Pair Copula Constructions for Insurance Experience Rating

ABSTRACT

In nonlife insurance, insurers use experience rating to adjust premiums to reflect policyholders’ previous claim experience. Performing prospective experience rating can be challenging when the claim distribution is complex. For instance, insurance claims are semicontinuous in that a fraction of zeros is often associated with an otherwise positive continuous outcome from a right-skewed and long-tailed distribution. Practitioners use credibility premium that is a special form of the shrinkage estimator in the longitudinal data framework. However, the linear predictor is not informative especially when the outcome follows a mixed distribution. In this article, we introduce a mixed vine pair copula construction framework for modeling semicontinuous longitudinal claims. In the proposed framework, a two-component mixture regression is employed to accommodate the zero inflation and thick tails in the claim distribution. The temporal dependence among repeated observations is modeled using a sequence of bivariate conditional copulas based on a mixed D-vine. We emphasize that the resulting predictive distribution allows insurers to incorporate past experience into future premiums in a nonlinear fashion and the classic linear predictor can be viewed as a nested case. In the application, we examine a unique claims dataset of government property insurance from the state of Wisconsin. Due to the discrepancies between the claim and premium distributions, we employ an ordered Lorenz curve to evaluate the predictive performance. We show that the proposed approach offers substantial opportunities for separating risks and identifying profitable business when compared with alternative experience rating methods. Supplementary materials for this article are available online.

KEYWORDS:

• Government insurance
• Mixed D-vine
• Mixture regression
• Predictive distribution
• Zero inflation

Supplementary Materials

The Appendix of the article can be found in the online supplementary materials.

Acknowledgment

The authors thank the editor, the associated editor, and two anonymous reviewers for helpful comments. Peng Shi gratefully acknowledges the financial support for this study by the Centers of Actuarial Excellence (CAE) Research Grant from the Society of Actuaries and the Wisconsin Naming Partners Fellowship from the University of Wisconsin-Madison.