Extreme value behavior of aggregate dependent risks

Consider a portfolio of n identically distributed risks with dependence structure modeled by an Archimedean survival copula. Wüthrich (2003) and Alink et al. (2004) proved that the probability of a large aggregate loss scales like the probability of a large individual loss, times a proportionality factor. This factor depends on the dependence strength and the tail behavior of the individual risk, denoted by qnF, qnW and qnG according to whether the tail behavior belongs to the maximum domain of attraction of the Fréchet, the Weibull or the Gumbel distribution, respectively. We investigate properties of the factors qnW and qnG with respect to the dependence parameter and/or the tail behavior parameter, and revisit the asymptotic behavior of conditional tail expectations of aggregate risks for the Weibull and the Gumbel cases by using a different method. The main results strengthen and complement some results in Alink et al. (2004, 2005)Barbe et al. (2006), and Embrechts et al. (2009).