No-life Insurance Pricing Research Based On Information Entropy Method

The insurance operation includes life and no-life insurance, in which both are relate to actuarial problems. The life actuarial science has been very perfect due to the stability of the loss distribution. The quantitative analysis on no-life actuarial science is more difficult than the one on life actuarial science because the former relates to more stochastic factors. So, the no-life actuarial science becomes the core problem that has being researched. On the other hand, the no-life insurance pricing problem, gotten the extensive recognition for many years, is one of no-life insurance actuarial science and is very important. The main object of this thesis is to analyze risk under incomplete information and give the reasonable price of insurance product to the question of no-life insurance pricing problem in insurance market.The main tools are the information entropy methods in this thesis. Based on the essence of entropy, the maximum entropy principle, the minimum cross-entropy principle and the large deviation entropy, the no-life insurance pricing problems including risk analysis and insurance pricing are researched from the different aspects. The main investigations and achievements are composed of following portions.1. Risk analysisWhen the incomplete information of loss distribution is obtained, the estimate of the maximum entropy loss distribution and the value of entropy function are acquired by the maximum entropy principle under these information constraints. Based on the essence of uncertainty, entropy is introduced into the risk measure. Entropy and variance are reinforced each other and used to decide to the size of risk. The numerical example examines the necessity of entropy risk measure. Based on the research, entropy is added to actual effect premium principle and the new variance-entropy premium principle is presented. The result indicates that the entropy of probability distribution is related to higher moment information and it changes with the moment information. Entropy risk decision embodies the more information of loss variable. Furthermore, the solution of the maximum entropy optimization model under moment information is easily obtained. So, the calculation difficulty of the adjusted premium principle is not added and the new premium principle embodies more insurance pricing information because of the existence of entropy. If the prior information on the loss distribution is obtained, the minimum cross-entropy transform is established by the minimum cross-entropy principle. The result of the minimum cross-entropy transform gives more weight to unfavorable events, which embodies the essence of risk change. The minimum cross-entropy transform is correlated to Esscher transform closely and their relationship is built. Based on the relationship, the new minimum cross-entropy transform way to deal with loss risk change is presented because Esscher transform usually used to deal with the risk change of aggregate risk. On the other hand, the useful information of the loss distribution contained in the current premium can be distilled by the minimum cross-entropy transform model. The results indicate that Esscher transform is the special minimum cross-entropy transform. The constraint, the transformed exception is larger than the mean loss, must be satisfied in order to obtain the risk compensate. The minimum cross-entropy transform embodies the insurer's attitude to loss risk more intuitively and is more convenient.2. Insurance pricingIn the incomplete insurance market, based on the risk-neutral pricing ideal, the minimum cross-entropy or the maximum entropy optimization principle is used to select the only unprejudiced risk-neutral density under the risk-neutral constraint and other market constraints. The minimum cross-entropy or maximum entropy risk-neutral premium is established. Further, the numerical example shows that the application of minimum cross-entropy or maximum entropy optimization principle is simple and convenient, which make the new premium principle more advantaged. The results indicate that the form of the constraint of risk-neutral is flexible in the built minimum cross-entropy or maximum entropy optimization model and the solution of the model can be obtained easily, which must be convenient. Under the prior information on loss variable, the risk loading disappears in the minimum cross-entropy or maximum entropy risk-neutral premium. So, the method is no parameter. Otherwise, this premium principle has instructional significance to insurant and insurer.The rate function, which is the cross-entropy to prior distribution, is very important in large deviation theory. Based on the Shannon entropy and Jaynes maximum entropy and large deviation entropy, the signification of information theory of large deviation rate function is explained. Thus, the rate function can be obtained easily by the entropy optimization principle. Otherwise, the large deviation probability measure is deduced from the cross-entropy form of large deviation rate function of probability distribution. Based on the large deviation probability measure, the insurance pricing problem is researched. The result indicates the premium based on the large deviation probability measure can estimate the large deviation probability for the different numbers insurance portfolio, which is the advantage that does not belong to other premium principle.