On Modeling Insurance Claims using Copulas

In this master's thesis, a copula approach is used to model the number of claims made by a customer holding three insurances. It is important for insurance companies to have good models for the risk proles of their customers, and the number of claims is a key element in calculating the expected cost for the company. Using copulas, multivariate distribution functions are allowed to have any desired marginal distributions and many dierent dependence structures, as these can be chosen separately. The data used consists of the number of claims made by 74 770 unique customers during one year. Dierent count data distributions are considered for the onedimensional marginal distributions, while four Archimedean copulas are tested as models for the dependence structure. To estimate the parameters of the nal model, full maximum likelihood is used, for which new implementations adapted to discrete data were created. 2-tests and likelihood ratio tests determined that negative binomial distribution and zero-in ated Delaporte distribution were the best distributions for the onedimensional marginals, while Cramer-von Mises method and Kendall's Cramer-von Mises method, using a parametric bootstrap, together with Akaike's Information Criterion, suggested Clayton copula to be the most suitable. The obtained model is compared to the empirical values and to investigate how well the model ts for dierent years, it is also tted to the corresponding data from the following year. The model provides a good t both compared to the empirical values for the year used for inference as well as for the year used for validation. However, the t is strongly in uenced by the values in the lower tail. Keywords: Insurances, Copulas, Count data, Negative binomial distribution, Delaporte distribution, Full maximum likelihood, Goodness of t