Source-linked AI summary
What is the best risk measure in practice? A comparison of standard measures
Susanne Emmer, Marie Kratz, Dirk Tasche
TL;DR
The paper examines whether ES, VaR, or Expectiles best satisfy desirable risk-measure properties and support practical risk management, especially backtesting. It compares their coherence, diversification, robustness, and elicitability, and evaluates capital-allocation implications. Despite estimation and backtesting caveats, the authors conclude that ES remains the best practical choice and that evidence does not justify replacing it wholesale with Expectiles.
Problem
ES is coherent but not elicitable, making direct backtesting less straightforward, while Expectiles have been proposed as alternatives to ES and VaR.
Method
The paper compares VaR, ES, Expectiles, and other standard measures across mathematical properties, forecasting, backtesting, diversification, and capital allocation.
Results
ES has feasible backtesting approaches, although achieving the same certainty requires more validation data than VaR.
Takeaways & Limitations
ES seems best for practical use, while Expectiles remain alternatives for specific applications rather than an all-inclusive replacement.
Takeaways & Limitations
Diversification indices can be misleading in the presence of undiversifiable risk, so unrealized diversification potential may be more useful than absolute index levels.
Abstract
from arXiv · showhide
Expected Shortfall (ES) has been widely accepted as a risk measure that is conceptually superior to Value-at-Risk (VaR). At the same time, however, it has been criticised for issues relating to backtesting. In particular, ES has been found not to be elicitable which means that backtesting for ES is less straightforward than, e.g., backtesting for VaR. Expectiles have been suggested as potentially better alternatives to both ES and VaR. In this paper, we revisit commonly accepted desirable properties of risk measures like coherence, comonotonic additivity, robustness and elicitability. We check VaR, ES and Expectiles with regard to whether or not they enjoy these properties, with particular emphasis on Expectiles. We also consider their impact on capital allocation, an important issue in risk management. We find that, despite the caveats that apply to the estimation and backtesting of ES, it can be considered a good risk measure. As a consequence, there is no sufficient evidence to justify an all-inclusive replacement of ES by Expectiles in applications. For backtesting ES, we propose an empirical approach that consists in replacing ES by a set of four quantiles, which should allow to make use of backtesting methods for VaR. Keywords: Backtesting; capital allocation; coherence; diversification; elicitability; expected shortfall; expectile; forecasts; probability integral transform (PIT); risk measure; risk management; robustness; value-at-risk
1 Introduction
The paper compares standard risk measures and asks which properties and practical effects should guide the choice of a single capital amount. It focuses on variance, VaR, ES, and Expectiles, including their implications for risk management and model validation.
- 1 Introduction: Risk measures map loss distributions or random variables to capital amounts used as buffers against unexpected future losses.This supports expressing risk with one number for financial decision-making.
- 1 Introduction: The paper asks what properties define a good or best risk measure for financial institutions.These questions motivate the comparison of standard measures.
- 1 Introduction: Variance became dominant in finance but requires finite variance and implicitly assumes approximately symmetric risk distributions.These limitations motivate examining alternative downside risk measures.
- 1 Introduction: ES replaced VaR in many institutions because ES is coherent, while VaR is not coherent in all cases.The Basel Committee also recommends replacing VaR with ES in internal market risk models.
- 1 Introduction: The study compares variance, VaR, ES, and Expectile through mathematical properties, risk management, capital allocation, and model validation.The paper presents the comparison for both academics and professionals.
2 Risk measures: definition and basic properties
The paper defines risk measures through their capital-assessment role and evaluates properties including coherence, comonotonic additivity, elicitability, and robustness. Expectiles are coherent and elicitable in relevant settings, but they are not comonotonically additive, while conditional elicitability provides a two-step forecasting route for ES.
- 2.1 Coherence and related properties: A risk measure assigns a capital amount that acts as a buffer against unexpected future losses.The paper treats risk measures as mappings from probability distributions or random variables into real numbers.
- 2.1 Coherence and related properties: Coherence comprises homogeneity, subadditivity, monotonicity, and translation invariance.These axioms formalize properties expected from risk measures used in regulation and management.
- 2.1 Coherence and related properties: Comonotonic additivity complements subadditivity by avoiding diversification benefits for comonotonic risks.Comonotonicity represents a strongest-possible dependence structure in the paper’s discussion.
- 2.2 Elicitability: Elicitability means that a functional has a strictly consistent scoring function, enabling optimal point-forecast estimation and forecast comparison.Weighted absolute error elicits quantiles, while weighted squared error elicits expectiles.
- 2.3 Conditional Elicitability: Conditional elicitability allows a non-elicitable risk measure to be forecast in two steps by eliciting one component and then the conditional second component.For ES, the quantile is forecast first, followed by ES conditional on that quantile.
3 Properties of the standard risk measures
The paper compares standard risk measures across coherence, diversification, robustness, and elicitability. It shows that VaR's properties depend on distributional assumptions, while ES is coherent but has estimation and backtesting caveats; ES and Expectiles remain continuous under Wasserstein distance, and Expectiles are elicitable.
- Coherence and subadditivity: VaR is not subadditive in general, so it is not coherent and may fail to provide a diversification benefit or an enterprise-wide risk bound.This complicates decentralized risk management based on aggregating VaR figures across portfolios or business units.
- Coherence and subadditivity: VaR is asymptotically subadditive for iid regularly varying losses if and only if the tail index β is at least 1.For β < 1, VaR is asymptotically superadditive.
- Coherence and subadditivity: For elliptical distributions, VaR is subadditive on the stated set of linear portfolios when 0.5 < α < 1.The paper also reviews Archimedean survival dependence as another setting relevant to VaR subadditivity.
- Robustness: Under Wasserstein distance, mean, VaR, ES, and Expectiles are continuous, whereas weak-topology robustness is more restricted for VaR and fails for ES and Expectiles.VaR is weak-topology robust at F0 when F0^-1 is continuous at α; Expectiles are Lipschitz-continuous in Wasserstein distance.
- Robustness: Historical ES is more sensitive than VaR to added observations and to their sizes, with historical ES at the 99% level especially sensitive relative to Gaussian and Laplace ES.The paper notes a potential conflict between subadditivity, and therefore coherence, and robustness of a risk-measure estimate; ES estimation may use larger subsamples than VaR estimation.
- Elicitability: ES is coherent and sensitive to losses beyond VaR, but it is not directly elicitable; conditional elicitability nevertheless supports a two-step forecasting procedure.Expectiles are elicitable for distributions with finite means and therefore conditionally elicitable.
4 Capital allocation and diversification benefits
The section examines capital allocation through risk contributions and diversification indices, comparing how risk measures represent portfolio concentration and diversification. It emphasizes derivative-based allocations, limitations of diversification indices, and differences between VaR and ES.
- Capital allocation: Risk contributions can be defined as partial derivatives of portfolio risk, and Euler’s theorem makes them add up to total risk for homogeneous measures.This motivates derivative-based capital allocation, although the derivatives need not always exist.
- Capital allocation: For Expected Shortfall and Expectiles, the paper derives risk contributions using derivative-based definitions and Delbaen’s subgradient method.The Expectile result applies for 1/2 ≤ τ < 1 when the relevant partial derivative exists.
- Capital allocation: If loss distributions are not smooth, subgradient sets may contain multiple elements, leaving no unique candidate vector for risk contributions.Risk contributions can still be defined, but uniqueness may fail.
- Diversification benefits: A diversification index near 1 indicates that portfolio positions are almost comonotonic and therefore receive little diversification benefit.Comparing marginal and portfolio indices can help identify unrealized diversification potential.
- Diversification benefits: Absolute diversification-index thresholds are difficult to interpret when undiversifiable risk remains, so unrealized diversification potential may be more informative.The indices also depend strongly on portfolio size, risk measure, and dependence structure.
- Diversification benefits: Standard deviation and Expectiles can show full diversification for perfectly linearly correlated positions but not for some comonotonic positions, risking underestimated concentration.This occurs because these measures are not comonotonically additive.
- Diversification benefits: VaR may report diversification benefits even when undiversifiable risk remains, whereas ES can correctly remain unchanged as more risks are added.This comparison illustrates a limitation of VaR for measuring diversification effects.
5 Backtesting: which methods can be used?
The section reviews backtesting methods for VaR, ES, and complete distribution forecasts. It proposes validating ES through multiple quantiles, while noting that ES requires longer samples and that the number of supporting quantiles must be chosen carefully.
- Forecast types: Backtesting checks whether ex post realized losses are consistent with risk forecasts, and the appropriate method depends on whether the forecast is a point, interval, or full distribution.VaR and ES are treated as interval or probability-range forecasts, while PIT methods evaluate complete distributions.
- Backtesting ES: ES is conditionally elicitable as a combination of two elicitable components: a quantile and a conditional mean.The proposed procedure backtests the quantile first, then treats it as fixed when scoring the ES component with quadratic error.
- Backtesting VaR: VaR backtesting typically tests whether violations are independent and occur with probability close to 1 − α.Under these conditions, violations follow independent Bernoulli trials and their count has a binomial distribution.
- Backtesting VaR: Unconditional historical VaR can be questionable because violation independence may fail in practice.Duration-based tests have been proposed to assess the independence assumption.
- Backtesting ES: ES can be approximated using four quantiles, enabling ES validation through VaR backtesting methods followed by manual inspection of the upper 0.25% tail.The approach avoids Monte Carlo simulation for the statistical test.
- Backtesting ES: The four-quantile procedure requires case-by-case selection of supporting points because test power declines with more points but increases with sample size.The number of supporting quantiles must therefore reflect the available observations.
- Backtesting ES: For equal certainty, validating ES requires a much longer sample than validating VaR.The Basel Committee’s variant tests violations at both 97.5% and 99% quantile levels.
- Distribution backtesting: PIT-based validation evaluates complete distribution forecasts and is especially relevant for tail-based measures such as ES.The method relies on the distributional behavior of the probability integral transform.
6 Conclusion
The paper compares VaR, ES, and Expectiles across coherence, robustness, elicitability, and comonotonic additivity, finding trade-offs among all three measures. Despite estimation and backtesting caveats, ES appears best for practical use, while the evidence does not justify universally replacing it with Expectiles.
- VaR lacks subadditivity in general, although this may not be serious when risks have finite variance or, in some cases, finite mean.Its failure to capture losses beyond VaR can nevertheless be serious when risks have different tails.
- ES addresses VaR’s lack of subadditivity and tail-loss sensitivity but is not elicitable, making its backtesting less straightforward.
- Several feasible ES backtesting approaches exist, but achieving the same certainty requires more validation data than for VaR.Suggested approaches use distribution forecasts, linear approximations based on VaR at different confidence levels, or Monte Carlo tests.
- Expectiles offer coherent and elicitable alternatives, but their concept is less intuitive and they are not comonotonically additive.This may cause them to miss risk concentrations arising from nonlinear dependencies.
- Among the measures considered, ES seems best for practical use despite estimation and backtesting caveats that can be carefully mitigated.The authors find insufficient evidence to justify an all-inclusive replacement of ES by Expectiles, while retaining Expectiles as alternatives for specific applications.