The Research Of Credibility Premium With Relative Entropy

Premium pricing is very important in the insurance industry. The credibility theory is usually used when the insurance company determines the premium, and the main method of the credibility theory is to predict or just the future premium of the claim in the past. In the 1960 s, B ü hlmann limited the claim data in the past into some kind of linear function and used the least square method to set up the credibility theory with free distribution, which is called the classical credibility theory. Then the credibility theory developed quickly based on this basis.In classical credibility theory and modern theory, the future premium is often obtained by using the least squares method to optimize the expected loss function. Consequently, the main method of the present paper is divided into two steps:(1)The history claims data is limited into some kind of linear function(For example, δ(X1, · · ·, Xn) =α0+n∑j=1αjXj). Optimize it for the ?rst time by minimizing the following expected loss function minαi∈R E[(δπ- α0-n∑j=1αjXj)2;(2) Based on the ?rst optimization, apply the relative entropy method to optimize for the second time. Thus, the expression of the future premium is established and the rationality of the credibility premium is explained.In addition, this kind of premium is the further optimization to the Bayesian estimator.If the insurance company apply the least square method to depict the degree between the premium and the risk, then the premium calculated does not have the property of a safety load. The exponential premium principle is one of the most important premium principle due to its simple mathematical expression and good properties. Thus, many researchers have discussed it before. Consequently, the exponential premium principle is introduced and the expression of the premium estimator applying the relative entropy method under the exponential premium principle is obtained in the third section.At last, the researchers only discussed the precision of the estimation and didn’t discuss the ?tting degree of the estimation in the classical credibility theory. In order to consider both precision and the ?tting degree of the estimation, the balanced loss function is introduced and the corresponding credibility formula is obtained applying the relative entropy method in the fourth section. The credibility premium formula under squared loss function is generalized further.