Recent crises in the financial industry have shown weaknesses in the modeling of Risk-Weighted Assets (RWAs). Relatively minor model changes may lead to substantial changes in the RWA numbers. Similar problems are encountered in the Value-at-Risk (VaR)-aggregation of risks. In this article, we...

Despite well-known shortcomings as a risk measure, Value-at-Risk (VaR) is still the industry and regulatory standard for the calculation of risk capital in banking and insurance. This paper is concerned with the numerical estimation of the VaR for a portfolio position as a function of different...

The problem of finding the best-possible lower bound on the distribution of a non-decreasing function of n dependent risks is solved when n=2 and a lower bound on the copula of the portfolio is provided. The problem gets much more complicated in arbitrary dimensions. When no information on the...

The Wasserstein barycenter is an important notion in the analysis of high dimensional data with a broad range of applications in applied probability, economics, statistics, and in particular to clustering and image processing. We state a general version of the equivalence of the Wasserstein...

We introduce a new algorithm, called the swapping algorithm, to approximate numerically the minimal and maximal expected inner product of two random vectors with given marginal distributions. As a direct application, the algorithm computes an approximation of the L2-Wasserstein distance between...

We introduce the concepts of φ-complete mixability and φ-joint mixability and we investigate some necessary and sufficient conditions to the φ-mixability of a set of distribution functions for some supermodular functions φ. We give examples and numerical verifications which confirm our findings