The Application Of Kalman Filter In Credibility Models With Risk Dependence

Credibility theory is an important technique towards ratemaking in non-life insurance practice. The classical credibility theory adjusted future premiums based on the insurer’s risk characteristics and past experiences, which can be written by:Credibility premium = Z × experience +(1- Z) × collective premium,where Z ∈ [0, 1] is called the credibility factor. Bühlmann(1967) did lots of remarkable contributions to the modern credibility theory, he derived a credibility formula which has the form of weighted sum of the individual mean and the overall mean in a distribution-free way by using a least-square criterion and thereby placed credibility ratemaking on a firm modeling foundation.Currently the calculations of the credibility premiums generally use Bayesian estimation methods, because there are strong connections between Bayesian theory and Kalman filter, so Kalman filter can be introduced into credibility theory, and provide a new approach towards credibility premiums calculating–Kalman filtering.In classical credibility models, a common assumption is that all the risk individuals random vectors are independent, however, there existed some corrections among all risks in some practical circumstances. This paper study the model which with dependence induced by common effects(such as earthquake and financial crisis et al.) The recursive forecasts of future premiums were generated under Kalman framework. The recursive result showed that the credibility estimation was obtained by Kalman filtering still has weighted form like classical credibility premium, and it was consistent with which was obtained by means of orthogonal projection method,so, Kalman filter can be introduced into credibility theory, and provide a new approach for calculating credibility premiums, and this model can be viewed as an extension for the classical credibility model.On the other hand, in order to avoid bankruptcy, the insurance company needs to consider the positive safety loading of the premium, but the net premium is not appropriate in this situation. So, we introduce the exponential premium principle,and consider the credibility model with equal correlation over risk under exponential premium principle. At last, the error was considered, the credibility model with error uniform dependence also with equal correlation over risk under exponential premium principle dependence was studied under Kalman filter framework. The recursive results showed that Kalman filter is also suitable for the model, and the model also can be viewed an an extension for the classical credibility model.