Risk Constraint and Optimal Insurance:An Insurance Contract That Better Meets Expectations of the Insured
For a long time,an optimal insurance design has always been a hot and difficult issue in insurance theory research,and has attracted widespread attention from theoretical and industrial circles.The pioneering research by Arrow,a Nobel laureate in economics,provides a model basis and research ideas for optimal insur-ance design.He assumes that risk-neutral insurance companies charge excess premiums in line with the current development level of the insurance market under the principle of expected premiums,while the insured belong to the risk-averse type and have a von Neumann-Morgenstern utility function,and designs insurance products according to the maximum expected utility of the insured.However,Arrow's research and subsequent related studies ignore the risk constraint needs of the insured.In reality,the insured usually hope to obtain sufficient com-pensation from the insurance company after an accident and control their own losses within their expected accepta-ble range.Therefore,if Arrow's insurance cannot meet the risk constraint needs of the insured,how to design an insurance contract that meets the risk constraint needs of the insured?This issue needs to be further studied.On the basis of the Arrow model,when the loss of the insured is no more than a certain non-negative value,the net loss constraint of the insured is introduced to study the optimal insurance problem of the insured.This is because:(1)Setting the net loss constraint of the insured in the partial range rather than the total loss range is to achieve utility improvement.(2)Setting the net loss constraint of the insured in the low loss range rather than the high loss range,the optimal insurance contract can motivate the insured to avoid risks reasonably.In addition,the constructed model also includes the continuity of the compensation function,which prevents insurance com-panies from refusing to provide insurance due to concerns about the moral hazard of the insured.According to the research ideas of RAVIV(1979)and GOLLIER(1987),the model is solved in two steps.First,we study the optimal insurance contract with fixed premium under the assumption of fixed premium,and then let go of the assumption of fixed premium to further study the optimal insurance contract with general premium.The study shows that if the solution of the Arrow model satisfies the net loss constraint of the insured,then the solution of the Arrow model is the solution of this model,and the optimal policy is a partial insurance contract with only one deductible.Otherwise,there will be a special solution to this model,and the optim al pol-icy is a partial insurance contract with two deductibles.Drawing on the research methods of MA Benjiang and JIANG Xuehai(2024),this paper also proves a sufficient condition that the excess premium is strictly positive for the deductible by using the intermediate value,and according to the intermediate value,the key quantitative characteristics of the special solution of the model are obtained.For example,when the utility of the insured is optimal,the first deductible should be strictly less than the second deductible,and the sum of the optimal premi-um and the first deductible should be equal to the upper limit of the net loss of the insured,but the sum of the optimal premium and the second deductible should be strictly greater than this upper limit,and so on.In addi-tion,the utility of the insured is related to the upper limit of net loss and the cut-off point of small loss,that is,the expected utility of the insured will increase with an increase in the upper limit of net loss and a decrease in the cut-off point of small loss.However,when the upper limit of net loss increases or the cut-off point of small loss decreases to a certain extent,so that the solution of the Arrow model satisfies the net loss constraint of the low loss interval of the insured,the utility of the insured reaches the maximum and will not be further improved.Future research can be expanded from the following two aspects:(1)Since Arrow's optimal insurance is a deductible insurance contract,the maximum loss of the insured is the deductible.However,in the proportional insurance contract,the net loss of the insured will increase with an increase in the loss,so the introduction of the net loss constraint of the insured in the deductible insurance contract will have greater research value.(2)Since the risk-neutral assumption of insurance companies,the expected utility function of the insured,and the calcula-tion principle of expected premiums all have certain limitations,under the assumption of insurance companies'risk aversion,introducing other more reasonable and complex premium principles and the expected utility func-tions of the insured to build models will greatly enrich and deepen this study.
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