Risk Measures and Optimal Reinsurance

Abstract

In this thesis, we study the optimal reinsurance design problem and extend the classical model in three different directions:

(1) In the first framework, we add the additional assumption that the reinsurer can default on its obligations. If the indemnity is beyond the reinsurer's payment ability, the reinsurer fails to pay for the exceeding part and this induces a default risk for the insurer. In our model, the reinsurer is assumed to measure the risk of an insured loss by Value-at-Risk regulation and prepares the same amount of money as the initial reserve. As soon as the indemnity is larger than this value plus the premium, default occurs. From the insurer's point of view, two optimization problems are going to be considered when the insurer: 1) maximizes his expectation of utility; 2) minimizes the VaR of his retained loss.

(2) In the second framework, the reinsurance buyer (insurer) adopts a convex risk measure to control his total loss while the reinsurance seller (reinsurer) price the reinsurance contract by Wang's premium principle with a distortion. Without specifying a particular convex risk measure and distortion, we obtain a general expression for the optimal reinsurance contract that minimizes the insurer's total risk exposure.

(3) In the third framework, we study optimal reinsurance designs from the perspectives of both an insurer and a reinsurer and take into account both an insurer's aims and a reinsurer's goals in reinsurance contract designs. We develop optimal reinsurance contracts that minimize the convex combination of the VaR risk measures of the insurer's loss and the reinsurer's loss under two types of constraints, respectively. The constraints describe the interest of both the insurer and the reinsurer. With the first type of constraints, the insurer and the reinsurer each have their limit on the VaR of their own loss. With the second type of constraints, the insurer has a limit on the VaR of his loss while the reinsurer has a target on his profit from selling a reinsurance contract. For both types of constraints, we derive the optimal reinsurance form for a wide class of reinsurance policies and under the expected value reinsurance premium principle.