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Jointly Exchangeable Collective Risk Models: Interaction, Structure, and Limit Theorems

Daniel Gaigall, Stefan Weber

TL;DR

This work extends the classical collective risk model by introducing jointly exchangeable arrays to capture contagion and network interactions in insurance portfolios. It derives expectation, variance, and central limit theorems under multiple asymptotic regimes (increasing portfolio size and/or time horizon), and shows that total losses can be approximated by mixtures of normals in large settings. The authors validate the theoretical results through simulation-based case studies, illustrating how dependence and contagion alter tail behavior and risk measures. By connecting to the standard collective model as a special case, the framework offers a principled approach for risk management and pricing in cyber, operational, and systemic risk contexts.

Abstract

We introduce a framework for systemic risk modeling in insurance portfolios using jointly exchangeable arrays, extending classical collective risk models to account for interactions. We establish central limit theorems that asymptotically characterize total portfolio losses, providing a theoretical foundation for approximations in large portfolios and over long time horizons. These approximations are validated through simulation-based numerical experiments. Additionally, we analyze the impact of dependence on portfolio loss distributions, with a particular focus on tail behavior.

Jointly Exchangeable Collective Risk Models: Interaction, Structure, and Limit Theorems

TL;DR

This work extends the classical collective risk model by introducing jointly exchangeable arrays to capture contagion and network interactions in insurance portfolios. It derives expectation, variance, and central limit theorems under multiple asymptotic regimes (increasing portfolio size and/or time horizon), and shows that total losses can be approximated by mixtures of normals in large settings. The authors validate the theoretical results through simulation-based case studies, illustrating how dependence and contagion alter tail behavior and risk measures. By connecting to the standard collective model as a special case, the framework offers a principled approach for risk management and pricing in cyber, operational, and systemic risk contexts.

Abstract

We introduce a framework for systemic risk modeling in insurance portfolios using jointly exchangeable arrays, extending classical collective risk models to account for interactions. We establish central limit theorems that asymptotically characterize total portfolio losses, providing a theoretical foundation for approximations in large portfolios and over long time horizons. These approximations are validated through simulation-based numerical experiments. Additionally, we analyze the impact of dependence on portfolio loss distributions, with a particular focus on tail behavior.

Paper Structure

This paper contains 20 sections, 6 theorems, 83 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A collective loss model with interaction
  3. Contagious losses
  4. Jointly exchangeable arrays
  5. Mixtures and limit theorems
  6. Examples
  7. The number of loss events
  8. The loss of the insurance company
  9. Limit theorems
  10. Expectation and variance
  11. Central limit theorems as the number of insurance contracts increases
  12. Central limit theorems as the number of insurance contracts and the point in time increase
  13. Central limit theorems as the point in time increases
  14. Relation to the collective risk model
  15. Approximations and interaction effects -- case studies
  16. ...and 5 more sections

Key Result

Theorem 7

Suppose that Assumptions a1--a4 hold. a) The expected total loss satisfies where $\mu_N(t)=E(N(t))$, $\mu_L(n)=\mathrm {E}(L_{1}(n))$ and b) The variance of the total loss is given by where $\sigma^2_N(t)={\rm Var}(N(t))$ and $\sigma^2_L(n)={\rm Var}(L_1(n))$ with

Figures (5)

  • Figure 1: Agents, transmissions and infections as a directed graph.
  • Figure 2: Transmissions and infections in the model specifications in Example \ref{['exa1']}.
  • Figure 3: Plots related to Theorem \ref{['thm2']}.
  • Figure 4: Plots related to Theorem \ref{['thm3']}.
  • Figure 5: Plots related to Theorem \ref{['thm4b']}.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Remark 3
  • Example 4
  • Example 5
  • Example 6
  • Theorem 7
  • Theorem 8
  • Remark 9
  • Theorem 10
  • ...and 11 more