On mixed Erlang reinsurance risk: aggregation, capital allocation and default risk

Abstract

In this paper, we address the aggregation of dependent stop loss reinsurance risks where the dependence among the ceding insurer(s) risks is governed by the Sarmanov distribution and each individual risk belongs to the class of Erlang mixtures. We investigate the effects of the ceding insurer(s) risk dependencies on the reinsurer risk profile by deriving a closed formula for the distribution function of the aggregated stop loss reinsurance risk. Furthermore, diversification effects from aggregating reinsurance risks are examined by deriving a closed expression for the risk capital needed for the whole portfolio of the reinsurer and also the allocated risk capital for each business unit under the TVaR capital allocation principle. Moreover, given the risk capital that the reinsurer holds, we express the default probability of the reinsurer analytically. In case the reinsurer is in default, we determine analytical expressions for the amount of the aggregate reinsured unpaid losses and the unpaid losses of each reinsured line of business of the ceding insurer(s). These results are illustrated by numerical examples.

On Some Multivariate Sarmanov Mixed Erlang Reinsurance Risks: Aggregation and Capital Allocation

“…It should be noted that Ratovomirija [12] provided a general expression for the joint density of S in the particular case when k 1 = . .…”

“…The following results have already been developed in [3] and [12]. is equal to the pdf f (x, β, G(Q)) of a mixed Erlang distribution with mixing probabilities G(Q) = (g 1 , g 2 , .…”

“…For the last equality, apart the definition of H, we also used the fact that the convolution of two Erlang distributions having the same scale parameter is again an Erlang distribution with the same scale parameter, while its shape parameter equals the sum of the shape parameters of the convoluted distributions. Inserting now (33) into (32) and the result into (31) yields (12). To obtain the formula of U, we use…”

“…The aim of this contribution is to address risk aggregation and TVaR capital allocation for insurance and reinsurance mixed Erlang risks whose dependency is governed by the Sarmanov distribution with a certain expression of the kernel functions. This study comes along the lines of some recent contributions: Vernic [15] considered capital allocation based on the TVaR rule for the Sarmanov distribution with exponential marginals; Cossette et al [3] used the Farlie-Gumbel-Morgenstern (FGM) distribution to model the dependency between mixed Erlang distributed risks and applied it to capital allocation and risk aggregation; Hashorva and Ratovomirija [6] and Ratovomirija [12] presented aggregation and capital allocation in insurance and reinsurance for mixed Erlang distributed risks joined by the Sarmanov distribution with a specific kernel function different from the one considered in this study. Note that the choice of the Sarmanov and mixed Erlang distributions is not incidental, these distributions gained a lot of interest in the actuarial literature lately: for the Sarmanov distribution, see e.g., [19], [7], [1], [16], while for the mixed Erlang distribution we refer to [9], [17], [10] or [18].…”

Frequency and Severity Dependence in the Collective Risk Model: An Approach Based on Sarmanov Distribution

“…Starting from an increasing need of flexible multivariate distributions, Sarmanov distribution recently gained interest in the actuarial literature and, fitted to some real insurance data in its bivariate and trivariate forms, provided better fits than other distributions, including Copula ones. In this sense, we mention its applications in modeling continuous claim sizes see [6], modeling discrete claim frequencies see [7,8], in the evaluation of ruin probabilities see [9,10] or in capital allocation see [11][12][13].…”