The Optimal Insurance Contract Design Under Risk Constraints

In recent years,with the development of the insurance industry,the academic study of insurance is booming,the topic on "optimal insurance" has been paid more and more attention.Current academic researches focus on the optimal insurance mainly for the insured,namely,to maximize the expected utility of the insured,and obtain different form of optimal insurance under different premium calculation principle,but little attention has been paid on the restraint of the insurer’s risk exposure.As a special financial institution,which manages risk,risk management and control is one of the most concerns for insurance company.Therefore,the insurance company should not only maximize the insureds’ wealth utility,but also control its risk within a certain range in the process of designing the optimal insurance contracts.In this paper,we impose the insurer’s risk constraint on Arrow’s optimal insurance model.This article is divided into seven chapters,the first chapter is introduction.Firstly,we have introduced the insurance basic concept,the insurance premium standards as well as the rudimentary knowledge related to insurance such as criterion for expected utility and risk attitudes,etc.Then we have summed up the previous studies about optimal insurance on the basis of the expected utility theory,and for further discussion.In this paper,we extend the work of Arrow model and impose the insurer’s risk constraint on the Arrow’s optimal insurance model.The insured aims to maximize his/her expected utility of terminal wealth,under the constraint that the insurer wishes to ensure the expected earnings of his/her terminal wealth which cannot below some specific level.In the third chapter,we set the risk constraints in the form of the value at risk.In the fourth chapter with reference to the Zhou(2007)who control s insurance external expected loss within a certain tolerance,we set the risk constraints in the form of expectations.In the fifth chapter,we set the risk constraints in the form of absolute maximum loss.In most of the previous literature,little attention has taken underwriting business and investment business into consideration in the optimal insurance contract design.Therefore,we will take the comprehensive consideration of underwriting risks and investment risks,designing the optimal insurance contract under the constraint condition to maximize the wealth utility of the insured.On this basis,we establish the corresponding mathematical model,and discussed and analysis the model in detail.In the fourth chapter of this paper,the optimal insurance model is applied to the vehicle insurance market and medical insurance market.It is shown that the insurer will not only increase the compensation for small loss of the insured,but also increase the compensation for large losses of the insured under the VAR risk constraint model,when the big risk happens,the underwriter’s risk exposure is the largest among these three models.Under the expected risk of loss constraint models,the insurer may increase the compensation for small loss of the insured,but reduce the compensation for large losses of the insured,which make insurer’s risk exposure more strictly compared with the VAR risk constraint model.If the solution of Arrow model satisfies the insurer’s risk constraints,the increase of insurer’s risk tolerance has no effect on the insured’s expected utility.At the same time,if the solution does not satisfy insurer’s risk constraints,and there exists the optimal solution,the insured’s optimal expected utility will increase if the insurer increases his/her risk tolerance.Under the absolute maximum loss risk constraint model5 the underwriter will increase the compensation for small loss of the insured,but compensate losses that more than limits.That is to say,the insurer will only compensate for the costs of the insured within the coverage limit,and will not compensate the part that exceeds the insured amount limit.It is clearly that the absolute maximum risk constraint model is more realistic than the expected risk of loss constraint model and the Var model.