Comparative and qualitative robustness for law-invariant risk measures

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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Acknowledgements

The authors thank Freddy Delbaen, Paul Embrechts, Marco Frittelli and two anonymous referees for comments, which helped to improve a previous draft of the paper.

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Correspondence to Volker Krätschmer.

Appendix: Auxiliary results on the ψ-weak topology

Appendix: Auxiliary results on the ψ-weak topology

First recall from Corollary A.45 in [21] that the ψ-weak topology on \(\mathcal{M}_{1}^{\psi}(\mathbb{R})\) is separable and metrizable. The following lemma provides some useful characterizations of ψ-weak convergence; see [26, Lemma 3.4] for a proof.

Lemma A.1

The following statements are equivalent:

  1. (i)

    μ n μ ψ-weakly.

  2. (ii)

    f n →∫f for every \(f\in C_{\psi}(\mathbb {R})\).

  3. (iii)

    f n →∫f for every continuous f with compact support and for f=ψ.

  4. (iv)

    μ n μ weakly andψ n →∫ψ.

The following lemma gives a transparent characterization of the ψ-weakly compact subsets of \({\mathcal{M}}_{1}^{\psi}\). Recall that a set \({\mathcal{N}}\subset{\mathcal{M}}_{1}^{\psi}\) is called uniformly ψ-integrating if it satisfies (2.12).

Lemma A.2

A set \({\mathcal{N}}\subset{\mathcal{M}}_{1}^{\psi}\) is relatively compact for the ψ-weak topology if and only if there exists a measurable function \(\phi:\mathbb{R}\to[0,\infty)\) such that ϕ(x)/ψ(x)→∞ as |x|→∞ and such that

$$ \sup_{\nu\in{\mathcal{N}}}\int\phi\,d\nu<\infty. $$
(A.1)

In this case, \({\mathcal{N}}\) is uniformly ψ-integrating.

Proof

The first statement is an immediate consequence of Corollary A.47 in [21]. For bounded ψ, the second statement is trivial. To prove it for unbounded ψ, we assume without loss of generality that ϕ>0. Fix ε>0, and denote by K the left-hand side of (A.1). Choosing M 1>0 so large that ψ(x)/ϕ(x)≤ε/K when |x|≥M 1, and choosing M 0>0 so large that ψ(x)≥M 0 implies |x|≥M 1, we obtain

for all MM 0. That is, (2.12) holds. □

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Krätschmer, V., Schied, A. & Zähle, H. Comparative and qualitative robustness for law-invariant risk measures. Finance Stoch 18, 271–295 (2014). https://doi.org/10.1007/s00780-013-0225-4

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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

Mathematics Subject Classification (2010)

  • 62G35
  • 60B10
  • 60F05
  • 91B30
  • 28A33

JEL Classification

  • D81