On Sarmanov Mixed Erlang Risks in Insurance Applications

Abstract

In this paper we consider an extension to the aggregation of the FGM mixed Erlang risks, proposed by Cossette et al. (2013 Insurance: Mathematics and Economics, 52, 560–572), in which we introduce the Sarmanov distribution to model the dependence structure. For our framework, we demonstrate that the aggregated risk belongs to the class of Erlang mixtures. Following results from S. C. K. Lee and X. S. Lin (2010 North American Actuarial Journal, 14(1) 107–130), G. E. Willmot and X. S. Lin (2011 Applied Stochastic Models in Business and Industry, 27(1) 8–22), analytical expressions of the contribution of each individual risk to the economic capital for the entire portfolio are derived under both the TVaR and the covariance capital allocation principle. By analysing the commonly used dependence measures, we also show that the dependence structure is wide and flexible. Numerical examples and simulation studies illustrate the tractability of our approach.

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

“…defines the FGM distribution as explored in [2] and φ i (x i ) = e −xi − E e −Xi introduced by Hashorva and Ratovomirija [7] for mixed Erlang marginals. In the rest of the paper we refer to the latter as the Laplace case.…”

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

“…In the context of risk aggregation and capital allocation, Hashorva and Ratovomirija [6] assumed that g(x) = e −x , Vernic [15] studied the case where the marginals are exponentially distributed, while Cossette et al [3] used the FGM distribution with mixed Erlang marginals (the FGM is a special case of the Sarmanov distribution for g(x) = 2(1 − F (x)), with F denoting the distribution function of the marginal). Thus, in the sequel, we consider the following kernel function…”

“…We consider g(x) = e −tx for t = 1, hence the corresponding kernel function is given by φ(x) = e −x − E e −X . Pearson's correlation coefficient along with its lower and upper bounds can be found in [6], where the particular case of mixed Erlang marginals is emphasized.…”

“…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].…”

EFFICIENT ESTIMATION OF ERLANG MIXTURES USING iSCAD PENALTY WITH INSURANCE APPLICATION

“…The multivariate model not only inherits most of the desirable distributional properties but also offers a non-copula approach for dependence modeling. Also see Hashorva and Ratovomirija (2015), Verbelen, Antonio and Claeskens (2015), Willmot and Woo (2015), and Badescu et al (2015).…”