Computing and Approximation Methods for the Distribution of Multivariate Aggregate Claims

Insurance companies typically face multiple sources (types) of claims. Therefore, modeling dependencies among different types of risks is extremely important for evaluating the aggregate claims of an insurer. In the first part of this thesis, we consider three classes of bivariate counting distributions and the corresponding compound distributions introduced in a 1996 paper by Hesselager. We implement the recursive methods for computing the joint probability functions derived by Hesselager and then compare the results with those obtained from fast Fourier transform (FFT) methods. In applying the FFT methods, we extend the concept of exponential tilting for univariate FFT proposed by Grubel and Hermesmeier to the bivariate case. Our numerical results show that although the recursive methods yield the exact compound distributions if the floating-point representation error is ignored, they generally consume more computation time than the FFT methods. On the other hand, although FFT methods are in general very fast, they suffer from the so called alias error. However, the alias error can be effectively reduced via the introduced exponential tilting. Therefore, the FFT methods constitute viable alternatives to the recursive methods for computing the joint probabilities. In the second part of the thesis, we introduce a multivariate aggregate claims model, which allows dependencies among claim numbers as well as dependencies among claim sizes. This model makes practical sense because insurance companies typically write multi lines (types) of insurance policies and the claims from different lines of businesses are usually dependent. For example, in auto insurance, insurance companies have to pay claims due to property damages and bodily injuries. The numbers of claims from property damages and bodily injuries are typically dependent. In addition, one would expect that the sizes of the two types of claims are dependent because some accidents cause two types of claims simultaneously. For this proposed model, we derive recursive formulas for the joint probability functions of different types of claims. In addition, we show that the concepts of exponential tilting in the multivariate FFT can be applied to compute the joint probability functions of the various types of claims in the introduced multivariate aggregate claims model. Numerical examples are provided to compare the accuracy and efficiency of the two computation methods. In the third part of the thesis, we apply a moment-based technique to approximate the distribution of univariate and bivariate aggregate claims. The numerical examples presented herein indicate that the proposed approximation method constitutes another viable alternative to the recursive and FFT methods