Risk aggregation in multivariate dependent Pareto distributions

On the Evaluation of Multivariate Compound Distributions With Continuous Severity Distributions and Sarmanov's Counting Distribution

“…In this case, a special attention must be paid to the choice of the discretization span: a large span can generate important errors, while a very small span can lead to a very long running time, especially in the multivariate case. In this respect, it is unfortunate that closedtype formulas for compound distributions with continuous type claim sizes are so scarce; in the univariate case, apart the Gamma severity distribution (which also includes the well-known exponential case) leading to the so-called Tweedie compound distribution (see, e.g., Dunn and Smyth, 2005), we mention the recent work of Sarabia et al (2016), who went even further on by considering a Pareto-type dependency between the aggregated claim sizes. In this paper, we propose closed-type formulas for some multivariate compound distributions with Sarmanov counting distribution and Erlang severity distributions; furthermore, inspired by Sarabia et al (2016), we also include some dependency between the claim sizes.…”

“…In this respect, it is unfortunate that closedtype formulas for compound distributions with continuous type claim sizes are so scarce; in the univariate case, apart the Gamma severity distribution (which also includes the well-known exponential case) leading to the so-called Tweedie compound distribution (see, e.g., Dunn and Smyth, 2005), we mention the recent work of Sarabia et al (2016), who went even further on by considering a Pareto-type dependency between the aggregated claim sizes. In this paper, we propose closed-type formulas for some multivariate compound distributions with Sarmanov counting distribution and Erlang severity distributions; furthermore, inspired by Sarabia et al (2016), we also include some dependency between the claim sizes. Our formulas are expressed mainly in terms of the special hypergeometric function already implemented in the existing mathematical software, hence making the related calculations numerically feasible without involving other techniques.…”

“…We shall call by "inside-type independency" the independency assumption between the claims of same type, while by "between-types independency" we designate the independency assumptions between claims of different types. Then, similarly with Sarabia et al (2016), we shall relax the inside-type independency condition by considering a certain type of dependency in each set of claims X jl l≥1 . Note that in the univariate case, dependency between the individual risks has already been considered especially in the individual model, see, e.g., Goovaerts and Dhaene (1996), Genest et al (2003), Denuit et al (2005) and the references therein.…”

“…Regarding the claim sizes distributions, we choose the Erlang distribution in the inside-type independency case and, when relaxing this condition, the same multivariate type-II Pareto distribution as in Sarabia et al (2016). This choice has been driven by the interest in obtaining closed formulas numerically computable with the existing software, but also by the fact that both distributions have been intensively studied in the actuarial literature lately; for the Erlang distribution, see, e.g., Willmot and Lin (2011) or Willmot and Woo (2015), while for the multivariate type-II Pareto distribution, see Asimit et al (2013) or the generalizations in Asimit et al (2010) and Guillén et al (2013).…”

Aggregate claim models with one-way and two-way dependence among individual claims

“…[2] used a simple idea of mixing to establish dependence among claim sizes and among claim inter-arrival times in collective risk models and obtained a number of explicit formulas for ruin probabilities and related quantities. More recently, [16] derived closed-form expressions for the probability distribution function of aggregated risks with multivariate dependent Pareto type II distributions proposed by [4] and [5], which is widely used in insurance and risk analysis.…”