Optimal Reinsurance Under The Standard Deviation Calculation Principle

This paper concerns the problem of how to purchase the reinsurance in order to make the insurer and the reinsurer's total risk least under the standard deviation calculation principle. Sufficient conditions for optimality of reinsurance contract are given within a class of admissible contracts under the standard deviation calculation principle. It is assumed that the reinsurer's risk must be less than a given constant and in such a condition the optimal reinsurance contract in order to minimize the insurer's risk is found out. Here we adopt variance risk measure, semivariance risk measure and L₁ risk measure to survey the insurer's risk respectively and find out sufficient conditions for optimality of reinsurance arrangement of the three risk measures above. This paper also provides optimal solutions under the expectation calculation principle.There are three parts in this paper:1.The introduction of reinsurance, including the kinds of reinsurance strategy and premium calculation principles.2. In this chapter we introduce Convex function, Gateaux derivative and subgradient, which are useful for the proof of Lemmas and Theorems.3.This chapter is the core of the paper, we obtain Optimal reinsurance under the standard deviation calculation principle under some mild conditions and provide sufficient conditions for optimality of reinsurance contract. We also give some examples to illustrate our results. In this chapter, change loss reinsurance arrangement is the optimal strategy when the risk measure of the insurer taken by variance risk measure or semi-variance risk measure, which implies the importance of change loss reinsurance arrangement.4.In this chapter we compare the optimality of the three risk measures above and obtain the optimal arrangement under the expectation calculation.This paper gives theories with proof and provides some numerical to illustrate the results, may be it can make sense to reinsurance in practice to some extent.