Optimal Insurance Pricing Based On Rank Dependent Utility Theory And Moral Constraints

As an important part of social development and progress,insurance plays an important role in ensuring social stability and promoting sustained and steady economic development.Through the research on the premium pricing of various types of insurance,insurance companies not only improve their competitiveness with peers in the market,but also provide a guarantee for the stable operation of the entire financial market.In the aspect of insurance,what is reflected is a kind of risk sharing,that is,a method to reduce risk by spreading the burden of risk among various parties.From the mathematical point of view,the risk sharing of insurance can be understood as an insurance contract designed between the insurer and the insured to achieve Pareto optimality.Taking motor vehicle insurance as an example,as the main business of each major insurance company,it is always faced with a series of problems such as high loss ratio,rising operating cost and insufficient profit.As the premium rate market reform accelerates,motor vehicle total premium volume is bound to decline,the traditional way rely on lower premiums and gain market share space more and more difficult to survive,in the context of severe market competition,how the insurance company as soon as possible out of the traditional business model,establishment of scientific and reasonable premium to achieve utility insurer and the insured both the advantage will become an important issue.In today's automobile market,most insurance companies adopt absolute deductibles max E{?-I(x)-C0[I(x)]}s.t.E{u[w0-?-x+I(x)]}? E[u(w0-x)].The optimal premium pricing is discussed under the condition that the compensation function I(x)is normally distributed.But here,in order to highlight the possible kurtosis of the compensation function,we change the distribution of I(x)following,and change the compensation amount function I(x)under the expected utility theory from the traditional normal distribution to the condition that the compensation amount function I(x)follows the Gamma(?,?)distribution,then the probability density function of I(x)is f(x,?,?)=q(??/?(?))x(?-1)e-?x,(x>0).The resulting equation for premium and deductible d isAccording to the results,it is not hard to find,insurance company's franchise system is absolutely can significantly reduce the number of small claims,reduce claims expenditures,but provides the post moral hazard of the insured motivation,because of the insurer shall bear the compensation and the insured shall bear the remaining losses on a global scale did not increase,so the insured can completely hide or exaggerate the real loss,cause the insurance to pay for high compensation,compensation cost greatly increase the insurance company.In addition,for the premium pricing method of the expected utility theory model applied by insurance companies,the weight function given by it is too linear and monotonous.Leading to the inability to explain many practical financial insurance phenomena such as Allais' paradox,which is not conducive to the calculation and effective operation of the optimal premium of insurance companies.Bernard,He,Yan and Xu et al,studied an optimal premium design problem,in which the individual preference is the type of RDU with rank dependence,showing an insurance contract with optimal deductibles.However,their results have two defects.In the first paper,the optimal insurance pricing is not considered when the loss x=0,and the insured has serious moral hazard in the premium pricing,which provides sufficient motivation for the insured to falsely report the actual loss,so that the insurance company may face high frequency or high amount of compensation.Second,the relation between the compensation function I(x)and the premium under the security additional coefficient is given without detailed discussion on the operating cost of insurance companies.This article to optimize the formulation of insurance contract,first consider the optimal deductible under expected utility theory,under the previous EU theory,change the probability density function of the sum of compensation to obey the Gamma distribution optimizing suggestions according to the results of insurance both sides,secondly,based on Bernard,He,Yan and Xu and others rely on utility theory to improve the rank,increasing C0[I(x)]said the insurance company for claim I(x)to claim the cost,(I(x)consists of fixed cost C0 and variable cost)#12To this end,we change the optimal premium pricing problem into the following level dependent utility theory description#12#12#12The following definition is given For EU(expected utility)theory,the weight function T(x)is congruent to x,then Here,we just assume that T is differentiable,thenndividual preferences are assigned to graded dependent utility RDU types in order to ncrease the indemnity function of the insurer and the retention function of the insured globally.In order to avoid possible moral hazard of the insured,we impose mora constraints to make both parties reach Pareto optimality at the same time,that is,under certain given conditions,any change can't make at least one person's situation worse,to achieve such a state of resource allocation,and then solve these problemsHowever,in the class of the utility of the deformation in the process of the insurance model,from the point of view of optimization,we have big problems,namely in the presence of the general weighted function T(x),even if u(x)is the concave,RDU preferences is not as concave,to this end,we use the quantile formulation and change the method of differential and integral calculus to solve the concave of the model optimization the key idea is to decision variables from the quantile functions of wealth W0 into itself,then solve the concave optimization problem,through the Lagrange theorem to characterize the optimal solution of the franchiseThe general necessary and sufficient conditions of the optimal solution are obtained Finally,Yarri criterion is used to simplify the problem,and the optimal insurance contract is obtained as triple contract,including small loss,large loss and compensation as constants of the median loss rangeThe form expression of the optimal compensation function I(·)for the above problem given[1]if ??{1+?)E[x],then I*(z)=z,(?)z ?[0,M][2]if ?c<?<(1+?)E[x],then#12 of which(d,e)is the only one that satisfies 0 ?d<a<e ?c,f(d)?f(e)and E[I*(x)]=?/(1+?)[3]if 0????c,then I*(z)=(?)where q satisfies c ?q and E[I*(x)]=?/(1+?)When the premium is small(0 ????c),the contract only compensates for larger amounts over a certain amountWhen the premium is in the medium range(?c<?<(1+?)E[x]),the contract is a triple contract,containing small and large losses Full coverage is provided when the insurance premium is large enough(??(1+?)E[x])Based on the above results,we use numerical simulation to discuss the figure of compensation function I(x)corresponding to loss x under different premium.Firstly through the different values of basic parameters,the insured's risk aversion coefficient and premium value are drawn,respective the image of the influence of the size of security additional coefficient on the insured's risk aversion is analyzed.Then give loss x and weighted function T? in the form of expression,in view of the ?=0.6,respectively,to discuss the premium value ?=1.5,?=3,?=4.5,different loss compensation function of x corresponds to the I(x)of images,when the premium ?==3 is obtained by image comparison,for the effectiveness of insurance both sides can achieve optimal at the same timeAt the end of this paper,we present the premium T=3,deductibles and based on the theory of the EU and RDU loss under the theory of relationship between x and I(x)compensation function image comparison,proved that under a range of loss,based on the RDU theory of insurance contract can better avoid the possible moral risk insured,but the corresponding data also show that the compensation function under EU theory I(x)of the optimal value of 1.219,RDU compensation function under the theory of the I(x)of the optimal value of 1.227,shows that the insurance company at the same time of avoid the risk of moral hazard.It also has to pay an extra 0.006 in compensation to balance out the additional costs and the increased share of compensation due to moral constraints.