Credibility premiums for the zero-inflated Poisson model and new hunger for bonus interpretation

The purpose of this paper is to explore and compare the credibility premiums in generalized zero-inflated count models for panel data. Predictive premiums based on quadratic loss and exponential loss are derived. It is shown that the credibility premiums of the zero-inflated model allow for more flexibility in the prediction. Indeed, the future premiums not only depend on the number of past claims, but also on the number of insured periods with at least one claim. The model also offers another way of analysing the hunger for bonus phenomenon. The accident distribution is obtained from the zero-inflated distribution used to model the claims distribution, which can in turn be used to evaluate the impact of various credibility premiums on the reported accident distribution. This way of analysing the claims data gives another point of view on the research conducted on the development of statistical models for predicting accidents. A numerical illustration supports this discussion.

[1]  L. Bermúdez,et al.  Bonus-malus system using an exponential loss function with an Inverse Gaussian distribution , 2003 .

[2]  Stephen E. Fienberg,et al.  Credibility: Theory and Applications. , 1975 .

[3]  Michel Denuit,et al.  Bonus-Malus scales using exponential loss functions , 2001 .

[4]  V. Young.  Credibility and Persistency , 1996, ASTIN Bulletin.

[5]  N. L. Johnson,et al.  Discrete Multivariate Distributions , 1998 .

[6]  Kelvin K. W. Yau,et al.  On modeling claim frequency data in general insurance with extra zeros , 2005 .

[7]  Virginia R. Young.  Credibility using a loss function from spline theory: Parametric models with a one-dimensional sufficient statistic , 1998 .

[8]  Z. Griliches,et al.  Econometric Models for Count Data with an Application to the Patents-R&D Relationship , 1984 .

[9]  V. Young,et al.  Credibility in Favor of Unlucky Insureds , 2000 .

[10]  Michel Denuit,et al.  Fixed versus Random Effects in Poisson Regression Models for Claim Counts: A Case Study with Motor Insurance , 2006, ASTIN Bulletin.

[11]  Michel Denuit,et al.  Number of Accidents or Number of Claims? An Approach with Zero-Inflated Poisson Models for Panel Data , 2009 .

[12]  W. S. Jewell,et al.  The Use of Collateral Data in Credibility Theory: A Hierarchical Model , 1975 .

[13]  Jean Lemaire.  La Soif du Bonus , 1977, ASTIN Bulletin.

[14]  Michel Denuit,et al.  Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems , 2007 .

[15]  Hans Bühlmann,et al.  Experience rating and credibility , 1967, ASTIN Bulletin.