A marginal indemnity function approach to optimal reinsurance under the Vajda condition

To manage the risk of insurance companies, a reinsurance transaction is among the myriad risk management mechanisms the top ranked choice. In this paper, we study the design of optimal reinsurance contracts within a risk measure minimization framework and subject to the Vajda condition. The Vajda condition requires the reinsurer to take an increasing proportion of the loss when it increases and therefore imposes constraints on the indemnity function. The distortion-risk-measure-based objective function is very generic, and allows for, for example, an objective to minimize the risk-adjusted value of the insurer’s liability, and for heterogeneous beliefs regarding the loss distribution by the insurer and reinsurer. Under a mild condition, we propose a backward-forward optimization method that is based on a marginal indemnity function formulation. To show the applicability and simplicity of our strategy, we provide three concrete examples with the Value-at-Risk: one with the risk-adjusted value of the insurer’s liability, one with an objective function that follows from imposing Pareto optimality, and one with heterogeneous beliefs. We conclude this paper with an empirical application with Danish fire insurance losses and the Value-at-Risk and the Tail Value-at-Risk.