1,721,027 research outputs found
Sharp bounds on the expected shortfall for a sum of dependent random variables
No full textUsing a connection between the rearrangement algorithm introduced in Puccetti and Rüschendorf (2012) and convex order, we show how to compute the best-possible expected shortfall for the sum of d random variables having fixed marginal distributions
An algorithm to approximate the optimal expected inner product of two vectors with given marginals
No full textWe 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 two multivariate measures. The algorithm is simple to implement, accurate and less computationally expensive than the algorithms generally used in the literature for this problem. The algorithm also provides a discretized image of optimal measures and can be extended to more general cost functionals
Extremal Dependence Concepts
Full text linkThe 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 devel- opments 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 ques- tions 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 de- pendence given in literature and introduce a novel notion, which in a gen- eral 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. Re- arrangement 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
Computation of sharp bounds on the expected value of a supermodular function of risks with given marginals
No full textWe show that the rearrangement algorithm (RA) introduced in Puccetti and Rüschendorf (2012) to compute distributional bounds can be used also to compute sharp lower and upper bounds on the expected value of a supermodular function of d random variables having fixed marginal distributions. Compared to the analytical methods existing in the literature the algorithm is widely applicable, more easily obtained and gives insight into the dependence structures attaining the bounds
Bounds on total economic capital : the DNB case study
No full textMost banks use the top-down approach to aggregate their risk types when computing total economic capital. Following this approach, marginal distributions for each risk type are first independently estimated and then merged into a joint model using a copula function. Due to lack of reliable data, banks tend to manually select the copula as well as its parameters. In this paper we assess the model risk related to the choice of a specific copula function. The aim is to compute upper and lower bounds on the total economic capital for the aggregate loss distribution of DNB, the largest Norwegian bank, and the key tool for computing these bounds is the Rearrangement Algorithm introduced in Embrechts et al. (J. Bank. Financ. 37(8):2750–2764, 2013). The application of this algorithm to a real situation poses a series of numerical challenges and raises a number of warnings which we illustrate and discuss
Aggregating operational risk across matrix structured loss data
No full textWe study the problem of evaluating the risky position involved in a matrix of random losses with some given probabilistic structure. In the Basel II regulatory setup for operational risk in banking, we analyze how interde- pendencies between individual loss random variables within the matrix may influence different estimates for the minimum capital charge required
Bounds for joint portfolios of dependent risks
No full textIn this paper, we survey, extend and improve several bounds for the distri- bution function and the tail probabilities of portfolios, where the dependence structure within the portfolio is completely unknown or only partially known. We present various methods for obtaining bounds based on rearrangements, duality theory, conditional moments and reduction techniques. In particular, we consider the case where only the simple marginal distributions are known, the general overlapping marginals case where certain joint distributions are known and the case of additional restrictions on the dependence structure, as, for example, the restriction to positive dependence. Some of the bounds pose a considerable numerical challenge. We discuss the quality of the bounds and numerical aspects in some examples
Detecting complete and joint mixability
No full textWe introduce the Mixability Detection Procedure (MDP) to check whether a set of d distribution functions is jointly mixable at a given confidence level. The procedure is based on newly established results regarding the convergence rate of the minimal variance problem within the class of joint distribution functions with given marginals. The MDP is able to detect the complete mixability of an arbitrary set of distributions, even in those cases not covered by theoretical results. Stress-tests against borderline cases show that the MDP is fast and reliable
Sharp bounds for sums of dependent risks
No full textSharp 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 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
Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates
No full textWe give a new sufficient condition for a continuous distribution to be completely mixable, and we use this condition to show that the worst-possible value-at-risk for the sum of d inhomogeneous risks is equivalent to the worst-possible expected shortfall under the same marginal assumptions, in the limit as d→∞. Numerical applications show that this equivalence holds also for relatively small dimensions d
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