Paper: arXiv 2308.11828

Authors: Emma Kroell, Sebastian Jaimungal, Silvana M. Pesenti

Abstract

We study a reinsurer who faces multiple sources of model uncertainty. The reinsurer offers contracts to $n$ insurers whose claims follow compound Poisson processes representing both idiosyncratic and systemic sources of loss. As the reinsurer is uncertain about the insurers’ claim severity distributions and frequencies, they design reinsurance contracts that maximise their expected wealth subject to an entropy penalty. Insurers meanwhile seek to maximise their expected utility without ambiguity. We solve this continuous-time Stackelberg game for general reinsurance contracts and find that the reinsurer prices under a distortion of the barycentre of the insurers’ models. We apply our results to proportional reinsurance and excess-of-loss reinsurance contracts, and illustrate the solutions numerically. Furthermore, we solve the related problem where the reinsurer maximises, still under ambiguity, their expected utility and compare the solutions.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a sophisticated theoretical framework using continuous-time Stackelberg games, compound Poisson processes, and relative entropy penalties to address model uncertainty in reinsurance. However, it is purely theoretical and numerical in nature, completely lacking real-world empirical backtesting or historical market data validation.

Research Flowchart

  flowchart TD
    A["Research Goal: Design optimal robust reinsurance contracts for n insurers under model uncertainty"] --> B["Inputs: Insurer claim data (Compound Poisson: Idiosyncratic & Systemic)"]
    B --> C["Methodology: Continuous-Time Stackelberg Game (Ambiguous Reinsurer vs. Non-ambiguous Insurers)"]
    C --> D["Optimization: Maximize Reinsurer Wealth/Utility with Entropy Penalty for Uncertainty"]
    D --> E["Applications: Solve for General, Proportional, and Excess-of-Loss Contracts"]
    E --> F["Key Outcome: Reinsurer prices risk via a distortion of the barycentre of insurers' models"]