Optimal Reinsurance Design: A Mean-Variance Approach

Abstract

In this article, we study an optimal reinsurance model from the perspective of an insurer who has a general mean-variance preference. In order to reduce ex post moral hazard, we assume that both parties in a reinsurance contract are obligated to pay more for a larger realization of loss. We further assume that the reinsurance premium is calculated only based on the mean and variance of the indemnity. This class of premium principles is quite general in the sense that it includes many widely used premium principles such as expected value, mean value, variance, and standard deviation principles. Moreover, to protect the insurer's profit, a lower bound is imposed on its expected return. We show that any admissible reinsurance policy is dominated by a change-loss reinsurance or a dual change-loss reinsurance, depending upon the coefficient of variation of the ceded loss. Further, the change-loss reinsurance is shown to be optimal if the premium loading increases in the actuarial value of the coverage; while it becomes decreasing, the optimal reinsurance policy is in the form of dual change loss. As a result, the quota-share reinsurance is always optimal for any variance-related reinsurance premium principle. Finally, some numerical examples are applied to illustrate the applicability of the theoretical results.

Notes

¹Throughout this article, the terms “increasing” and “decreasing” mean “nondecreasing” and “nonincreasing,” respectively.

²In this article, a dual change-loss reinsurance is defined as the difference between full reinsurance and a change-loss reinsurance. The formal definition will be given in Section 3.

An increasing function g₁(x) is said to up-cross an increasing function g₂(x), if there exists an such that

A random variable Y₁ is said to be smaller than a random variable Y₂ in the convex order (denoted by Y₁ ⩽ _cxY₂), if for any real number d. See Shaked and Shanthikumar (2007) for more details on stochastic orders.