P>The hunger for bonus is a well-known phenomenon in insurance, meaning that the insured does not report all of his accidents to save bonus on his next year's premium. In this article, we assume that the number of accidents is based on a Poisson distribution but that the number of claims is generated by censorship of this Poisson distribution. Then, we present new models for panel count data based on the zero-inflated Poisson distribution. From the claims distributions, we propose an approximation of the accident distribution, which can provide insight into the behavior of insureds. A numerical illustration based on the reported claims of a Spanish insurance company is included to support this discussion.

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Number of Accidents or Number of Claims? An Approach with Zero-Inflated Poisson Models for Panel Data

When actuaries face with the problem of pricing an insurance contract that contains different types of coverage, such as a motor insurance or homeowner's insurance policy, they usually assume that types of claim are independent. However, this assumption may not be realistic: several studies have shown that there is a positive correlation between types of claim. Here we introduce different multivariate Poisson regression models in order to relax the independence assumption, including zero-inflated models to account for excess of zeros and overdispersion. These models have been largely ignored to date, mainly because of their computational difficulties. Bayesian inference based on MCMC helps to solve this problem (and also lets us derive, for several quantities of interest, posterior summaries to account for uncertainty). Finally, these models are applied to an automobile insurance claims database with three different types of claims. We analyse the consequences for pure and loaded premiums when the independence assumption is relaxed by using different multivariate Poisson regression models and their zero-inflated versions.

/pdf/modelling-dependence-in-a-ratemaking-procedure-with-1b40qrw2hu.pdf

Modelling Dependence in a Ratemaking Procedure with Multivariate Poisson Regression Models

When actuaries face the problem of pricing an insurance contract that contains different types of coverage, such as a motor insurance or a homeowner’s insurance policy, they usually assume that types of claim are independent. However, this assumption may not be realistic: several studies have shown that there is a positive correlation between types of claim. Here we introduce different multivariate Poisson regression models in order to relax the independence assumption, including zero-inflated models to account for excess of zeros and overdispersion. These models have been largely ignored to date, mainly because of their computational difficulties. Bayesian inference based on MCMC helps to resolve this problem (and also allows us to derive, for several quantities of interest, posterior summaries to account for uncertainty). Finally, these models are applied to an automobile insurance claims database with three different types of claim. We analyse the consequences for pure and loaded premiums when the independence assumption is relaxed by using different multivariate Poisson regression models together with their zero-inflated versions.

/pdf/bayesian-multivariate-poisson-models-for-insurance-25kvxsb8nz.pdf

Bayesian multivariate Poisson models for insurance ratemaking

Vehicle sensor data enable novel, usage-based insurance premium models known as 'Pay-As-You-Drive' (PAYD) insurance, but pose substantial challenges for actuarial decision-making because of their inherent complexity and volume. Based on a large real-world sample of location data from 1572 vehicles, the present study proposes a classification analysis approach that addresses (i) the selection of predictor variables, (ii) the presence of class skew and time-variant prior distributions, and (iii) the suitability of classifier scores as an aggregated actuarial rate factor. Using raw location data, we derive a set of 15 predictor variables that we use to train and compare logistic regression, neural network, and decision tree classifiers. We find that while neural networks exhibit superior classification performance, logistic regression is better suited from an actuarial viewpoint in several ways. In sum, our results clearly demonstrate the potential of high-resolution exposure data for reducing the complexity of PAYD insurance pricing in practice. DSS may help to reduce the complexity of Pay-As-You-Drive (PAYD) insurance ratemaking.A plethora of potential actuarial rate factors can be derived from GPS-based vehicle data.The present study uses a real-world dataset and classification analysis to evaluate rate factors.We find that neural networks exhibit the best classification performance.However, logistic regression models seem more suitable from an actuarial viewpoint.

Evaluation and aggregation of pay-as-you-drive insurance rate factors

A data set from a Belgian telematics product aimed at young drivers is used to identify how car insurance premiums can be designed based on the telematics data collected by a black box installed in the vehicle. In traditional pricing models for car insurance, the premium depends on self-reported rating variables (e.g. age, postal code) which capture characteristics of the policy(holder) and the insured vehicle and are often only indirectly related to the accident risk. Using telematics technology enables tailor-made car insurance pricing based on the driving behavior of the policyholder. We develop a statistical modeling approach using generalized additive models and compositional predictors to quantify and interpret the effect of telematics variables on the expected claim frequency. We find that such variables increase the predictive power and render the use of gender as a discriminating rating variable redundant.

Jean-Franois Walhin

Papers

Actuarial Modelling of Claim Counts