Sharp Bounds for Sums of Dependent Risks

Abstract

Sharp tail bounds for the sum of d random variables with given marginal distributions and arbitrary dependence structure have been known since Makarov (1981) and Rüschendorf (1982) for d = 2 and, in some examples, for d ≥ 3. Based on a duality result, dual bounds have been introduced in Embrechts and Puccetti (2006b). In the homogeneous case, F 1 = · · · = F n , with monotone density, sharp tail bounds were recently found in Wang and Wang (2011). In this paper we establish the sharpness of the dual bounds in the homogeneous case under general conditions which include, in particular, the case of monotone densities and concave densities. We derive the corresponding optimal couplings and also give an effective method to calculate the sharp bounds.

A new approach to assessing model risk in high dimensions

“…Furthermore, in many practical cases one can construct a dependence such that the quantile function of the sum becomes approximately flat; see e.g., Embrechts et al (2013) and Bernard et al (2015) for illustrations. In this regard, we note that a large class of distributions exhibits asymptotic mixability implying that in high-dimensional problems the lower bound A that is stated in Proposition 3.1 is expected to be approximately sharp; see e.g., and Puccetti and Rüschendorf (2013). Hence, to achieve the minimum bound .…”

Bounds on total economic capital: the DNB case study

“…Only tails matter As a consequence of Theorem 2.1 in Puccetti and Rüschendorf (2013), the worst-possible α-VaR for a sum of random variables only depends on the upper (1 − α)-parts of the supports of the fixed marginal distributions. Maximization of the VaR of L + 6 can be equivalently seen as the maximization of the tail function P (L + 6 ≥ s), for some suitably chosen threshold s. Since the probability P L + 6 ≥ s is increasing on the mass of the upper parts of the fixed marginals and no more than (1 − α) probability mass can be allocated to the upper part of the optimal solution, it is intuitively obvious that the optimal solution should use only the largest (1−α) part of each marginal component.…”

Distribution functions, extremal limits and optimal transport

“…For this purpose it is possible to utilize variants of the optimal transport problem: try to minimize a risk measure of a portfolio of several risks with respect to their distribution while preserving their marginals. Some recent publications studying problems from risk management are Rüschendorf [40], Puccetti & Rüschendorf [36] or Bernard et al [11]. A paper dealing with model independent bounds on option prices using theory of optimal transport is Beiglböck et al [7].…”