Experience Rating And Their Applications Of Risk Premiums In Non-Life Insurance And Actuarial Science

In the insurance and actuarial science,risk is defined as the loss that an insured may suffer.In order to transfer this uncertainty to the insurance company,the policyholder needs to pay a fixed amount money or premium.For policyholders,they want to pay as little as possible under the condition that risk is transferred as much as possible.For insurance companies,it is necessary to formulate an appropriate premium pricing strategy based on the characteristics of the risk itself.If the premium rate is too low,the insurance company will suffer large losses or even go bankrupt.If the premium rate is too high,the insured will choose other insurance companies that can provide lower premium,causing the insurance company to lose the policy.Therefore,premium pricing is one of the most important tasks for actuaries in insurance companies.In probability theory and mathematical statistics,the non-negative random variables are often used to characterize risk.The probability distribution of a random variable fully characterizes all the properties of the risk.At this time,premium is defined as a function of the set of non-negative random variables to the set of non-negative real numbers.Commonly used premium calculation principles include expected value premium principle,variance premium principle,standard deviation premium principle,modified variance premium principle,exponential premium principle,Esscher premium principle,Kamps premium principle,Dutch premium principle,and so on.For a given premium calculation principle,the actuary needs to accurately estimate the premium based on the available data,so that the premium can precisely reflect the characteristics of the risk.Then the insurance company can operate normally and be more competitive in the market.There are two types of information available in the non-life insurance premium estimation process.One type of information is the loss observations of the insured over several years,and the other type of information is the prior information formed by the same industry premium data or the actuary's knowledge.In early actuarial science,actuaries established the Bayesian model of policy losses,and obtained the estimators of premiums under the squared-error loss function by limiting the estimators of net premium to a linear function of the samples.The result is exactly the weight of experience premium and collective premium,where the weight is called credibility factor.In multi-contract policy data,an empirical Bayesian method can be used to obtain an estimate of the credibility factor,thereby an empirical Bayesian estimate of future premium is derived.This method of premium pricing is also known as credibility theory or experience rating technique.However,from the point of mathematical statistics,the principle of net premium is actually the conditional mean of risk under given risk parameters.Therefore,the net premium principle only reflects the mathematical expected characteristic of risk,and has nothing to do with the volatility of random variables such as variance.More importantly,the net premium principle cannot meet the positive safety-loading of the premium calculation principle.Therefore,the early credibility theory cannot be directly applied in practice.In modern credibility theory,there have been more literatures discussing the credibility estimation and experience rating of risk premiums in some commonly used premium calculation principles.From the research methods and results of the existing literatures,most of them are based on the loss function method to study the experience rating of risk premiums in some special premium calculation principles.However,this method cannot be extended to other premium calculation principles.On this basis,this paper studies the credibility estimation of risk premium and its experience rating under the premium calculation principles such as Esscher's premium principle,expected utility principle,and moment-related premium principle and so on.This paper not only studies the experience rating of risk premiums in most of the commonly used premium calculation principles in non-life insurance,but also gives a unified experience rating method for certain premium calculation principles.Because the premium calculation principles discussed in this article all have a positive safety-loading property,they can be used directly in practice.The research in this article can provide new ideas and theoretical basis for the decision-making process of actuaries.In summary,the research content of this article includes the following parts.(1)The classic credibility theory model is introduced in detail,and the efficiencies of the maximum likelihood estimation,the Bayesian estimation and the credibility estimation of risk premium under the net premium principle are compared.The premium principles are commonly used in non-life insurance and their characteristics are introduced.Based on the expected value premium principle and Va R premium principle,the optimality of premium estimation is studied by numerical simulation and Bootstrap method.Taking the exponential premium principle as an example,the hypothesis test of the change point of the premium is discussed.The estimation of the position of the change point is given,and the convergence speed of the change point estimation is proved.(2)The experience rating of risk premium under the Esscher premium principle is studied.In fact there have been more research literatures on the credibility estimation under the Esscher premium principle.Different from the traditional research methods,by analyzing the structure of risk premium,the risk premium is divided into two parts.Using the idea of credibility theory to linearly estimate independently,and the credibility estimation of risk premium is obtained by minimizing the weighted expected squared loss.Compared with the existing research conclusions,the consistency of the credibility estimates obtained by this method is easily proved,and the structural parameters are easy to estimate.The results of numerical simulations show that the estimates obtained in this paper have smaller mean square error than existing estimates.Furthermore,the actual data of insurance companies are used to verify and compare the results of this paper.(3)Combined with the expected utility principle in economics,the expected utility premium principle is defined.In addition,the definition of risk premium under the expected utility premium principle is given.Furthermore,the Bayesian estimation and the two forms of credibility estimation of risk premium are given,and their consistency and asymptotic normality are proved.In the multi-contract policy combination model,the estimation of structural parameters is proposed,and the empirical Bayesian estimation of risk premium is obtained.By choosing different utility functions,the results of this part can be seen as a generalization of the credibility estimation under net premium principle,exponential premium principle and Esscher premium principle.The results of simulation show that the credibility estimation not only have good statistical properties,but also are easy to interpret.The estimated efficiency is verified by using the data of the third party liability insurance of an insurance company.(4)This paper proposes a new premium calculation principle—the moment-related premium principle,which includes at least six commonly used premium calculation principles.In addition,a Bayesian model of risk premium is established,and the credibility estimators of the moment generating function are obtained by combining the ideas of credibility theory.Finally,the credibility estimators and experience rating of risk premium under the moment-related premium principle are obtained.The research in this part unifies the experience rating method for six commonly used premium calculation principles.In the numerical simulation,the credibility estimates under net premium principle,variance premium principle,Esscher premium principle,and exponential premium principle are compared with already existing estimates in other literatures.The conclusion shows that the estimators obtained in this chapter not only have good statistical properties,but also have strong practicability.Furthermore,the credibility estimation method of the moment generating function is applied to the estimation of distribution function.Therefore the credibility estimators of distribution function and distortion risk premium are obtained.Under the distortion risk premium principle,the large sample properties of the two credibility estimators are proved,and their efficiencies are compared.(5)The experience rating method of the claim amount is extended to the aggregate risk model.It is assumed that the amount of claims and the number of claims depend on different risk parameters,then a joint Bayesian model of aggregate risk is established.Based on the principle of variance-related premiums,the risk premium for the total risk,the corresponding Bayesian estimator and credibility estimator are derived.Furthermore,in the Poisson aggregation risk model,the Bayesian estimation and credibility estimation under the variance-related premium calculation principles are discussed.The mean square errors of Bayesian estimator and credibility estimator are compared by numerical simulation methods,and the statistical properties of these two estimators are verified..(6)The application of the experience rating method in the liability reserve is researched.A Bayesian model of the random progress factor in the liability reserve is established,and the credibility estimator of the random progress factor is obtained.Finally,the credibility estimator of the liability reserve is obtained.Compared with the traditional chain ladder method,the research in this paper assumes that the progress factor is a random variable,and the estimated credibility reserve not only uses the prior information of the progress factor,but also does not require the specific distribution assumptions of the sample distribution and the prior distribution.In addition,the results can be taken as a generalization of the non-randomized chain ladder method.The simulation results show that the results in this paper are more accurate than the traditional chain ladder method.