Impact of dependence on some multivariate risk indicators

The minimization of some multivariate risk indicators may be used as an allocation method, as proposed in Cénac et al. [9]. The aim of capital allocation is to choose a point in a simplex, according to a given criterion. In a previous paper [17] we proved that the proposed allocation technique satisfies a set of coherence axioms. In the present one, we study the properties and asymptotic behavior of the allocation for some distribution models. We analyze also the impact of the dependence structure on the allocation using some copulas. Introduction Natural phenomena, financial events and risks are usually modeled through random vectors or processes. In these fields, considering dependencies between random variables is necessary. In actuarial science, this issue has led to the development of multivariate risk theories. Indeed, an insurance company is generally exposed to several risks which cannot be assumed to be independent. It is therefore necessary to adopt a multivariate approach that takes into account both the marginal structures of risks and their dependence structure. Multivariate risk theory is based on dependence modeling and includes multivariate ruin probabilities and multivariate risk measures. In the univariate case, ruin probability has been widely studied since the beginning of the 20th century (Lundberg (1903) [16], Cramer (1930) [7]). In multivariate contexts, several definitions are proposed for ruin probability. Hult and Lindskog (2006) [14] defined a multivariate ruin probability based on a notion of ruin sets. Cai and Li (2007) [4] defined different finite-time ruin probabilities. Risk measure theory was also enriched by some multivariate risk measures definitions. Jouini et al. (2004) [15] defined vector-valued coherent risk measures. Cousin and Di Bernardino (2013) [6] introduced some multi-dimensional extensions of usual univariate risk measures such as the Value-at-Risk. These measure are multi-dimensional valued, and thus cannot take benefit from a full order. Dhaene et al. (2012) introduced of real valued family for risk measures for mono-periodic multivariate processes ([11]). In multi-periodic context, Cénac et al. (2012) [9] defined new multivariate risk indicators as sum of expected local ruin amounts using penalty functions. Capital allocation may be seen as a direct application of multivariate risk theory. Allocating a capital in actuarial contexts means distributing a positive quantity representing an allocation capital in a simplex. For insurance industry the calculation of the regulatory economic capital, which is called the Solvency Capital Requirement, is well controlled and its methodology is almost imposed by the supervisory authorities of the sector. Nevertheless, the allocation of this capital may be considered as an internal exercise for each company, and constitutes a management choice whose success is a key factor for firm performance optimization. It can also be seen as an indicator Date: July 5, 2015. 2010 Mathematics Subject Classification. 62H00, 62P05, 91B30 .