Optimal Risk Allocation in Reinsurance Networks

TL;DR

This paper investigates the optimal allocation of reinsurance in a macroeconomic context, demonstrating that layer reinsurance treaties are socially optimal under certain risk measures and dependence assumptions, without requiring risk dependence structures.

Contribution

It establishes conditions under which layer reinsurance treaties are socially optimal, extending previous results to Range-Value-at-Risk and positive risk dependence scenarios.

Findings

Layer reinsurance treaties are socially optimal under Value-at-Risk with monotonic premium principles.

Optimality of layer treaties extends to Range-Value-at-Risk with positive risk dependence.

Discussion of differences between socially and individually optimal reinsurance treaties.

Abstract

In this paper we consider reinsurance or risk sharing from a macroeconomic point of view. Our aim is to find socially optimal reinsurance treaties. In our setting we assume that there are $n$ insurance companies each bearing a certain risk and one representative reinsurer. The optimization problem is to minimize the sum of all capital requirements of the insurers where we assume that all insurance companies use a form of Range-Value-at-Risk. We show that in case all insurers use Value-at-Risk and the reinsurer's premium principle satisfies monotonicity, then layer reinsurance treaties are socially optimal. For this result we do not need any dependence structure between the risks. In the general setting with Range-Value-at-Risk we obtain again the optimality of layer reinsurance treaties under further assumptions, in particular under the assumption that the individual risks are positively dependent through the stochastic ordering. At the end, we discuss the difference between socially optimal reinsurance treaties and individually optimal ones by looking at a number of special cases.