关键词: 最优保险问题, 净损失约束, Arrow模型, 期望效用

Abstract: Optimal insurance design has always been a hot and difficult issue in insurance theory research. The Arrow model, as a classical model to study the insured’s optimal insurance problem, has been widely used in theoretical and practical circles. Since the Arrow model was put forward, the improvement and optimization around it have never stopped. The improvement and optimization does not mean the utility improvement in information economics, but the design of a more realistic and reasonable insurance model to fully reflect the actual needs of the insured, so as to improve their enthusiasm for purchasing insurance. However, the existing research rarely pays attention to the need of the insured’s risk constraint, and cannot guarantee the expected compensation level of the insured. In the case of incomplete insurance, the insured usually hopes to get enough compensation from the insurance company after loss occurs, so that their actual loss can be controlled within the expected acceptable range, which will be more in line with their original intention of insurance purchase for risk transfer and the types of risk avoidance of most insured. Therefore, this paper intends to set the insured’s risk constraint conditions on the basis of the Arrow model, and study the optimal insurance problem with net loss constraint in the high loss interval of the insured. The designed insurance contract can effectively meet the needs of the insured’s risk constraint and deepen the social management function of insurance transfer risk. Therefore, this study is of far-reaching significance not only for the theoretical expansion of the Arrow model, but also for the long-term development of the insurance market.
   Firstly, the model is solved from fixed premium to general premium. It is pointed out that if the solution of the Arrow model satisfies the net loss constraint of the insured, the solution of this model is the same as that of the Arrow model, and the optimal policy is a partial insurance contract with only one deductible. Otherwise, the model will have a special solution, and the optimal policy is a partial insurance contract with two deductibles. For the special solution of this model, we use an intermediate value to prove a sufficient condition that the excess premium is strictly positive, and thus obtain the key quantitative characteristics among the relevant variables of the optimal insurance contract. Because the intermediate variable has significant economic meaning (that is, the change of deductible at this point will cause the reverse equal change of premium), the above proof method is universal and can be popularized. Then, comparing the special solution of this model with the solution of the Arrow model, we find that when the insured utility is optimal, the compensation level provided by this model is always not lower than that of the Arrow model when dealing with high losses, while the compensation level provided by this model for IARA (/DARA/CARA) insured is lower than (/higher than/equal to) the Arrow model in turn when dealing with low losses. In addition, when the solution of Arrow model is the solution of this model, the utility of the insured will be optimal. Moreover, the expected utility of the insured will gradually increase with the increase in the specific value of loss and the upper limit of net loss. The difference is that the solution of the Arrow model is the solution of this model until the former is close to positive infinity, while the latter only needs to be increased to a certain extent so that the solution of the Arrow model satisfies this constraint, the solution of the Arrow model is the solution of this model, and then the utility of the insured is optimal and no longer increases.
   Finally, it should be pointed out that the introduction of the net loss constraint of the insured into the Arrow model may face the moral hazard problem of the insured. The Arrow model describes deductible insurance. Obviously, introducing this constraint into proportional compensation insurance will have more research value, and there will be no moral hazard problem of the insured, so this is an important direction of future research. Considering that the Arrow model plays an important role in the research of insurance theory, the next research can also consider introducing the upper limit of the expected net loss of the insured into the Arrow model. On the one hand, it can guarantee the expected compensation level of the insured as well as this model, and on the other hand, there is no moral hazard problem of the insured, so this is also an important direction of future research.

Key words: optimal insurance, net loss constraint, Arrow model, expected utility