Optimal reinsurance combination of quota-share and stop-loss reinsurance based on conditional-tail-expectation (CTE) optimization

Abstract

To maintain financial stability and to effectively manage risk, an insurer will partially reinsure the loss to a reinsurance company. Two of the most often used reinsurance contracts are quota-share and stop-loss. In quota-share, the loss will be split based on a fixed proportion and the reinsurance premium depends on the value of the proportion, while in stop-loss the loss will be split depending on the retention value. In the hope that these two types of reinsurance can cover each other weaknesses, this study combines both quota-share and stop-loss reinsurance. Subsequently, to get a good coverage for the insurer, it is necessary to find the optimal proportion and retention value. One way to do so is using risk measure optimization. The smaller the value of the risk measure, the smaller the loss borne by the insurer. The risk measure used in this paper is Conditional-Tail-Expectation (CTE), where it involves Value-at-Risk (VaR) in its calculation. Calculated using the expected value principle, the reinsurance premium is used as a constraint in the CTE optimization for each of the reinsurance combinations, which are stop-loss after quota-share and quota-share after stop-loss. By optimizing CTE, it is found that each combination produces the same minimal CTE, so both reinsurance combinations are optimal for use by the insurer. By using different distributions, it is seen that the minimal CTE depends on the distribution’s tail behavior. Furthermore, in determining the minimal value, the conditions that are used in optimization using CTE are different from VaR.