Optimal Catastrophe Risk Pooling

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.

💡 Research Summary

The paper addresses the challenge of pooling catastrophe risk, which is characterized by rare but extremely large losses that follow heavy‑tailed distributions. Traditional insurance often cannot cover such extreme events, so the authors propose a risk‑pooling mechanism that allocates diversification benefits among participants in a Pareto‑optimal way. They introduce a new Diversification Ratio (DR) that compares a participant’s Value‑at‑Risk (VaR) before and after joining the pool, incorporating both the retained loss and the portion of risk shared through the pool. A DR below one indicates a net risk reduction.

Directly solving the high‑dimensional Pareto‑optimal allocation problem (minimising the sum of DRs across all participants) is computationally prohibitive. To overcome this, the authors examine the limit as the VaR confidence level p approaches 1, where tail behavior dominates. They derive closed‑form asymptotic expressions for DR_i(1)=lim_{p→1}DR_i(p) and define an “asymptotic optimal pool” that minimizes these limits.

Two models are studied. Model 1 assumes all losses share a common tail index α but may have different scale factors θ_i. Attachment points d_i and limits l_i are set as multiples of VaR: d_i≈ξ·VaR_p(X_i) with ξ<1, and l_i≈λ_i·d_i with λ_i>1. Theorem 2.1 provides a piece‑wise formula for DR_i(1) involving parameters δ_i and Δ_{ξ,λ}, which capture overall pool diversification. The analysis shows that if ξ≥1 the pool provides no benefit, so practical designs require ξ<1.

Model 2 relaxes the common‑tail assumption, allowing each loss X_i to have its own tail index α_i. Here the attachment points are chosen so that each loss has the same tail probability at d_i, while λ_i are tuned to give every participant their maximal possible DR_i(1). This construction achieves Pareto optimality and simultaneously delivers each participant’s individual optimum, a rare result in the literature.

Simulation experiments compare the asymptotic optimal pool with numerically obtained true optimal pools for finite p (e.g., 0.99, 0.995). The asymptotic solutions closely match the exact optima, confirming the practical accuracy of the approximation. An algorithmic comparison (Appendix B) demonstrates that the proposed global‑optimization routine outperforms four alternative methods in both speed and precision.

The empirical section applies the methodology to flood loss data from the U.S. National Flood Insurance Program. Three Pareto‑optimal pools are constructed, each reflecting different regional loss characteristics. The resulting DRs range from 0.70 to 0.85, indicating substantial risk reduction. Premiums and coverage limits derived from the asymptotic framework align with observed insurance pricing, and the authors provide concrete guidance on setting attachment points and limits for policymakers.

Overall, the paper contributes a theoretically rigorous yet computationally tractable framework for catastrophe risk pooling. By leveraging heavy‑tail asymptotics, it transforms an intractable high‑dimensional optimization into a solvable problem, while preserving Pareto efficiency and delivering tangible benefits in realistic insurance settings.