Value-at-Risk answers one question: what loss is exceeded with probability alpha. It ignores everything beyond that quantile, and it can report that a diversified book is riskier than its parts. The literature on risk measures is a thirty-year effort to fix those two defects without breaking estimability. The archive holds 35 papers with the risk-measures tag and over a hundred on neighbouring keywords, mostly theoretical. Here is the map, what the corpus adds, and the decision guide.

Coherent and convex measures

Artzner, Delbaen, Eber and Heath (1999) defined a coherent risk measure by four axioms: monotonicity, translation invariance, positive homogeneity and subadditivity. Subadditivity is the one VaR fails; it is the formal statement that diversification should never increase measured risk. Expected shortfall (ES, also CVaR), the average loss beyond VaR, satisfies all four. A 2023 note in the archive gives an elementary proof of the dual representation of ES and recovers subadditivity as a by-product, which is the cleanest route to understanding why ES aggregates sensibly. On the VaR side, a 2025 paper shows that VaR is subadditive at every level if and only if the losses are comonotonic, that is, exactly when there is no diversification to measure.

Convex risk measures relax homogeneity and subadditivity into a single convexity axiom, so risk can grow faster than linearly with position size, the right behaviour for illiquid positions. Beyond convexity, a 2025 paper introduces Lambda Expected Shortfall as the smallest quasi-convex, law-invariant measure dominating Lambda-VaR, with an explicit formula.

Spectral and distortion measures

Spectral risk measures weight the quantile function with a non-increasing density, so worse outcomes count more; ES is the special case of a flat weight on the tail. Distortion risk measures (the Choquet-integral view) reweight the distribution function with a distortion g. A 2024 paper constructs a functional that quantifies the degree of risk aversion of a spectral measure, and a 2024 estimation paper builds a nonparametric spectral risk estimator for truncated and censored data, the insurance setting of deductibles and limits, with bootstrap accuracy results and real loss datasets. The price of comonotonic additivity is laid out in a 2022 literature review: such measures cannot be surplus invariant.

Elicitability and why ES is hard to backtest

A functional is elicitable if it minimizes the expectation of some scoring function; that is what makes forecast comparison and regression-based estimation possible. VaR is elicitable (pinball loss). ES is not elicitable on its own (Ziegel 2016), but the pair (VaR, ES) is jointly elicitable (Fissler and Ziegel 2016). In practice you cannot rank ES forecasts by a loss on ES alone, and there is no standalone ES exception test comparable to a VaR exception count.

This is the FRTB context. Basel’s market-risk framework moved internal-model capital to ES at 97.5%, yet desk-level backtesting still runs on VaR exceptions at 99% and 97.5%, precisely because of the elicitability gap. A 2024 paper extends joint identifiability and elicitability to general tail risk measures, nesting the Fissler-Ziegel scores. A 2024 backtesting paper separates the duration and severity of VaR violations with a model-free Wald test, so that an ES rejection can be traced to the frequency or the size of exceedances. A 2024 paper on gradient allocations of ES backtests the whole vector of ES contributions using multi-objective elicitability. On the forecast-combination side, two empirical papers in the archive, one using the Model Confidence Set over nine indices and one combining 90 candidate VaR/ES forecasts, find that trimmed means and performance-weighted combinations of a small diverse pool beat individual models under the joint FZ loss. A 2025 paper gives a constructive recipe for new elicitable risk measures via multiplicative scoring functions, including convex-but-not-coherent ones.

Robust measures and model uncertainty

Every measure above is evaluated under one distribution you do not know. Robust risk measurement replaces the point distribution with a set. The standard move is worst-case over a Wasserstein or divergence ball; a 2026 paper solves robust distortion riskmetrics under Wasserstein ambiguity with computable error bounds and applies it to portfolio selection. A 2026 alternative averages the baseline measure over nearby payoffs instead of taking the worst case, yielding a convex measure. A 2025 paper by Buehler, Horvath, Limmer and Schmidt proposes a model-agnostic subsampling framework that superimposes an uncertainty measure over models, validated on real data. Robust versions of elicitable functionals exist too, via KL-divergence neighbourhoods.

What Ruodu Wang’s papers add

Wang is the most prolific author in this archive, with 25 papers forming a coherent programme. On diversification: the diversification quotient based on VaR and ES (2023) gives an axiomatic diversification index, and the 2024 expectile-based version avoids the small-sample tail problem that VaR- and ES-based indices face, with a linear-programming optimizer. On heavy tails: Infinite-mean models in risk management (2024) argues that many finite-mean intuitions, including diversification itself, fail or reverse when means do not exist. On structure: Submodular risk measures (2026) shows law-invariant coherent measures are submodular exactly when they are coherent distortion measures, and its empirical check on daily US equities finds no ES violations and many VaR violations. On estimation: adaptive window selection (2026) uses bootstrap thresholds on realized scores to pick the look-back window online, outperforming fixed rolling windows under structural change. Further entries cover partial law invariance, quantiles under ambiguity and risk sharing with distortion riskmetrics.

Which measure for which decision

  • Regulatory and economic capital. ES at a high confidence level, because capital must aggregate across desks and subadditivity is non-negotiable. Attribute it with Euler or Shapley allocation; a 2025 paper testing allocation schemes under Basel 2.5 and FRTB finds Shapley the best compromise, and a 2024 note shows Euler allocation under VaR is not monotone. Backtest with the VaR exception tests you can actually run.
  • Position and desk limits. A short-horizon VaR or ES limit plus a drawdown limit. Limits are operational; consistency of computation matters more than axiomatic purity. Keep volatility targeting in view, since a vol-scaled book moves its own VaR.
  • Portfolio optimization. CVaR is convex and linear-programmable, which is why it dominates the optimization literature. But every tail measure is estimated from a handful of observations, and estimation error usually dwarfs the difference between measures. Robust or shrinkage versions, or plain minimum variance and risk parity, are often the better practical choice.

The estimation-error caveat

ES at 97.5% on a 250-day window averages roughly six observations. Its sampling error and single-observation sensitivity are large, and its confidence interval is wide enough to swallow most model comparisons. Theoretical gains from a refined measure can vanish at estimation time. A 2025 paper on coherent estimation of risk measures attacks this directly by characterizing estimators that inherit the axioms of the measure they estimate, with an ES study under overlapping samples relevant to FRTB. Until that kind of work matures, treat any claimed improvement in tail-risk measurement as a hypothesis about estimators, not just about functionals.

How to read these papers

Most of this list sits in the Lab Rats quadrant: high math, low empirical content. That is appropriate for axiomatic work, but a paper can be correct and irrelevant to your desk at the same time. Read the theory for the properties you need (subadditivity, convexity, elicitability), then look for the few empirical papers that test whether those properties survive estimation: the ES backtesting, forecast combination, adaptive window and FRTB allocation papers above. The broader hub is Risk Management & Tail Risk; the related keyword page is FRTB.

References: Artzner, Delbaen, Eber & Heath, “Coherent Measures of Risk” (1999); Föllmer & Schied (2002) and Frittelli & Rosazza Gianin (2002) on convex risk measures; Acerbi (2002) on spectral measures; Rockafellar & Uryasev (2000) on CVaR optimization; Gneiting (2011) on elicitability; Ziegel (2016); Fissler & Ziegel (2016).

Broader area: Risk Management · All topics: research topics → · Full archive: every paper →