Bowley solution under the reinsurer's default risk

In this paper, we examine how a monopolistic reinsurer designs a Bowley reinsurance contract, under the assumption that the reinsurer will default on payment if the compensated loss exceeds the sum of the initial capital and the premium charged from the contract. The problem is divided into two subproblems faced by the insurer and the reinsurer in turn. The optimal reinsurance contract is analyzed when both the insurer and the reinsurer minimize their retained risks, as quantified by the VaR measure, and the optimal ceded loss function and the optimal pricing function are provided. Explicit expressions are then derived when the reinsurer adopts either VaR- or TVaR-based regulation capital and charges premiums by the expected -value premium principle. Numerical examples using exponential and Pareto distributions are provided to illustrate the sensitivity effect generated by the confidence levels of the VaR for both parties, as well as those for the initial capitals on the set of Bowley reinsurance contracts.