tail equivalent (same tail index) and another where losses have different tail indices. They derive asymptotic expressions for the Diversification Ratio (DR) under both models. Simulation studies are used to explore the implications of the results. Finally, an empirical analysis using data from the U.S. National Flood Insurance Program (NFIP) illustrates the practical application of the framework. The key finding is a method for constructing an asymptotic optimal pool that provides a reasonably accurate approximation to the practical optimal pool. This method involves selecting the optimal loss layer for each participant. Notably, the asymptotic optimal pool not only achieves Pareto optimality but also allows each participant to obtain the maximum possible diversification benefit. This is significant because it offers a computationally efficient alternative to solving the complex high-dimensional optimization problem directly, making optimal risk pooling more accessible and practical.
Key Insights
- •Asymptotic Optimal Pool Derivation: The paper provides a method to derive an asymptotic optimal pool by evaluating diversification benefits at the limit as the VaR level approaches 1. This simplifies the complex optimization problem.
- •Pareto Optimality and Maximum Diversification: The derived asymptotic optimal pool achieves Pareto optimality, meaning no participant can improve their diversification benefit without reducing another's. Furthermore, each participant attains their individual maximum diversification benefit.
- •Tail Equivalence vs. General Losses: The paper considers two models: one with tail-equivalent losses (same tail index) and another with general heavy-tailed losses (different tail indices), providing a flexible framework for different risk profiles.
- •DR Asymptotic Expressions: The authors derive asymptotic expressions for the Diversification Ratio (DR) under both models, allowing for the quantification of diversification benefits as the VaR level approaches 1. The DR is defined in Equation 2.2.
- •Simulation Validation: Simulation studies demonstrate that the asymptotic optimal pool provides a reasonably accurate and reliable approximation to the practical optimal pool. The GSA algorithm is used for the numerical evaluation of the practical problem.
- •Empirical Application: An empirical analysis using U.S. National Flood Insurance Program (NFIP) data illustrates how the framework can be applied in practice, showing that the resulting pools are consistent with the theoretical findings.
- •Computational Efficiency: The asymptotic approximation significantly reduces computational cost compared to solving the high-dimensional optimization problem directly. The simulation studies show that each line in Tables 1 and 2 requires between 3.74 and 187 hours depending on the available computing power.
Practical Implications
- •Catastrophe Risk Management: The research provides a practical framework for managing catastrophe risk through optimal risk pooling. It allows for more efficient allocation of diversification benefits among participants.
- •Insurance Sector Benefit: Insurance companies and risk managers can benefit from this research by using the proposed method to design more efficient catastrophe risk pools.
- •Policy Design Tool: The results can be used to design insurance policies and risk-sharing mechanisms that are more resilient to extreme events. Practitioners can use the asymptotic optimal pool as a first-order approximation for designing risk pools.
- •Future Research: The paper opens up avenues for future research, such as extending the framework to incorporate dependencies among losses and exploring alternative risk measures. It would be useful to extend this research to cases where ξ = 0.
- •Tail Index Estimation Importance: The work emphasizes the importance of accurate tail index estimation for determining optimal pooling structures, thus highlighting the need for robust statistical methods.