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Comparative and qualitative robustness for law-invariant risk measures

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  • Volker Krätschmer
  • Alexander Schied
  • Henryk Zähle

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. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Volker Krätschmer & Alexander Schied & Henryk Zähle, 2014. "Comparative and qualitative robustness for law-invariant risk measures," Finance and Stochastics, Springer, vol. 18(2), pages 271-295, April.
    • Handle: RePEc:spr:finsto:v:18:y:2014:i:2:p:271-295
    DOI: 10.1007/s00780-013-0225-4
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    References listed on IDEAS

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    More about this item

    Keywords

    Law-invariant risk measure; Convex risk measure; Coherent risk measure; Orlicz space; Qualitative robustness; Comparative robustness; Index of qualitative robustness; Hampel’s theorem; ψ-Weak topology; Distortion risk measure; Skorohod representation; 62G35; 60B10; 60F05; 91B30; 28A33; D81;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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