Multivariate reinsurance designs for minimizing an insurer’s capital requirement

Abstract

This paper investigates optimal reinsurance strategies for an insurer with multiple lines of business under the criterion of minimizing its total capital requirement calculated based on the multivariate lower-orthant Value-at-Risk. The reinsurance is purchased by the insurer for each line of business separately. The premium principles used to compute the reinsurance premiums are allowed to differ from one line of business to another, but they all satisfy three mild conditions: distribution invariance, risk loading and preserving the convex order, which are satisfied by many popular premium principles. Our results show that an optimal strategy for the insurer is to buy a two-layer reinsurance policy for each line of business, and it reduces to be a one-layer reinsurance contract for premium principles satisfying some additional mild conditions, which are met by the expected value principle, standard deviation principle and Wang’s principle among many others. In the end of this paper, some numerical examples are presented to illustrate the effects of marginal distributions, risk dependence structure and reinsurance premium principles on the optimal layer reinsurance.