Comparative and qualitative robustness for law-invariant risk measures

Krätschmer, Volker; Schied, Alexander; Zähle, Henryk

doi:10.1007/s00780-013-0225-4

Cited by 146 publications

(171 citation statements)

References 35 publications

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“…Proof. The claim follows as a special case from [16], Proposition 2.22. We provide an independent proof in the Appendix B.…”

Section: Propositionmentioning

confidence: 78%

“…Following the observation of [22] that Orlicz spaces or Orlicz hearts (see Proposition 2 below) are appropriate in this case, [16] show that for p ∈ [0, ∞), L p is an Orlicz heart and thus can serve as a space of financial positions. In fact, based on [23], Theorem 2.10 of [16] shows that law-invariant convex risk measures on L ∞ has a unique extension to L 1 (convex, monotone, and lower semicontinuous with respect to the L 1 -norm), which inherits continuity properties.…”

Section: A Suitable Distance For Risk Managementmentioning

confidence: 95%

“…Recently, [16] have investigated the robustness properties of convex risk measures, in particular regarding the Wasserstein metric.…”

Section: Examplementioning

confidence: 99%

“…Ref. [16] propose the alternative notion of ψ-robustness. Here, ψ : P + → [0, ∞] is a left-continuous, nondecreasing convex (Young) function, which we call weight function in the following.…”

Section: Propositionmentioning

confidence: 99%

“…of forecast distributions. Our framework allows for utilisation of several results that have been established recently in [16].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Wasserstein Metric and Robustness in Risk Management

Kiesel

Rühlicke

Stahl³

et al. 2016

Risks

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Abstract:In the aftermath of the financial crisis, it was realized that the mathematical models used for the valuation of financial instruments and the quantification of risk inherent in portfolios consisting of these financial instruments exhibit a substantial model risk. Consequently, regulators and other stakeholders have started to require that the internal models used by financial institutions are robust. We present an approach to consistently incorporate the robustness requirements into the quantitative risk management process of a financial institution, with a special focus on insurance. We advocate the Wasserstein metric as the canonical metric for approximations in robust risk management and present supporting arguments. Representing risk measures as statistical functionals, we relate risk measures with the concept of robustness and hence continuity with respect to the Wasserstein metric. This allows us to use results from robust statistics concerning continuity and differentiability of functionals. Finally, we illustrate our approach via practical applications.

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“…Proof. The claim follows as a special case from [16], Proposition 2.22. We provide an independent proof in the Appendix B.…”

Section: Propositionmentioning

confidence: 78%

Section: A Suitable Distance For Risk Managementmentioning

confidence: 95%

“…Recently, [16] have investigated the robustness properties of convex risk measures, in particular regarding the Wasserstein metric.…”

Section: Examplementioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

“…of forecast distributions. Our framework allows for utilisation of several results that have been established recently in [16].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Wasserstein Metric and Robustness in Risk Management

Kiesel

Rühlicke

Stahl³

et al. 2016

Risks

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A comparison of Range Value at Risk (RVaR) forecasting models

Maria Müller,

Weber Gössling,

Solgon Santos

et al. 2023

Journal of Forecasting

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Risk forecasting is an important and helpful process for investors, fund managers, traders, and market makers. Choosing an inappropriate risk forecasting model can trigger irreversible losses. In this context, this study aims to evaluate the quality of different models to forecast the Range Value at Risk (RVaR) in univariate and multivariate analyses. The forecasts for other important measures like Value at Risk (VaR) and Expected Shortfall (ES) are also obtained. To assess the performance of both the univariate and multivariate models to RVaR forecasting, we consider an empirical exercise with different asset classes, rolling window estimations, and significance levels. We evaluated the empirical forecasts with the score functions of each risk measure. We identified that different models forecast different assets better, and the GARCH model with Student's and skewed Generalized Error distribution overcame the other distributions. We observed the RVine and CVine copulas as better models in the multivariate study. Besides, we noted that the models with Student's marginal distribution perform better according to realized loss (score function). We also note that RVaR forecasts follow the evolution of financial returns, showing an interesting measure to be used in industry and empirical investigations.

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Utility‐based shortfall risk: Efficient computations via Monte Carlo

Zhang

2018

Naval Research Logistics

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With the development of financial risk management, the notion of convex risk measures has been proposed and has gained increasing attentions. Utility‐based shortfall risk (SR), as a specific and important class of convex risk measures, has become popular in recent years. In this paper we focus on the computational aspects of SR, which are significantly understudied but fundamental for risk assessment and management. We discuss efficient estimation, optimization, and sensitivity analysis of SR, based on Monte Carlo techniques and stochastic optimization methods. We also conduct extensive numerical studies on the proposed approaches. The numerical results further demonstrate the effectiveness of these approaches.

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Comparative and qualitative robustness for law-invariant risk measures

Cited by 146 publications

References 35 publications

The Wasserstein Metric and Robustness in Risk Management

The Wasserstein Metric and Robustness in Risk Management

A comparison of Range Value at Risk (RVaR) forecasting models

Utility‐based shortfall risk: Efficient computations via Monte Carlo

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