On a Multiplicative Multivariate Gamma Distribution With Applications in Insurance

In a recent paper [Albrecher, Constantinescu and Loisel (2011). Explicit ruin formulas for models with dependence among risks. Insurance: Mathematics and Economics 48(2), 265 – 270] Professors Hansjörg Albrecher, Corina Constantinescu and Stephane Loisel noted – in passing – a way to employ exponential mixtures for formulating multivariate probability distributions with dependent univariate margins distributed gamma. The main message of our report is that the probabilistic construction in ibid., which has been arguably overlooked by the actuarial community, should be given very serious considerations when modelling dependent risks. In order to convey this message, we conduct a systematic study of the aforementioned construction. Specifically, we show, among other findings, that the model in Albrecher et al. (2011): (1) admits the interpretation of the Multiplicative Background Risk Model with risk components distributed gamma, and as such is easy to communicate to upper management; (2) is remarkably tractable, e.g., the risks aggregation within it is significantly simpler than in the case when the risk components are independent and distributed gamma; and (3) enjoys rich (tail) dependence characteristics