Source-linked AI summary
Comparative and qualitative robustness for law-invariant risk measures
Volker Krätschmer, Alexander Schied, Henryk Zähle
TL;DR
The paper examines robustness when estimating risk from historical data or Monte Carlo simulation, arguing that Hampel’s notion is unsuitable for risk measurement. It develops a refined robustness framework for law-invariant convex risk measures on Orlicz spaces, quantifies robustness with an index, and derives related continuity, consistency, and representation results.
Problem
Hampel’s classical qualitative robustness is not suitable for risk measurement, where stable estimates and the tradeoff between robustness and tail sensitivity matter.
Method
The paper analyzes law-invariant convex risk measures on Orlicz spaces using a refined notion of qualitative robustness and its numerical index.
Results
The analysis compares risk measures by robustness degree and establishes continuity, estimator consistency, and a Skorohod representation theorem linking ψ-weak convergence with norm convergence in Orlicz space.
Takeaways & Limitations
The robustness index provides a way to compare risk measures while capturing their differing balance between robustness and tail sensitivity.
Takeaways & Limitations
The framework uses Orlicz spaces for potentially unbounded risks, and consistency results require assumptions such as a strong law of large numbers for the data sequence.
Abstract
from arXiv · showhide
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 space. This concept of robustness 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 that are of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for ψ-weak convergence.
1 Introduction
The paper argues that Hampel’s qualitative robustness is ill-suited to risk measurement because it ignores tail sensitivity and imposes an artificial robust/non-robust split. It develops a graded robustness framework for law-invariant convex risk measures, alongside consistency, continuity, and representation results.
- Motivation: Hampel’s robustness can reject all law-invariant coherent risk functionals, even ordinary expectation, while essentially accepting Value at Risk.This sharp contrast motivates replacing the classical criterion for risk measurement.
- Motivation: Weakly close P&L laws may have very different tails, so robustness that suppresses tail sensitivity can underestimate risk.The paper links this concern to the consequences of faulty tail assessment during financial crises.
- Contribution: The refined notion assigns most risk measures a degree of robustness, capturing a continuum and the tradeoff between robustness and tail sensitivity.The degree is quantified by an index ranging from 0 to infinity, with extremes corresponding to full tail sensitivity and Hampel’s robustness.
- Setting: The analysis studies law-invariant convex risk measures on Orlicz spaces, which accommodate potentially unbounded P&Ls.The paper emphasizes continuity properties and the role of the Δ2-condition for the supporting Orlicz space.
- Results: The paper proves estimator consistency for stationary and ergodic data and develops stronger robustness results for statistical functionals.It also presents a Skorohod representation theorem connecting ψ-weak convergence with norm convergence in Orlicz space.
2 Statement of main results
The paper develops consistency, continuity, and refined robustness results for law-invariant convex risk measures on Orlicz spaces. It replaces a binary robustness classification with comparative degrees that quantify the tradeoff between robustness and tail sensitivity.
- 2.1 Setup: The framework studies law-invariant convex risk measures on Orlicz spaces, including standard cash-additive risk measures and possible relaxations of cash additivity.Orlicz spaces accommodate unbounded P&Ls, while the presentation retains the standard cash-additive framework.
- 2.2 Consistency: Strong consistency holds for the empirical estimator when observations form a stationary and ergodic sequence with the target law.The estimator converges almost surely to the risk ρ(X).
- 2.3 Continuity properties of Rρ: Continuity of Rρ in the Ψ-weak topology for every law-invariant convex risk measure is equivalent to Ψ satisfying the ∆2-condition.For Ψ_p(x)=x^p/p, this yields continuity under the Wasserstein metric of order p.
- 2.4 Qualitative and comparative robustness: The same ∆2-condition is equivalent to Ψ-robustness of every associated risk functional on M(HΨ).For risk measures on L∞, robustness is likewise characterized through the corresponding continuity conditions.
- 2.4 Qualitative and comparative robustness: Robustness can be compared across risk measures through an implication-based ordering defined on L∞ risk functionals.A measure is more robust when it remains Ψ-robust for some finite Young function where the comparison measure does not.
- 2.4 Qualitative and comparative robustness: The qualitative-robustness index is at most 1, equals 1/p for one-sided-moment risk measures, and spans the full range for the stated distortion family.For gβ(t)=(t/α)^β∧1, the index is β; AV@Rα has index 1.
3 Some general results
This section develops ψ-weak versions of Hampel’s robustness results and characterizes ψ-weak convergence through norm convergence in Orlicz spaces. These tools support the paper’s robustness analysis and the Skorohod representation theorem.
- General framework: The paper replaces weak-topology robustness with a finer ψ-weak topology to obtain a more balanced robustness picture.The ψ-weak framework is designed for robust statistics and ψ-weak convergence.
- Robustness criteria: ψ-robustness is defined using uniformly ψ-integrating neighborhoods and eventual control of estimator distributions under the Prohorov metric.The definition quantifies over every uniformly ψ-integrating set containing the reference measure.
- Robustness criteria: ψ-weak continuity implies ψ-robustness, while ψ-robustness plus local weak consistency implies ψ-weak continuity.The two results provide Hampel-type and converse criteria for the ψ-weak topology.
- Skorohod representation: For a finite Young function Ψ, Ψ-weak convergence is equivalent to the existence of coupled random variables converging in the Luxemburg norm, provided Ψ satisfies the Δ2-condition.The equivalence links law convergence with norm convergence in HΨ.
- Skorohod representation: The proof propagates uniform integrability from Ψ(|X_n|) to Ψ(2^m|X_n−X_0|) and then uses Vitali’s theorem to obtain norm convergence.The argument proceeds through the uniform-integrability lemma and the two implications of Theorem 3.5.
4 Proofs of the results from Section 2
These proofs establish continuity and robustness properties of risk functionals by combining ergodic and Skorohod representations, Orlicz-space convergence, and structural conditions such as the Δ2-condition.
- Continuity and representation: Skorohod coupling and uniform integrability yield Luxemburg-norm convergence, after which norm continuity transfers convergence to the risk functional.The proof obtains convergence of Ψ(a|X_n−X_0|) expectations and then applies continuity of ρ.
- Continuity and representation: If Ψ fails the Δ2-condition, a utility-based shortfall risk measure can make Rρ discontinuous for Ψ-weak convergence.The construction uses a random variable with finite Ψ(Y) expectation but infinite Ψ(2Y) expectation.
- Continuity and representation: A sequence of laws can converge Ψ-weakly to δ0 while the corresponding risk values are unbounded, proving failure of Ψ-weak continuity.The proof constructs bounded random variables whose laws converge to δ0 but whose risk values do not remain bounded.
- Robustness consequences: Robustness combined with strong consistency forces continuity of Rρ and, consequently, the Δ2-condition on Ψ.This implication is used in the proof of the characterization theorem.
- Risk-measure structure: For distortion-type risk measures, finiteness on HΨ is characterized through the relevant quantile function belonging to the dual Orlicz space.The proof uses the Banach–Steinhaus theorem and the upper Hardy–Littlewood inequality.