Optimal Excess-of-Loss Reinsurance Contract in a Dynamic Risk Model

Abstract

This paper studies the optimal excess-of-loss reinsurance contract between an insurer and a reinsurer in a dynamic risk model. The risk process is assumed to be a diffusion approximation process of the classical Cramer-Lundberg model which is perturbed by a Brownian motion. In addition to reinsurance, we assume that the insurer is allowed to invest his/her surplus into a financial market containing one risk-free rate of return and determines the reinsurance strategy by a self-reinsurance function. Our aim is to obtain the simultaneous equilibrium strategy in this reinsurance dynamic risk setting using the objective functions of insurer and reinsurance. By employing the dynamic programming approach, we derive the minimization of insurer’s ruin probability and maximization of reinsurance’s expected aggregate discounted net profits to have the optimal portfolio for the two parties treaties in a fixed term insurance contract. In order to provide a more explicit reinsurance contract and to facilitate our quantitative analysis, we study the case when the reinsurance premium function is based on the standard-deviation principle from the integro-differential equations. A numerical example is given to investigate the effects of model parameters on the equilibrium strategy.