A Copula Approach to Modeling Insurance Claims

Abstract: It is crucial for the insurance business to create risk profiles for their customers to be able set a fair price for their insurances. This thesis presents an alternative method for measuring the dependence between the numbers of claims made by a customer holding two insurances. The group of insurance holders consisted of 74770 unique customers holding at least two insurances during one year. The method uses copulas to model the dependence between the numbers of claims made in each insurance. An advantage using copulas to model the dependence is that the margins and the dependence structure can be modeled separately. The copulas used in the analysis were four Archimedean copulas namely Clayton, Frank, Gumbel and Ali-Mikhail- Haq. These copulas were chosen for their simple explicit expressions and variety in dependence structure. A flag has to be raised regarding the fact that the margins were discrete and the largest part of applied copula theory handles continuous margins. This led to complications in the modeling that were not expected. The marginal parameters were estimated using ML-method and a χ2-test decided that the negative binomial distribution respectively the zero-inflated negative binomial distribution were the best fits for the insurances. Regarding the copula, the parameter was estimated using inverse Kendall's tau. By performing a parametric bootstrap using Craḿer von Mises method the Gumbel copula was shown to provide the best fit. When a bivariate distribution was obtained from the model, it was compared to the empirical counterpart. Conditional distributions and conditional expected values were calculated from the bivariate model and they were compared to their empirical equivalent. The conclusion was that the model provided an overall good fit but the best fit was in the lower tail.