Optimal investment–reinsurance policy for an insurance company with VaR constraint

This paper investigates an investment–reinsurance problem for an insurance company that has a possibility to choose among different business activities, including reinsurance/new business and security investment. Our main objective is to find the optimal policy to minimize its probability of ruin. The main novelty of this paper is the introduction of a dynamic Value-at-Risk (VaR) constraint. This provides a way to control risk and to fulfill the requirement of regulators on market risk. This problem is formulated as an infinite horizontal stochastic control problem with a constrained control space. The dynamic programming technique is applied to derive the Hamilton–Jacobi–Bellman (HJB) equation and the Lagrange multiplier method is used to tackle the dynamic VaR constraint. Closed-form expressions for the minimal ruin probability as well as the optimal investment–reinsurance/new business policy are derived. It turns out that the risk exposure of the insurance company subject to the dynamic VaR constraint is always lower than otherwise. Finally, a numerical example is given to illustrate our results.