Optimizing Multiple Reinsurance Contracts

An insurance contract implies that risk is ceded from ordinary policy holders to companies. Companies do the same thing between themselves, and this is known as reinsurance. The problem of determining reinsurance contracts which are optimal with respect to some reasonable criterion has been studied extensively. Different contract types are considered such as stop-loss contracts where the reinsurance company covers risk above a certain level, and insurance layer contracts where the reinsurance company covers risk within an interval. The contracts are then optimized with respect to some risk measure, such as value-at-risk or conditional tail expectation. In the present paper we investigate this problem further and show that the optimal solution depends on the tail hazard rates of the risk distributions. If the tail hazard rates are decreasing, which is the case for heavy tailed distributions like lognormal and pareto distributions, the optimal solution is balanced. That is, reinsurance contracts for identically distributed risks should be identical insurance layer contracts. However, if the tail hazard rate is increasing, which is the case for light tailed distributions like truncated normal distributions, the optimal solution is typically not balanced. Even for identically distributed risks, some contracts should be insurance layer contracts, while others should be stop-loss contracts. In the limiting case, where the hazard rate is constant, i.e., when the risks are exponentially distributed, we show that a balanced solution is optimal. We also present an efficient importance sampling method for estimating optimal contracts.