Frequency and Severity Dependence in the Collective Risk Model: An Approach Based on Sarmanov Distribution

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.

Bivariate Mixed Poisson and Normal Generalised Linear Models with Sarmanov Dependence—An Application to Model Claim Frequency and Optimal Transformed Average Severity

“…, while the other two expectations can be directly deduced from the Proposition 5 of Bolancé and Vernic [7]. The results are:…”

“…From Proposition 3 in Bolancé and Vernic [7], the following correlation for each i can be easily deduced:…”

“…Because it provides a flexible structure in joining different types of marginals, the bivariate Sarmanov distribution has recently found its place in actuarial studies likemodeling continuous claim costs [5], modeling discrete claim frequencies [1,6], modeling dependence between discrete frequency and continuous claims [7], ruin probability calculation [8,9] or modeling telematic variables [10].…”

“…To model the claim cost r.v. (severity), Gamma and Lognormal distributions are the most common choices (see Reference [7] for comparing both cost distributions in a real motor data set). However, it may be the case that neither of these two distributions fits the data well enough.…”

“…On the other hand, Czado et al [25] proposed bivariate models based on copulae (see also References [26] for a more general approach). Furthermore, the bivariate Sarmanov model has also been analyzed in the collective model framework [7].…”