Risk transference constraints in optimal reinsurance

Abstract

This paper will deal with the optimal reinsurance problem and will involve the goals of both insurer and reinsurer. In particular, the study will incorporate the initial (before reinsurance) risk that the reinsurer uses in order to diversify (or hedge) the risk ceded by the insurer, general methods to prevent the reinsurer moral hazard will be extended, and a new constraint will have to be satisfied by the selected reinsurance contract, namely, "the reinsurer increment of risk will have to be lower than the contract premium". Simultaneously, since the contract must be attractive to the insurer too, "the contract premium will have to be lower than the insurer risk reduction". Integrating both ideas, "the contract premium will have to be higher than the reinsurer risk growth and lower than the insurer risk mitigation". Bearing in mind both requirements, that is, the protection against the moral hazard and the spread containing the contract premium, the optimal reinsurance problem will be studied under very general conditions about the risk measures and premium principles involved, general solutions will be provided, and a particular case with the conditional value at risk will be presented in more detail.