TVaR-based capital allocation with copulas

Abstract

Because of regulation projects from control organizations such as the European solvency II reform and recent economic events, insurance companies need to consolidate their capital reserve with coherent amounts allocated to the whole company and to each line of business. The present study considers an insurance portfolio consisting of several lines of risk which are linked by a copula and aims to evaluate not only the capital allocation for the overall portfolio but also the contribution of each risk over their aggregation. We use the tail value at risk (TVaR) as risk measure. The handy form of the FGM copula permits an exact expression for the TVaR of the sum of the risks and for the TVaR-based allocations when claim amounts are exponentially distributed and distributed as a mixture of exponentials. We first examine the bivariate model and then the multivariate case. We also show how to approximate the TVaR of the aggregate risk and the contribution of each risk when using any copula.

CMPH: a multivariate phase-type aggregate loss distribution

“…The right-hand side of the last equation naturally leads (Bargès, Cossette and Marceau [2]) to the TVaR-based allocation principle. Namely, the allocated loading to type k loss is…”

Impact of Dependence on Some Multivariate Risk Indicators

“…Then, 2 For (x, y) ∈ R 2+ , we shall denote y = Θ(x) if there exist 0 < C 1 ≤ C 2 < ∞, such that for x large enough,…”

“…Now, relations: (2.8), (2.6), and (2.9), imply that T 1 , T 2 , and T 3 , go to zero, and this is a contradiction. (1) there exist 0 < κ 1 < κ 2 < 1 such that for all i = 1, . .…”

“…In [8] properties and asymptotic behavior of the allocation are studied in some bivariate cases. The impact of dependence on capital allocation was studied for certain cases of Euler's method by Bargès et al (2009) [2] and Cossette et al (2012) [5].…”

Bayesian Modelling, Monte Carlo Sampling and Capital Allocation of Insurance Risks

“…Even though, in general, these expectations are not available in closed form, some exceptions exist, such as the multivariate Gaussian model, first discussed in this context in Panjer (2001) and extended to the case of multivariate elliptical distributions in Landsman and Valdez (2003) and Dhaene et al (2008); the multivariate gamma model of Furman and Landsman (2005); the combination of the Farlie-Gumbel-Morgenstern (FGM) copula and (mixtures of) exponential marginals from Bargès et al (2009) or (mixtures of) Erlang marginals Cossette et al (2013); and the multivariate Pareto-II from Asimit et al (2013).…”