On some multivariate Sarmanov mixed Erlang reinsurance risks: aggregation and capital allocation

TL;DR

This paper derives new closed-form formulas for risk aggregation and capital allocation in multivariate Sarmanov mixed Erlang risks, exploring different kernel functions and their dependency structures, with numerical illustrations in reinsurance contexts.

Contribution

It introduces novel formulas using previously unstudied kernel functions for Sarmanov's distribution in risk management, expanding the analytical tools available.

Findings

New closed-form formulas for risk aggregation and capital allocation.

Analysis of dependency structures via Pearson's correlation coefficient.

Numerical illustrations in stop-loss reinsurance scenarios.

Abstract

Following some recent works on risk aggregation and capital allocation for mixed Erlang risks joined by Sarmanov's multivariate distribution, in this paper we present some closed-form formulas for the same topic by considering, however, a different kernel function for Sarmanov's distribution, not previously studied in this context. The risk aggregation and capital allocation formulas are derived and numerically illustrated in the general framework of stop-loss reinsurance, and then in the particular case with no stop-loss reinsurance. A discussion of the dependency structure of the considered distribution, based on Pearson's correlation coefficient, is also presented for different kernel functions and illustrated in the bivariate case.