Comparative and qualitative robustness for law-invariant risk measures |
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Authors: | Volker Krätschmer Alexander Schied Henryk Zähle |
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Affiliation: | 1. Faculty of Mathematics, University of Duisburg-Essen, Thea Leymann Str. 9, 45127, Essen, Germany 2. Department of Mathematics, University of Mannheim, A 5, 6, 68131, Mannheim, Germany 3. Department of Mathematics, Saarland University, Postbox 151150, 66041, Saarbrücken, Germany
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Abstract: | When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel’s classical notion of qualitative robustness is not suitable for risk measurement, and we propose and analyze a refined notion of robustness that applies to tail-dependent law-invariant convex risk measures on Orlicz spaces. This concept captures the tradeoff between robustness and sensitivity and can be quantified by an index of qualitative robustness. By means of this index, we can compare various risk measures, such as distortion risk measures, in regard to their degree of robustness. Our analysis also yields results of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for ψ-weak convergence. |
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