Bounds for Functions of Dependent Risks

Embrechts, Paul; Puccetti, Giovanni

doi:10.1007/s00780-006-0005-5

Cited by 130 publications

(117 citation statements)

References 13 publications

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“…Recent results can be found in Albrecher et al (2006), Klüppelberg and Resnick (2008), Wang and Tang (2006), Asmussen and Rojas-Nandayapa (2008), Alink et al (2004), Embrechts and Puccetti (2006), and Ko and Tang (2008). Approximating this probability helps us evaluate risk measures for investment portfolios as well as estimating credit risk.…”

Section: Introductionmentioning

confidence: 99%

Aggregation of rapidly varying risks and asymptotic independence

Mitra¹,

Resnick²

2009

Adv. Appl. Probab.

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We study the tail behavior of the distribution of the sum of asymptotically independent risks whose marginal distributions belong to the maximal domain of attraction of the Gumbel distribution. We impose conditions on the distribution of the risks (X, Y ) such that P(X +Y > x) ∼ (constant) P(X > x). With the further assumption of nonnegativity of the risks, the result is extended to more than two risks. We note a sufficient condition for a distribution to belong to both the maximal domain of attraction of the Gumbel distribution and the subexponential class. We provide examples of distributions which satisfy our assumptions. The examples include cases where the marginal distributions of X and Y are subexponential and also cases where they are not. In addition, the asymptotic behavior of linear combinations of such risks with positive coefficients is explored, leading to an approximate solution of an optimization problem which is applied to portfolio design.

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Section: Introductionmentioning

confidence: 99%

Aggregation of rapidly varying risks and asymptotic independence

Mitra¹,

Resnick²

2009

Adv. Appl. Probab.

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“…The worst-case values of VaR for n = 2 were given in Makarov (1981) based on early results in multivariate probability theory. Embrechts and Puccetti (2006) gave a dual bound for the worst-case VaR for n 3 in the homogeneous model, i.e. all marginal risks have the same distribution.…”

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confidence: 99%

Aggregation-robustness and model uncertainty of regulatory risk measures

2015

Self Cite

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Research related to aggregation, robustness, and model uncertainty of regulatory risk measures, for instance, Value-at-Risk (VaR) and Expected Shortfall (ES), is of fundamental importance within quantitative risk management. In risk aggregation, marginal risks and their dependence structure are often modeled separately, leading to uncertainty arising at the level of a joint model. In this paper, we introduce a notion of qualitative robustness for risk measures, concerning the sensitivity of a risk measure to the uncertainty of dependence in risk aggregation. It turns out that coherent risk measures, such as ES, are more robust than VaR according to the new notion of robustness. We also give approximations and inequalities for aggregation and diversification of VaR under dependence uncertainty, and derive an asymptotic equivalence for worst-case VaR and ES under general conditions. We obtain that for a portfolio of a large number of risks VaR generally has a larger uncertainty spread compared to ES. The results warn that unjustified diversification arguments for VaR used in risk management need to be taken with much care, and potentially support the use of ES in risk aggregation. This in particular reflects on the discussions in the recent consultative documents by the Basel Committee on Banking Supervision.

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“…In order to assess its quality from a numerical point of view, we need to know (at least some) optimal solutions to which we can compare the RA algorithm. The paper [12] presents a formula for obtaining VaR α (L + ) in the homogeneous case; see also [13,Proposition 4] or [14,Proposition 1]. In this section, we address the corresponding numerical aspects and algorithmic improvements.…”

Section: Known Optimal Solutions In the Homogeneous Case And Their Trmentioning

confidence: 99%

Improved algorithms for computing worst Value-at-Risk

Hofert

Memartoluie

Saunders

et al. 2017

Statistics &Amp; Risk Modeling

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Numerical challenges inherent in algorithms for computing worst Value-at-Risk in homogeneous portfolios are identified and solutions as well as words of warning concerning their implementation are provided. Furthermore, both conceptual and computational improvements to the Rearrangement Algorithm for approximating worst Value-at-Risk for portfolios with arbitrary marginal loss distributions are given. In particular, a novel Adaptive Rearrangement Algorithm is introduced and investigated. These algorithms are implemented using the R package qrmtools and may be of interest in any context in which it is required to find columnwise permutations of a matrix such that the minimal (maximal) row sum is maximized (minimized).

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Bounds for Functions of Dependent Risks

Abstract: Copulas, Dependent risks, Dependence bounds, Fréchet bounds, 60E15, 60E05, G10,

Cited by 130 publications

References 13 publications

Aggregation of rapidly varying risks and asymptotic independence

Aggregation of rapidly varying risks and asymptotic independence

Aggregation-robustness and model uncertainty of regulatory risk measures

Improved algorithms for computing worst Value-at-Risk

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