A unifying approach to risk-measure-based optimal reinsurance problems with practical constraints

The design of optimal reinsurance treaties in the presence of multifarious practical constraints is a substantive but underdeveloped topic in modern risk management. To examine the influence of these constraints on the contract design systematically, this article formulates a generic constrained reinsurance problem where the objective and constraint functions take the form of Lebesgue integrals whose integrands involve the unit-valued derivative of the ceded loss function to be chosen. Such a formulation provides a unifying framework to tackle a wide body of existing and novel distortion-risk-measure-based optimal reinsurance problems with constraints that reflect diverse practical considerations. Prominent examples include insurers' budgetary, regulatory and reinsurers' participation constraints. An elementary and intuitive solution scheme based on an extension of the cost-benefit technique in Cheung and Lo [Cheung, K.C. & Lo, A. (2015, in press). Characterizations of optimal reinsurance treaties: a cost-benefit approach Scandinavian Actuarial Journal. doi:10.1080/03461238.2015.1054303.] is proposed and illuminated by analytically identifying the optimal risk-sharing schemes in several concrete optimal reinsurance models of practical interest. Particular emphasis is placed on the economic implications of the above constraints in terms of stimulating or curtailing the demand for reinsurance, and how these constraints serve to reconcile the possibly conflicting objectives of different parties.