Analysis of the Expected Shortfall of Aggregate Dependent Risks

Abstract

We consider d identically and continuously distributed dependent risks X 1 , …, X d . Our main result is a theorem on the asymptotic behaviour of expected shortfall for the aggregate risks: there is a constant c d such that for large u we haveMoreover we study diversification effects in two dimensions, similar to our Value-at-Risk studies in [2].

Archimedean copulas with applications to $${{\mathrm{VaR}}}$$ VaR estimation

“…The form of the constant q F n (α, β) for n > 2 is quite complicated and it is difficult to use (6) for n > 2. Alink et al (2004Alink et al ( , 2005 and Wüthrich (2003) also obtained similar results for Gumbel or Weibull extreme-value families. Using the results by de Haan and Resnick (1977), Barbe et al (2006) extend (6) beyond Archimedean dependence structures, where X i is multivariate regularly varying (for definition see Barbe et al 2006).…”

“…The asymptotic distribution of such sums under some technical assumptions was given in Alink et al (2004Alink et al ( , 2005 and Wüthrich (2003). Let S n = n i=1 X i .…”

“…In Wüthrich (2003) and Alink et al (2004Alink et al ( , 2005 the asymptotic distribution of (5) with an application to Value-at-Risk (VaR) estimation was given. More precisely if (X 1 , .…”

Multivariate Tweedie distributions and some related capital-at-risk analyses

“…However, in the last years the concept of dependence has received its merited attention, which has resulted in numerous papers (cf., e.g., Pfeifer and Neslehova, 2004;Bauerle and Grubel, 2005;Alink et al, 2005;Laeven et al, 2005;Centeno, 2005;Kijima, 2006). Also, some new multivariate distributions have been proposed (cf., e.g.…”

Impact of correlation crises in risk theory: Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed

“…The dependence between claim amounts has been investigated by Ignatov et al (2001) and by Lefèvre and Loisel (in press), who provide recursive formulas that involve Appell-type properties for the finite-time ruin probability with dependent claim amounts. Sums of dependent random variables have been studied by many authors, in particular by Barbe et al (2006), Alink et al (2005), Wüthrich (2003 and by Kortschak and Albrecher (in press). Dependence and non-stationarity have been partially taken into account for infinite-time ruin probabilities by Boudreault et al (2006), and Albrecher and Boxma (2004), who assume that the…”