Optimal Reciprocal Reinsurance With Constraints Under Distorted Risk Measures

Reinsurance plays a vital role in insurance activities because of its advantages of dis-persing risks,stabilizing operation and optimizing resource allocation.Because there are conflicting interests between insurers and reinsurers,and the signing of reinsurance con-tracts is decided jointly by both insurers and reinsurers.This paper studies the reciprocal reinsurance from the interests of both insurers and rcinsurers.In recent years,many scholars use risk measurement tools to study optimal reinsurance strategies.Because VaR and TVaR are two special cases of distortion risk measures,the expected premium principle and Wang's premium principle are two special cases of distortion risk premium principles.In order to make the models more general this paper studies the reciprocal reinsurance under the distortion risk measures with the distortion risk premium principles.In many optimization problems,the complexity of the optinization problem can be significantly increased by merely imposing a constraint.In particular,one often discovers that while an optimization procedure can be used to solve an unconstrained optimization problem analytically,the same procedure may no long be applicable when a constraint is imposed on the model.In this paper,we study two classes of optimal reinsurance models from perspectives of both insurers and reinsurers by minimizing their convex combination of the total losses where the risks arc measured by the distortion risk measures and the reinsurance premium is calculated by distortion premium principles.In the first place,we show that how to formulate the unconstrained optimization problem nd constrained optimization problem in a unified way.Then,we propose a geometrie approach to solve optimal reinsurance problems directly.This paper considers a class of increasing convex ceded loss functions and derives the explicit solutions of the optimal rcinsurance,which can be in forms of quota-sharestop-loss,change-loss,the combination of quota-share and change-loss or the combination of change-loss and change-loss with different retentions.Finally,we consider two specific cases of the distortion risk measures:Value at Risk(VaR)and Tail Value at Risk(TVaR).