Optimal Reinsurance Under The Concave Distortion Risk Measure

In recent years, minimizing some general risk measures has been one of topic on reinsurance. The insurer can control its risk by ceding part of its risk to a reinsurer. So the insurer will choose the optimal reinsurance to minimize the risk by balancing claim and reinsurance. There are three criteria for value the measure of reinsurance: minimizing risk, maximizing utility, minimizing the ruin probability.With the development of reinsurance industry, minimizing the general risk measure has been introduced to the research of optimal reinsurance issue.Moreover motivated by capital requirement of Basel II agreement, some studies on optimal reinsurance focus on Value-at-Risk(VaR) and Tail Value-at-Risk(TVaR) as their risk measures. With the reinsurance premium calculated by expected premium principle, by minimizing either VaR or TVaR of the insurer’s total risk, Cai et al.(2008)showed that stop-loss reinsurance is optimal when the ceded loss functions are convex and increasing, and Cheung(2010) obtained the same results by a geometric approach. Cheung(2010) also discussed the VaR optimization problem under Wang’s premium principle. Tan et al.(2011) studied TVaR optimization problem under an enlarged class of ceded loss functions. Chi and Tan (2011) and Chi and Tan (2012) investigated the VaR and TVaR optimization reinsurance models by focusing on different classes of ceded loss functions and different premium principle. Cui and Yang (2013) showed the optimal reinsurance minimizing the distortion risk under general reinsurance premium principles.This paper will focus on the following two optimization problems:· The optimization problem without premium constraint· The optimization problem with premium constraintTo facilitate the discussion of the optimal ceded loss function in (?) of the above optimization problems, we introduce the following class of ceded loss functions: By analyzing, we prove that the optimal ceded loss functionsh*in (?),whichminimize the concave distortion risk measure of the total costTh(X) forh∈(?), alsooptimally minimize the concave distortion risk measure of the total cost Tf(X) for Then the resulting total cost or the total risk exposure of the insurer in thepresence of reinsurance:Given a confidence levell-α,0<α<1, lead toThen it follows from the translation invariance property of VaR that:and by definingit is easy to show: Thus So by seeking extreme value method, show the optimal ceded loss function of theoptimization problem with premium constraint and without premium constraint.This is the important results of the paper: and the minimumTheorem2.For the concave distortion risk measure optimization problem (2.3), then f*(x) is one optimal solution; otherwise,...