Abstract
The aim of this paper is to introduce dependence between the claim frequency and the average severity of a policyholder or of an insurance portfolio using a bivariate Sarmanov distribution, that allows to join variables of different types and with different distributions, thus being a good candidate for modeling the dependence between the two previously mentioned random variables. To model the claim frequency, a generalized linear model based on a mixed Poisson distribution -like for example, the Negative Binomial (NB), usually works. However, finding a distribution for the claim severity is not that easy. In practice, the Lognormal distribution fits well in many cases. Since the natural logarithm of a Lognormal variable is Normal distributed, this relation is generalised using the Box-Cox transformation to model the average claim severity. Therefore, we propose a bivariate Sarmanov model having as marginals a Negative Binomial and a Normal Generalized Linear Models (GLMs), also depending on the parameters of the Box-Cox transformation. We apply this model to the analysis of the frequency-severity bivariate distribution associated to a pay-as-you-drive motor insurance portfolio with explanatory telematic variables.
Highlights
Calculating premiums is a fundamental task for an insurance company
The aim of this paper is to introduce dependence between the claim frequency and the average severity of a policyholder or of an insurance portfolio using a bivariate Sarmanov distribution, that allows to join variables of different types and with different distributions, being a good candidate for modeling the dependence between the two previously mentioned random variables
A simple procedure consists of considering the aggregate claims as the product of the random variable (r.v.) number of claims and of the r.v. average cost of these claims, of fitting appropriate distributions to these two random variables; if, the premium is evaluated for a given policyholder, some of its characteristics are often included in the calculation as covariates in Generalized Linear Models (GLMs) used for both variables
Summary
Calculating premiums is a fundamental task for an insurance company. To this purpose, a simple procedure consists of considering the aggregate claims as the product of the random variable (r.v.) number of claims and of the r.v. average cost of these claims, of fitting appropriate distributions to these two random variables; if, the premium is evaluated for a given policyholder, some of its characteristics are often included in the calculation as covariates in Generalized Linear Models (GLMs) used for both variables. As an alternative, generalized Poisson regression models have been considered in the literature (see Reference [12]) For this variable, in this paper a GLM based on the Negative Binomial distribution will be used. In this paper it is considered a bivariate Sarmanov distribution that allows us to join a Normal GLM distribution for the transformed average claim cost r.v. Tariffication based on the collective model has been proposed, that is, the premium is deduced from the distribution of the total cost variable, which in turn is deduced from the bivariate distribution of frequency and severity claims. This paper ends with Appendixes A and B containing the proofs and additional results
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