Abstract
In actuarial mathematics, the claims of an insurance portfolio are often modeled using the collective risk model, which consists of a random number of claims of independent, identically distributed (i.i.d.) random variables (r.v.s) that represent cost per claim. To facilitate computations, there is a classical assumption of independence between the random number of such random variables (i.e., the claims frequency) and the random variables themselves (i.e., the claim severities). However, recent studies showed that, in practice, this assumption does not always hold, hence, introducing dependence in the collective model becomes a necessity. In this sense, one trend consists of assuming dependence between the number of claims and their average severity. Alternatively, we can consider heterogeneity between the individual cost of claims associated with a given number of claims. Using the Sarmanov distribution, in this paper we aim at introducing dependence between the number of claims and the individual claim severities. As marginal models, we use the Poisson and Negative Binomial (NB) distributions for the number of claims, and the Gamma and Lognormal distributions for the cost of claims. The maximum likelihood estimation of the proposed Sarmanov distribution is discussed. We present a numerical study using a real data set from a Spanish insurance portfolio.
Highlights
The collective risk model is a basic classical actuarial risk model consisting of the sum of a random number of independent, identically distributed (i.i.d.) random variables (r.v.s) that represent costs.To facilitate computations related to this model, there is a classical assumption of independence between the random number of such random variables and the random variables themselves
Studies on real data emphasized in several cases the existence of a certain dependence that should be taken into account because it can affect important actuarial quantities like premiums and ruin probabilities
We propose the bivariate Sarmanov distribution to analyze the joint behavior of the number of claims and of each one of the individual claim amounts, instead of their average; i.e., we consider heterogeneity
Summary
The collective risk model is a basic classical actuarial risk model consisting of the sum of a random number of independent, identically distributed (i.i.d.) random variables (r.v.s) that represent costs. Starting from an increasing need of flexible multivariate distributions, Sarmanov distribution recently gained interest in the actuarial literature and, fitted to some real insurance data in its bivariate and trivariate forms, provided better fits than other distributions, including Copula ones In this sense, we mention its applications in modeling continuous claim sizes see [6], modeling discrete claim frequencies see [7,8], in the evaluation of ruin probabilities see [9,10] or in capital allocation see [11,12,13]. A numerical example is presented in Section 4: on a real data set, we compare the bivariate Sarmanov distribution relating the frequency and the individual severity with the simpler case in which it is assumed that the claim amounts are all equal to the mean cost per policyholder; this last assumption implies eliminating the heterogeneity within each insured, and has been used in the alternative works cited in the first paragraph of this introduction.
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