Optimal Robust Reinsurance with Multiple Insurers

TL;DR

This paper develops a model for a reinsurer managing multiple insurers under model uncertainty, designing optimal contracts that account for ambiguity and systemic risks, with solutions applied to common reinsurance types.

Contribution

It introduces a continuous-time Stackelberg game framework for reinsurance under multiple sources of model uncertainty, providing explicit solutions for proportional and excess-of-loss contracts.

Findings

Reinsurer prices are distorted versions of the barycentre of insurers' models.

Optimal reinsurance contracts depend on ambiguity preferences and systemic risk considerations.

Numerical illustrations demonstrate the practical implications of the theoretical results.

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.