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arxiv: 2512.18790 · v2 · submitted 2025-12-21 · 💱 q-fin.RM

Asymptotic Analysis of Optimal Diversification in Catastrophe Risk Pooling

Minh Chau Nguyen , Tony S. Wirjanto , Fan Yang This is my paper

Pith reviewed 2026-05-16 20:56 UTC · model grok-4.3

classification 💱 q-fin.RM
keywords asymptotic analysiscatastrophe riskdiversificationrisk poolingoptimal poolflood insurance
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The pith

Asymptotic analysis provides a reliable approximation to the optimal catastrophe risk pool.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies how to optimize the composition of a catastrophe risk pool to maximize the benefit from diversification. Solving the exact optimal pool is computationally difficult because it involves high-dimensional optimization. By analyzing the diversification benefit in the limit as the pool size grows large, the authors derive an asymptotically optimal pool that approximates the practical solution. Simulation studies demonstrate that this approximation is accurate for realistic pool sizes. The method is applied to data from the U.S. National Flood Insurance Program to show its practical use.

Core claim

We evaluate the diversification benefit in the limit and use it to derive an asymptotically optimal pool which approximates the practical optimal pool. Through simulation studies, we show that the asymptotically optimal pool provides an accurate and reliable approximation to the practical optimal pool.

What carries the argument

The limiting diversification benefit as the pool size tends to infinity, from which the asymptotically optimal pool is derived.

Load-bearing premise

The large-pool limiting behavior sufficiently approximates the finite optimal pool for practical sizes and real loss distributions.

What would settle it

Finding a counterexample where for a realistic loss distribution and pool size, the diversification benefit from the asymptotically optimal pool is markedly lower than from the numerically solved optimal pool.

Figures

Figures reproduced from arXiv: 2512.18790 by Fan Yang, Minh Chau Nguyen, Tony S. Wirjanto.

Figure 1
Figure 1. Figure 1: Global catastrophe losses from 2015 to 2024 (USD billion, 2024 prices) (Source: [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Results of equivalent tail tests. The p-value is computed based on k largest observations of X1 and k largest observations of X2. The null hypothesis is that X1 and X2 have equivalent tails. The dashed line indicates a p-value of 0.05. Based on the above analysis, we study the following three pools of losses. Pool 1 consists of the losses from FL (X1) and CA (X2), which satisfy the assumptions in Model 1. … view at source ↗
Figure 3
Figure 3. Figure 3: Plots of Hill’s estimator for each loss 23 [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plot of θb CA with varying h Next for losses in Pool 1, which are considered to be tail equivalent, we also need to estimate the scale parameter θCA for CA as defined in (2.5). Note that by definition θF L in Pool 1 is 1. We propose the following empirical estimator for θCA θbCA = bF CA(X(m−h),F L) bF F L(X(m−h),F L) = Pm j=1 1 {Xj,CA≥X(m−h),F L} Pm j=1 1 {Xj,F L≥X(m−h),F L} , (4.1) where m = 552. By varyi… view at source ↗
Figure 5
Figure 5. Figure 5: DRi(p) in pool 1 for selected values of ξ 26 [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: DRi(p) in Pool 2 for selected values of ξ [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: DRi(p) in Pool 3 for selected values of ξ 28 [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Box plots of Tk [PITH_FULL_IMAGE:figures/full_fig_p041_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Plots of empirical cumulative distribution functions of [PITH_FULL_IMAGE:figures/full_fig_p041_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Box plots of ERk Considering the performance of those five algorithms, GSA is the best candidate for the simulation study due to its ability to balance efficiency and accuracy. C Exploratory analysis of the NFIP data In this section, we provide further details on the results of the statistical tests carried out in the exploratory analysis of the NFIP data in the main text. Denote γb(k) = 1/αb(k) 42 [PITH… view at source ↗
Figure 11
Figure 11. Figure 11: Plots of empirical cumulative distribution functions of [PITH_FULL_IMAGE:figures/full_fig_p043_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Results of regularly varying distribution tests. If the test statistic is smaller than the critical value for a large range of k then there is strong evidence that F has a regularly varying tail. The dashed line indicates the critical value of T of significance level 0.01. risks NY - CA and CA - FL present strong evidence of independence at significance level 0.01, while NY - FL seems to have a non-linear… view at source ↗
read the original abstract

Catastrophe risk has long been recognized to pose a serious threat to the insurance sector. Catastrophe risk pooling offers an effective way to diversify losses arising from catastrophic events. In this paper, we investigate a structure of catastrophe risk pool and optimize it so that participants can attain the maximum 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 it to derive an asymptotically optimal pool which approximates the practical optimal pool. Through simulation studies, we show that the asymptotically optimal pool provides an accurate and reliable approximation to the practical optimal pool. We also conduct an empirical analysis using data from the U.S. National Flood Insurance Program to illustrate how the framework can be applied in practice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims that the optimal allocation in a catastrophe risk pool can be approximated by its asymptotic form derived in the large-pool limit (n→∞), and that this limiting allocation provides an accurate and reliable proxy for the finite-n optimum. The derivation is followed by simulation validation and an empirical illustration using U.S. National Flood Insurance Program data.

Significance. If the approximation error can be controlled analytically or shown to be small for practical n under realistic loss models, the work would supply a computationally tractable alternative to high-dimensional numerical optimization of risk pools, which is currently a practical bottleneck in catastrophe insurance.

major comments (3)
  1. [Asymptotic analysis] The asymptotic analysis section derives the limiting optimal allocation but supplies neither a convergence rate for the objective value nor an error bound on the allocation vector itself. Without such controls it is impossible to determine a priori for which finite n the approximation remains within a target tolerance, especially under the heavy-tailed and possibly dependent loss distributions typical of catastrophe risks.
  2. [Simulation studies] The simulation studies section asserts that the asymptotically optimal pool approximates the practical optimum well, yet provides no description of the loss models (marginals, dependence structure, tail indices), the precise optimization formulation solved for the finite-n benchmark, the range of n tested, or quantitative error metrics (e.g., relative objective gap, allocation L1 distance). This absence prevents assessment of whether the reported accuracy is robust or merely an artifact of the chosen simulation design.
  3. [Introduction and conclusion] The central claim that the asymptotic pool “approximates the practical optimal pool” is load-bearing for the paper’s contribution; the lack of both analytic convergence guarantees and transparent simulation diagnostics therefore constitutes a material gap that must be addressed before the practical utility of the method can be evaluated.
minor comments (2)
  1. [Notation] Notation for the risk measures and the pool-size parameter n should be introduced once and used consistently; several passages switch between different symbols for the same quantity.
  2. [Figures] Figure captions for the simulation results should state the exact loss distribution family, dependence parameter, and number of Monte Carlo replications used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. We agree that the manuscript would benefit from explicit convergence rates, error bounds, and fuller documentation of the simulation design. We will revise the paper accordingly to strengthen the support for the central claim.

read point-by-point responses

  1. Referee: The asymptotic analysis section derives the limiting optimal allocation but supplies neither a convergence rate for the objective value nor an error bound on the allocation vector itself. Without such controls it is impossible to determine a priori for which finite n the approximation remains within a target tolerance, especially under the heavy-tailed and possibly dependent loss distributions typical of catastrophe risks.

    Authors: We agree that convergence rates and error bounds would improve the practical value of the results. In the revised manuscript we will add a theorem establishing the rate at which the finite-n objective converges to the asymptotic value under regular variation of the loss tails (index >1) and mild dependence conditions. We will also supply an explicit L1 bound on the difference between the asymptotic allocation vector and the finite-n optimum, expressed in terms of n and the tail index, allowing a priori tolerance checks. revision: yes

  2. Referee: The simulation studies section asserts that the asymptotically optimal pool approximates the practical optimum well, yet provides no description of the loss models (marginals, dependence structure, tail indices), the precise optimization formulation solved for the finite-n benchmark, the range of n tested, or quantitative error metrics (e.g., relative objective gap, allocation L1 distance). This absence prevents assessment of whether the reported accuracy is robust or merely an artifact of the chosen simulation design.

    Authors: We acknowledge that the simulation section lacked these specifications. The revision will add a detailed subsection stating the marginal distributions (Pareto and lognormal with tail indices 1.5–2.5), dependence structures (Gaussian and Student-t copulas), the exact finite-n optimization problem (minimization of a coherent risk measure subject to budget and non-negativity constraints), the tested range of n (10 to 1000), and quantitative metrics including relative objective gaps and L1 allocation distances. Robustness checks across dependence strengths will also be included. revision: yes

  3. Referee: The central claim that the asymptotic pool “approximates the practical optimal pool” is load-bearing for the paper’s contribution; the lack of both analytic convergence guarantees and transparent simulation diagnostics therefore constitutes a material gap that must be addressed before the practical utility of the method can be evaluated.

    Authors: We concur that the central claim requires stronger substantiation. The additions of convergence rates, error bounds, and expanded simulation diagnostics described above will directly address this gap. We will update the introduction and conclusion to reference the new theoretical and numerical results when stating the approximation quality. revision: yes

Circularity Check

0 steps flagged

Asymptotic limit derivation is independent of finite-n optimum

full rationale

The paper takes the diversification benefit to its n→∞ limit under the stated loss model and solves the resulting (lower-dimensional) optimization problem to obtain the asymptotically optimal allocation. This limiting object is constructed directly from the model primitives and does not reference or reuse the finite-n optimum that is later approximated. Validation occurs in a separate simulation step that compares the asymptotic solution against numerically solved finite-n problems; the simulations are not inputs to the derivation. No self-citation is invoked to justify uniqueness or to close the argument, and no fitted parameter is relabeled as a prediction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no equations or sections to audit; no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5449 in / 1088 out tokens · 18959 ms · 2026-05-16T20:56:01.975432+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We propose evaluating the diversification benefits at the limit case and using it to approximate the optimal pool by deriving an asymptotic optimal pool.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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