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

Paper: arXiv 2504.06287

Authors: Daniel Gaigall, Stefan Weber

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. Joint exchangeability is a more general probabilistic symmetric than de Finetti’s exchangeability, characterized by the Aldous-Hoover-Kallenberg representation. We establish central limit theorems that asymptotically capture 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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a mathematically sophisticated framework extending classical actuarial collective risk models to network interactions via jointly exchangeable arrays and limit theorems. While it provides strong theoretical derivations and simulation-based validation, it lacks empirical backtesting on real-world insurance loss datasets.

Research Flowchart

  flowchart TD
    A[Research Goal: Model Systemic Risk & Interactions in Insurance Portfolios] --> B[Framework: Extend Collective Risk Models using Joint Exchangeability]
    B --> C[Theory: Apply Aldous-Hoover-Kallenberg Representation]
    C --> D[Analytical Derivation: Establish Central Limit Theorems for Total Losses]
    D --> E[Computational Processes: Run Simulation-Based Numerical Experiments]
    E --> F[Key Outcomes: Asymptotic Approximations for Large Portfolios & Tail Risk Analysis]