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Quantitative Statistical Robustness for Tail-Dependent Law Invariant Risk Measures

Quantitative Statistical Robustness for Tail-Dependent Law Invariant Risk Measures
Quantitative Statistical Robustness for Tail-Dependent Law Invariant Risk Measures
When estimating the risk of a financial position with empirical data or Monte Carlo simulations via a tail-dependent law invariant risk measure such as the Conditional Value-at-Risk (CVaR), it is important to ensure the robustness of the plug-in estimator particularly when the data contain noise. Krätschmer et al. [Comparative and qualitative robustness for law invariant risk measures. Financ. Stoch., 2014, 18, 271–295.] propose a new framework to examine the qualitative robustness of such estimators for the tail-dependent law invariant risk measures on Orlicz spaces, which is a step further from an earlier work by Cont et al. [Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance, 2010, 10, 593–606] for studying the robustness of risk measurement procedures. In this paper, we follow this stream of research to propose a quantitative approach for verifying the statistical robustness of tail-dependent law invariant risk measures. A distinct feature of our approach is that we use the Fortet–Mourier metric to quantify variation of the true underlying probability measure in the analysis of the discrepancy between the law of the plug-in estimator of the risk measure based on the true data and the one based on perturbed data. This approach enables us to derive an explicit error bound for the discrepancy when the risk functional is Lipschitz continuous over a class of admissible sets. Moreover, the newly introduced notion of Lipschitz continuity allows us to examine the degree of robustness for tail-dependent risk measures. Finally, we apply our quantitative approach to some well-known risk measures to illustrate our results and give an example of the tightness of the proposed error bound.
Fortet–Mourier metric, Quantitative robustness, admissible sets, index of quantitative robustness, tail-dependent law invariant risk measures
1469-7688
1669-1685
Wang, Wei
85862755-49c9-4c7d-a1f4-d838d35cb7b7
Xu, Huifu
67f2baf6-df5b-476a-95cd-3e7580635d39
Ma, Tiejun
1f591849-f17c-4209-9f42-e6587b499bae
Wang, Wei
85862755-49c9-4c7d-a1f4-d838d35cb7b7
Xu, Huifu
67f2baf6-df5b-476a-95cd-3e7580635d39
Ma, Tiejun
1f591849-f17c-4209-9f42-e6587b499bae

Wang, Wei, Xu, Huifu and Ma, Tiejun (2021) Quantitative Statistical Robustness for Tail-Dependent Law Invariant Risk Measures. Quantitative Finance, 21 (10), 1669-1685. (doi:10.1080/14697688.2021.1892171).

Record type: Article

Abstract

When estimating the risk of a financial position with empirical data or Monte Carlo simulations via a tail-dependent law invariant risk measure such as the Conditional Value-at-Risk (CVaR), it is important to ensure the robustness of the plug-in estimator particularly when the data contain noise. Krätschmer et al. [Comparative and qualitative robustness for law invariant risk measures. Financ. Stoch., 2014, 18, 271–295.] propose a new framework to examine the qualitative robustness of such estimators for the tail-dependent law invariant risk measures on Orlicz spaces, which is a step further from an earlier work by Cont et al. [Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance, 2010, 10, 593–606] for studying the robustness of risk measurement procedures. In this paper, we follow this stream of research to propose a quantitative approach for verifying the statistical robustness of tail-dependent law invariant risk measures. A distinct feature of our approach is that we use the Fortet–Mourier metric to quantify variation of the true underlying probability measure in the analysis of the discrepancy between the law of the plug-in estimator of the risk measure based on the true data and the one based on perturbed data. This approach enables us to derive an explicit error bound for the discrepancy when the risk functional is Lipschitz continuous over a class of admissible sets. Moreover, the newly introduced notion of Lipschitz continuity allows us to examine the degree of robustness for tail-dependent risk measures. Finally, we apply our quantitative approach to some well-known risk measures to illustrate our results and give an example of the tightness of the proposed error bound.

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More information

Accepted/In Press date: 12 February 2021
Published date: 26 March 2021
Keywords: Fortet–Mourier metric, Quantitative robustness, admissible sets, index of quantitative robustness, tail-dependent law invariant risk measures

Identifiers

Local EPrints ID: 449311
URI: http://eprints.soton.ac.uk/id/eprint/449311
ISSN: 1469-7688
PURE UUID: e717b21b-5408-4da1-a70f-ea9502ec220f

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Date deposited: 24 May 2021 16:31
Last modified: 27 Apr 2022 11:41

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Contributors

Author: Wei Wang
Author: Huifu Xu
Author: Tiejun Ma

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