Risk concentration of aggregated dependent risks: The second-order properties

Under the current regulatory guidelines for banks and insurance companies, the quantification of diversification benefits due to risk aggregation plays a prominent role. In this paper we establish second-order approximation of risk concentration associated with a random vector X:=(X1,X2,…,Xd) in terms of Value at Risk (VaR) within the methodological framework of second-order regular variation and the theory of Archimedean copula. Moreover, we find that the rate of convergence of the first-order approximation of risk concentration depends on the the interplay between the tail behavior of the marginal loss random variables and their dependence structure. Specifically, we find that the rate of convergence is determined by either the second-order parameter (ρ1) of Archimedean copula generator or the second-order parameter (ρ) of the tail margins, leading to either the so-called dependence dominated case or margin dominated case.