Diversification of aggregate dependent risks

We give a new proof of the central result in Wüthrich [Astin Bulletin 33 (1) (2003) 75]. For d identically and continuously distributed dependent risks X1,…,xd, the probability of a large aggregate loss of ∑i=1dXi scales like the probability of a large individual loss of X1, times a proportionality factor qd. This factor qd depends on the dependence strength and the tail behaviour of the individual risks Xi, i.e. whether this tail behaviour falls into the domain of attraction of the Fréchet, the Weibull or the Gumbel distribution; see Embrechts et al. [Modelling Extremal Events for Insurance and Finance, Springer, Berlin]. We give explicit formulas for q2 and describe their behaviour with respect to the dependence strength.