Matrix Analytic Methods for Portfolios of Multiplicative background Risks

Quantitative modeling a portfolio of dependent risks has drawn considerable attention in applied mathematics literature, particularly in actuarial science.  Under this context, practitioners often opt for such multivariate models that a.) allow for realistic positive dependence structures, b.) possess practically meaningful interpretations, and c.) guarantee a desirable level of analytical tractability.  The multiplicative background risks (MBR) models are precisely one such case.

This paper considers a class of MBR models with idiosyncratic risk factors distributed mixed Erlang, which belongs to the encompassing family of PH distributions. The models under investigation can be seen as a particular case of the PH-based MBR studied in Furman, E., Kye, Y., and Su, J. (2021).  Nevertheless, the proposed mixed Erlang-based MBR models still deserve special meaning for at least two reasons. Firstly, mixed Erlang is more parsimonious than the PH distribution, yet it is sufficiently versatile to carry over the universal approximation property. Secondly and perhaps more importantly, the mix Erlang-based MBR is more computable than the PH-based MBR. To shed light on the computation benefits, we will propose a recursive matrix analytic method for evaluating the distributional quantities of mix Erlang-based MBR, including joint/marginal distribution functions, moments, aggregation, and tail-based conditional expectations. Compared with the Jordan decomposition approach employed in Furman, E., Kye, Y., and Su, J. (2021), our proposed algorithm features efficient, transparent, and numerically stable and,  allowing the computation of previously barely computable risk measures, which is Tail-based Gini-functionals.