Quantitative statistical robustness for tail-dependent law invariant risk measures

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.

Keywords:

• Quantitative robustness
• tail-dependent law invariant risk measures
• Fortet–Mourier metric
• admissible sets
• index of quantitative robustness

JEL Classification:

• D81

Acknowledgments

The authors would like to thank the two anonymous referees for insightful comments and constructive suggestions which have significantly helped us strengthen the paper. They are also thankful to the associate editor for organizing an efficient review.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We note that the canonical model space for law invariant convex risk measure is $L^1$ (Filipović and Svindland 2012).

2 ζ-metric is needed because when we use the Kantorovich metric to measure the distance between two statistical estimators, the dual formulation of the metric enables us to reformulate the distance as the difference between $P^{\otimes N}$ and $Q^{\otimes N}$ under certain ζ-metric, see (35(35) $\begin{array}{rl}& {\mathsf{d}\mathsf{l}}_K\left(P^{\otimes N}\circ {\hat{\varrho }}_N^{-1},Q^{\otimes N}\circ {\hat{\varrho }}_N^{-1}\right)\\ & =\underset{\psi \in {\mathcal{F}}_1\left(\mathbb{R}\right)}{sup}\left|{\int }_{\mathbb{R}}\psi \left(t\right)P^{\otimes N}\circ {\hat{\varrho }}_N^{-1}\left(\mathrm{d}t\right)\right.\\ & -\left.{\int }_{\mathbb{R}}\psi \left(t\right)Q^{\otimes N}\circ {\hat{\varrho }}_N^{-1}\left(\mathrm{d}t\right)\right|\\ & =\underset{\psi \in {\mathcal{F}}_1\left(\mathbb{R}\right)}{sup}\left|{\int }_{{\mathbb{R}}^{\otimes N}}\psi \left(\varrho \right({\mathit{\xi }}^N\left)\right)P^{\otimes N}\left(\mathrm{d}{\mathit{\xi }}^N\right)\right.\\ & -\left.{\int }_{{\mathbb{R}}^{\otimes N}}\psi \left(\varrho \right({\mathit{\xi }}^N\left)\right)Q^{\otimes N}\left(\mathrm{d}{\mathit{\xi }}^N\right)\right|,\end{array}$(35) ) in the proof of Theorem 4.4.

Additional information

Funding

The work of the first author is partly carried out during his visit to the Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong supported by CUHK Direct Grant. His work is also supported by ESRC SCDTP [grant number ES/R501025/1]. The work of the second author is supported by a RGC [grant number 14500620].