Evaluating the Tail Risk of Multivariate Aggregate Losses

Abstract

In this paper, we study the tail risk measures for several commonly used multivariate aggregate loss models where the claim frequencies are dependent but the claim sizes are mutually independent and independent of the claim frequencies. We first develop formulas for the moment (or size biased) transforms of the multivariate aggregate losses, showing their relationship with the moment transforms of the claim frequencies and claim sizes. Then, we apply the formulas to compute some popular risk measures such as the tail conditional expectation and tail variance of the multivariate aggregated losses and to perform capital allocation analysis.

Risk analysis of a multivariate aggregate loss model with dependence

“…and the ≤ operation is defined piecewisely. Then, following Proposition 1 of Furman and Landsman (2005), Proposition 3.1 of Denuit and Robert (2022), or Lemma 2.1 in Jiang and Ren (2022), we have, for m ∈ M and k ∈ K,…”

“…As shown in Theorem 4.1 of Jiang and Ren (2022), for m, m 1 , m 2 ∈ M and m 1 = m 2 , we have L [m] = N [m] − 1 [m] ∼ HMN( W − 1, q 1 , . .…”

“…If one accident can only give rise to one type of claim, then M = K and the claim size variables are univariate. In this case, Theorem 3.1 was reduced to Theorem 3.1 by Jiang and Ren (2022). On the other hand, if the claim number random vectors N is univariate and the claim size random vector contains all risk types, Theorem 3.1 reduces to Theorem 3 in Ren (2022).…”

“…Determine the distributions of N, L [m] , and L [m 1 ,m 2 ] for m, m 1 , m 2 ∈ M. Some commonly used distribution functions of N, such as multinomial, additive common shock, and common Poisson mixture, were studied in the literature by, for example, Hesselager (1996) and Kim et al (2019). In these cases, as shown by Jiang and Ren (2022), the distributions of L [m] and L [m 1 ,m 2 ] are in fact mixture of some distributions in the same family as N and can be conveniently computed.…”

“…Based on this method, Denuit (2020) presented explicit expressions for TCE of some univariate compound distributions; Denuit and Robert (2022) illustrated how to apply moment transform to analyze TCE of multivariate random variables. Ren (2022) derived formulas for TCE and TV of multivariate compound models based on Sundt (1999), where claim frequency is onedimensional, and one claim can yield multiple dependent losses; Jiang and Ren (2022) provided methods for computing the TCE and TV of the multivariate aggregate losses, where the claim frequencies are dependent but the claim sizes are mutually independent and independent of the claim frequencies.…”