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 highlight some of the underlying issues, both methodologically, as well as through examples. In particular, we frame this discussion in the context of two recent regulatory documents we refer to as Basel 3.5.
The probabilistic characterization of the relationship between two or more
random variables calls for a notion of dependence. Dependence modeling leads to
mathematical and statistical challenges, and recent developments in extremal
dependence concepts have drawn a lot of attention to probability and its
applications in several disciplines. The aim of this paper is to review various
concepts of extremal positive and negative dependence, including several
recently established results, reconstruct their history, link them to
probabilistic optimization problems, and provide a list of open questions in
this area. While the concept of extremal positive dependence is agreed upon for
random vectors of arbitrary dimensions, various notions of extremal negative
dependence arise when more than two random variables are involved. We review
existing popular concepts of extremal negative dependence given in literature
and introduce a novel notion, which in a general sense includes the existing
ones as particular cases. Even if much of the literature on dependence is
focused on positive dependence, we show that negative dependence plays an
equally important role in the solution of many optimization problems. While the
most popular tool used nowadays to model dependence is that of a copula
function, in this paper we use the equivalent concept of a set of
rearrangements. This is not only for historical reasons. Rearrangement
functions describe the relationship between random variables in a completely
deterministic way, allow a deeper understanding of dependence itself, and have
several advantages on the approximation of solutions in a broad class of
optimization problems.Comment: Published at http://dx.doi.org/10.1214/15-STS525 in the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Li et al. [Distributions with Fixed Marginals and Related Topics, vol. 28, Institute of Mathematics and Statistics, Hayward, CA, 1996, pp. 198-212] provide bounds on the distribution and on the tail for functions of dependent random vectors having fixed multivariate marginals. In this paper, we correct a result stated in the above article and we give improved bounds in the case of the sum of identically distributed random vectors. Moreover, we provide the dependence structures meeting the bounds when the fixed marginals are uniformly distributed on the k-dimensional hypercube. Finally, a definition of a multivariate risk measure is given along with actuarial/financial applications.
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.
We describe a numerical procedure to obtain bounds on the distribution function of a sum of n dependent risks having fixed marginals. With respect to the existing literature, our method provides improved bounds and can be applied also to large non-homogeneous portfolios of risks. As an application, we compute the VaR-based minimum capital requirement for a portfolio of operational risk losses.
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