Coherence and elicitability

Johanna F. Ziegel

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Coherence and elicitability

Johanna F. Ziegel

arXiv:1303.1690v3Mar 7, 2013q-fin.RMmath.STq-fin.ST

TL;DR

The paper addresses how to verify and compare forecasts for risk measures when statistical decision theory requires elicitability. It analyzes law-invariant coherent and spectral risk measures, showing that spectral measures are generally not elicitable while certain expectiles are the only elicitable law-invariant coherent measures. Consequently, direct ranking of competing estimation procedures is unavailable for spectral measures through consistent scoring functions.

Problem
Risk measures estimated from historical data need decision-theoretically sound verification and comparison, but spectral measures such as expected shortfall are not elicitable.
Method
The paper analyzes elicitability for law-invariant coherent risk measures using necessary conditions and classes of probability distributions including two-point distributions.
Results
Expectiles are the only elicitable law-invariant coherent risk measures, while spectral risk measures are not elicitable except for minus the expected value.
Takeaways & Limitations
For spectral risk measures, backtesting a fixed procedure does not support quality ranking; comparisons should use consistent scoring functions, which do not exist for these measures.
Takeaways & Limitations
Spectral risk measures face a reported conflict between subadditivity and robustness, and the paper cautions that its non-elicitability result does not rule out backtesting under fixed model assumptions.

Abstract

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The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile based risk measures such as value at risk are elicitable. In this paper we show that law-invariant spectral risk measures such as expected shortfall are not elicitable unless they reduce to minus the expected value. Hence, it is unclear how to perform forecast verification or comparison. However, the class of elicitable law-invariant coherent risk measures does not reduce to minus the expected value. We show that it consists of certain expectiles.

1 Introduction

Financial risk measures must support verification and comparison of competing estimation procedures, but spectral measures face statistical and decision-theoretic limitations. The paper examines elicitability and identifies expectiles as the elicitable law-invariant coherent risk measures.

Risk measures: Value at Risk is widely used, while expected shortfall and other spectral measures offer coherence and sensitivity to losses beyond a threshold.ES generalizes VaR's loss-sensitive perspective, and spectral risk measures generalize ES.
Risk measures: Spectral risk measures have statistical drawbacks, including a reported conflict between subadditivity and robustness of risk measurement procedures.The paper argues that robustness should be considered alongside coherence when designing risk measurement procedures.
Motivation: Risk measures are estimated from historical data, so forecasting requires methods for verifying and comparing competing procedures.The paper frames backtesting as validating an estimation procedure and considers forecasts for future risk measures.
Elicitability: The paper asks whether any interesting law-invariant coherent risk measure is also elicitable and shows that only minus the expected value is both spectral and elicitable.Elicitability is presented as relevant to sound forecast verification and comparison.
Main results: Expectiles are the only elicitable law-invariant coherent risk measures, extending elicitability beyond minus the expected value.The paper notes that expectiles were introduced earlier, are elicitable from their definition, and are coherent risk measures.
Implications: Existing procedures can evaluate and test ES forecasts but do not permit direct comparison and ranking of competing forecasting methods.Related regression approaches for ES are discussed, but the forecast-comparison problem remains.

2 Elicitability

Elicitability links risk-measure forecasting to strictly consistent scoring rules, enabling optimal forecasts and comparison of competing procedures. The section introduces convex level sets as a necessary condition and contrasts elicitable quantiles with non-elicitable expected shortfall.

2 Elicitability: The paper studies these forecasting properties for functionals on probability distributions, with future observations modeled by a real-valued random variable.The framework treats law-invariant coherent risk measures as functionals on a class of distributions on R.
2 Elicitability: A scoring function is consistent for a functional when forecasts in the functional's value set minimize expected score, with strict consistency identifying the functional.An optimal forecast minimizes expected scoring loss, and consistent scores can rank competing forecast procedures.
2 Elicitability: Elicitability matters because inconsistent scoring functions can produce seriously misguided conclusions about forecast quality.Consistent scoring rules reward accurate forecasts of the target functional.
2 Elicitability: Value at risk is essentially a quantile and is elicitable, whereas expected shortfall is not elicitable.Consistent scoring functions for α-quantiles have been characterized under regularity and integrability conditions.
2 Elicitability: An elicitable functional has convex level sets: a common value for two distributions remains a valid value under their mixture.The paper proves this necessary condition using strict consistency and linearity of expected scores under mixtures.

3 Coherent risk measures

Coherent risk measures satisfy monotonicity, subadditivity, positive homogeneity, and translation invariance, and common applied measures are law-invariant. The section presents the Kusuoka representation and spectral-risk-measure formulation for law-invariant coherent risk measures.

3 Coherent risk measures: A coherent risk measure is monotone, subadditive, positively homogeneous, and translation invariant on bounded random variables.The definition specifies the domain L∞ and gives the corresponding inequalities and equalities for each property.
3 Coherent risk measures: Law invariance means that risk measures assign the same value to random variables sharing the same distribution.The paper notes that common coherent risk measures used in applications have this property.
3 Coherent risk measures: Kusuoka's theorem represents every law-invariant coherent risk measure through a closed convex set of probability measures on [0, 1].The theorem uses the weak topology on the set of probability measures over [0,1].
3 Coherent risk measures: For a probability measure m, the functional ν_m integrates the spectral family U_α(Y) over α, with U_0(Y) equal to the essential infimum.The representation is given for bounded random variables.
3 Coherent risk measures: Up to sign, the functionals ν_m are exactly Acerbi's spectral risk measures and also admit an alternative representation using a spectral function g_m.The spectral function is left-continuous and decreasing, and the resulting functional is finite on bounded random variables.

4 Coherence and elicitability

The paper characterizes which law-invariant coherent risk measures can be elicited: spectral risk measures generally fail elicitability, while certain expectiles satisfy it.

4.1 Coherent functionals with convex level sets: A functional that is not elicitable on two-point distributions cannot be elicitable on any larger distribution class.Two-point distributions therefore provide a sufficient test for disproving elicitability.
4.1 Coherent functionals with convex level sets: Theorem 4.1 derives necessary restrictions on the generating set M when a law-invariant coherent functional has convex level sets.The restrictions exclude measures assigning mass to zero and require a common structural lower bound involving some C ∈ (0, 1].
4.1 Coherent functionals with convex level sets: For two-point distributions, convexity of level sets constrains the functional to an affine form with coefficient α_p ∈ (0, 1), enabling an explicit formula in terms of C and p.The assumption δ0 ∉ M ensures α_p < 1 for at least one p, after which convexity yields the stated range for all p.
4.2 Spectral risk measures: Spectral risk measures other than minus the expected value are not elicitable on any class containing the two-point distributions.This includes the class of spectral risk measures considered in the paper; the proof also rules out minus the essential infimum.
4.3 Expectiles: The constructed spectral risk measure u_C is not elicitable unless C = 1, when it reduces to minus the expected value.Theorem 4.1 also yields ρ(X) = l_C(X), establishing the expectile representation for the characterized coherent risk measure.
4.3 Expectiles: The paper identifies certain expectiles as law-invariant coherent risk measures that are elicitable, with the coherent measure l_C equal to minus the τ-expectile.Proposition 4.4 gives τ = C/(C + 1), while expectile elicitability follows from the definition and consistent scoring functions.

5 Discussion

The paper discusses the difficulty of objectively comparing forecasts for spectral risk measures and considers probabilistic forecasts and expectiles as possible directions.

Spectral risk measures are not elicitable, making decision-theoretically sound ranking of their point forecasts unclear.The paper distinguishes this issue from backtesting a single procedure under fixed model assumptions.
Backtesting statistics should not be used to quality-rank competing estimation methods for spectral risk measures.Such rankings require consistent scoring functions, which do not exist for spectral risk measures.
Probabilistic forecasts could be assessed directly instead of reducing an estimated loss distribution to a single spectral-risk forecast.The proposed alternative evaluates the estimated distribution itself.
Exceedance residuals can serve as a backtesting score for ES but are not scoring functions in the paper’s sense because they depend on VaR and ES.The paper suggests extending elicitability, including the open question of whether the pair is 2-elicitable.
The discussion argues that robustness, elicitability, and the necessity of subadditivity should all inform risk-measure choice.It highlights expectiles’ potential given their statistical properties, while noting that subadditivity conflicts with robustness for spectral measures.