A mixed copula model for insurance claims and claim sizes

Abstract

A crucial assumption of the classical compound Poisson model of Lundberg for assessing the total loss incurred in an insurance portfolio is the independence between the occurrence of a claim and its claims size. In this paper we present a mixed copula approach suggested by Song et al. to allow for dependency between the number of claims and its corresponding average claim size using a Gaussian copula. Marginally we permit for regression effects both on the number of incurred claims as well as its average claim size using generalized linear models. Parameters are estimated using adaptive versions of maximization by parts (MBP). The performance of the estimation procedure is validated in an extensive simulation study. Finally the method is applied to a portfolio of car insurance policies, indicating its superiority over the classical compound Poisson model.

Keywords:

• GLM
• Copula
• Maximization by parts
• Number of claims
• Average claim size
• Total claim size

Acknowledgements

C. Czado is supported by the DFG (German Research Foundation) grant CZ 86/1-3. We like to thank Peter Song for contributing valuable ideas and information.

Notes

1. Deutsche Mark (DM) is the former German currency, which was replaced by the Euro in 2002 (1 DM=0.51 Euro). Here we also use abbreviation TDM for 1000 DM.

2. Note that the data aggregation in this paper is slightly different from the one performed in Kastenmeier (2008). In order to obtain more evenly spread age groups, we merged the youngest and the oldest two of the age groups in the above thesis, respectively.