Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates

Studying mixability with supermodular aggregating functions

“…In Section 4 we confirm the validity of the conditions given on some numerical examples obtained via an adaptation of the Mixability Detection Procedure described in Puccetti and Wang (2015). The joint mixability of f (x 1 + · · · + x d ), with f convex, is implied by the joint mixability of the sum, for which necessary and sufficient conditions have appeared in a variety of recent papers as for instance Wang andWang (2011), Puccetti et al (2012), Puccetti et al (2013) and Wang and Wang (2014). Conditions under which the joint mixability of the sum is always implied by the joint mixability of f (x 1 + · · · + x d ) are given in Section 5 along with some open problems for further research.…”

Reducing model risk via positive and negative dependence assumptions

“…for an inhomogeneous portfolio with d/2 Gamma(2,1/2) risks and d/2 Gamma(4,1/2) risks, k subgroups, n = d/k variables in each subgroup and dependence assumption Y ≤ wcs X. some trivial cases of limited applicability. Some kind of negative dependence allows to construct worst-case VaR distributions with VaR value bigger than in the comonotonic case (see Wang and Wang, 2011;Embrechts et al, 2013) and approaching asymptotically the worst ES bound (see Puccetti et al, 2013;Puccetti and Rüschendorf, 2014).…”

How Superadditive Can a Risk Measure Be?

“…Puccetti and Rüschendorf (2014) showed (2.4) under an assumption of complete mixability; Puccetti et al (2013) 1 For comonotonic additive ⇢, the denominator of (2.2) becomes the risk measure of the sum of comonotonic risks with corresponding marginal distributions; this is typically interpreted as a worst-case dependence scenario (see e.g. Dhaene et al, 2002Dhaene et al, , 2012.…”