Optimal Reinsurance Based On Stop-loss Model And Quota-share Model

Reinsurance policy is a very common tool to be used to protect insurers from large risk and to help insurers obtain profits. Based on different criterions, Reinsurance products can be classified into various categories. For instance, one reinsurance productcan be divided into life reinsurance and non-life reinsurance according to its risk coverage, and can also be divided into proportional reinsurance and non-proportional reinsurance based on its limitation of responsibility. There are a lot of risk measures too, like variance, value at risk, coefficient of variation, ruin probability and so on. Among the various reinsurance products and many risk measures, how to choose the best one to make the reinsurance policy reach its full potential in diversification of risk and get balanced between profit and risk becomes very important. This is the problem of reinsurance optimization.Reinsuranceoptimization includes two parts, one is to choose the best model to explain the product, and the other is to solve the parameters that make the model get optimized. This paper mainly talks about the latter one. During the whole discussion, we only talk about the optimization issue on two common reinsurance policy, stop-loss reinsurance and quota-share reinsurance. First of all, we build two different models for these two reinsurance, and the differences mainly lie in the number of risk types in assumption and the risk measure we apply. More specifically, we assume there is only one type of risk in the stop-loss reinsurance, and we apply value at risk to measure the risk in stop-loss reinsurance. Differently, for quota-share reinsurance, we assume there are more than one type of risks and we use variance to measure the risk. Based on different assumptions, we build two models, optimal stop-loss reinsurance model and optimal quota-share reinsurance model. Then, we solve the two models and get the explicit solutions of the parameters that make the two models optimized. According to the optimal solutions of the two models, we can get the conclusion that all the parameters appeared in no matter stop-loss reinsurance or quota-share reinsurance are determined by the distribution of the original risk before reinsurance policy and the relative safety loading introduced in the calculation of the reinsurance premium. It’s worth noting that for stop-loss reinsurance, there are some pre-conditions for the existence of the optimal parameters, and the condition puts some constraints on the confidence level of value at risk and the relative safety loading introduced in the calculation of reinsurance premiums.