Optimal reinsurance for both an insurer and a reinsurer under general premium principles

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A reinsurance contract should consider the conflicting interests of the insurer and the reinsurer. An optimal reinsurance contract for one party may not be optimal for another party and it might be unacceptable for another party. Therefore, in this paper, we study the optimal reinsurance models from the perspective of both the insurer and the reinsurer by minimizing their total costs under the criteria of loss function which is defined by the joint value-at-risk, assuming that the reinsurance premium principles satisfy risk loading and stop-loss ordering preserving. We derive the optimal reinsurance policies over three ceded loss function sets, the change-loss reinsurance is optimal among the class of increasing convex ceded loss functions; when the constraints on both ceded and retained loss functions are relaxed to increasing functions, the layer reinsurance is shown to be optimal; the quota-share reinsurance with a limit is always optimal when the ceded loss functions are in the class of increasing concave functions. We further use the expectation premium principle and Dutch premium principle to illustrate the application of our results by deriving the optimal parameters.