The complete mixability and convex minimization problems with monotone marginal densities

Wang, Bin; Wang, Ruodu

doi:10.1016/j.jmva.2011.05.002

Cited by 150 publications

(153 citation statements)

References 16 publications

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“…Most importantly, Puccetti and Rüschendorf [45] introduced the so-called rearrangement algorithm which is a fast procedure to numerically compute the bounds of interest. Under quite restrictive assumptions, analytical bounds can be computed based on the notion of complete mixability, which was introduced in Wang and Wang [54].…”

Section: Introductionmentioning

confidence: 99%

A Review on Ambiguity in Stochastic Portfolio Optimization

Pflug

Pohl

2017

Set-Valued Var. Anal

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In mean-risk portfolio optimization, it is typically assumed that the assets follow a known distribution P 0 , which is estimated from observed data. Aiming at an investment strategy which is robust against possible misspecification of P 0 , the portfolio selection problem is solved with respect to the worst-case distribution within a Wasserstein-neighborhood of P 0 . We review tractable formulations of the portfolio selection problem under model ambiguity, as it is called in the literature. For instance, it is known that high model ambiguity leads to equally-weighted portfolio diversification. However, it often happens that the marginal distributions of the assets can be estimated with high accuracy, whereas the dependence structure between the assets remains ambiguous. This leads to the problem of portfolio selection under dependence uncertainty. We show that in this case portfolio concentration becomes optimal as the uncertainty with respect to the estimated dependence structure increases. Hence, distributionally robust portfolio optimization can have two very distinct implications: Diversification on the one hand and concentration on the other hand.

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Section: Introductionmentioning

confidence: 99%

A Review on Ambiguity in Stochastic Portfolio Optimization

Pflug

Pohl

2017

Set-Valued Var. Anal

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“…all marginal risks have the same distribution. Partial solutions for the worst-case and best-case values of VaR are to be found in Wang et al (2013), Puccetti and Rüschendorf (2013) and Bernard et al (2014), based on the notion of complete mixability introduced in Wang and Wang (2011). A fast algorithm to numerically calculate the worst-case and best-case values of VaR under general conditions was introduced in Embrechts et al (2013).…”

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confidence: 99%

Aggregation-robustness and model uncertainty of regulatory risk measures

2015

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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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“…If the distribution of S = X 1 + X 2 + · · · + X d has minimum variance it must actually hold that 3 Indeed var d k=1 X k = var X j + k = j X k , and a necessary condition for var d k=1 X k to become minimum is that each X j is anti-monotonic with k = j X k . 4 The concept of mixability has been introduced in Wang and Wang (2011) and further extended in Wang and Wang (2016); see also Gaffke and Rüschendorf (1981) for some early results that are connected. 5 Each time a column is effectively rearranged, the variance of the sum strictly decreases.…”

Section: Standard Rearrangement Algorithm (Ra)mentioning

confidence: 99%

Rearrangement algorithm and maximum entropy

2017

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We study properties of the block rearrangement algorithm (BRA) in the context of inferring dependence among variables given their marginal distributions and the distribution of their sum. We show that when all distributions are Gaussian the BRA yields solutions that are "close to each other" and exhibit almost maximum entropy, i.e., the inferred dependence is Gaussian with a correlation matrix that has maximum possible determinant. We provide evidence that, when the distributions are no longer Gaussian, the property of maximum determinant continues to hold. The consequences of these findings are that the BRA can be used as a stable algorithm for inferring a dependence that is economically meaningful.

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The complete mixability and convex minimization problems with monotone marginal densities

Cited by 150 publications

References 16 publications

A Review on Ambiguity in Stochastic Portfolio Optimization

A Review on Ambiguity in Stochastic Portfolio Optimization

Aggregation-robustness and model uncertainty of regulatory risk measures

Rearrangement algorithm and maximum entropy

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