Pareto-optimal Reinsurance Policies with Maximal Synergy

Abstract

Optimal reinsurance policies have been studied extensively in the economics and insurance literature. Two types of optimality criteria are commonly used: maximizing the expected utility (EU) or minimizing risks. Understandably, applying the two types of criteria usually will result in different “optimal” policies. To strike a balance between maximizing utility and minimizing risk, Borch (1960b) derived the EU-maximizing reinsurance policies assuming that the admissible policies are those that minimize the total variance of the losses borne by the two parties. This in fact implies that only quota-share policies are admissible and greatly simplifies the problem. In this paper, we follow the approach in Borch (1960b). However, we assume that the two parties apply distortion risk measures instead of variance. We first identify a set of reinsurance policies that minimize the total risk shared by the two parties, then we take this set of policies as admissible and determine the Pareto-optimal policies that maximize the EU of the two parties. In contrast to the results in Borch (1960b), we show that applying risk measures such as the Value at Risk (VaR) and the Tail Value at Risk (TVaR) results in multi-layered policies.

Keywords: Optimal Reinsurance, Maximal Synergy, Pareto-Optimal Reinsurance, Nash Bargaining Solution, Multi-Layered Policy

JEL Classification: C71, G22