Flexible Expected Shortfall (FES)

• FES is an umbrella framework that generalizes expected shortfall by incorporating features like covariate dependence and flexible aggregation of tail losses.
• It employs methodologies such as joint quantile–ES regression, robust block-based estimation, and distortion risk measures to improve stability and accuracy in tail risk estimation.
• These flexible approaches enhance forecasting, risk allocation, and counterparty risk management while better addressing heavy tails and outlier sensitivity.

Searching arXiv for recent and foundational papers on Flexible Expected Shortfall, expected shortfall regression, and related generalized ES frameworks. Flexible Expected Shortfall (FES) denotes a family of expected-shortfall-based constructions that make tail-risk assessment adaptable to covariates, aggregation rules, benchmark profiles, or robustness requirements. In the literature considered here, FES appears as joint quantile–ES regression, block-based robust ES estimation, semi-parametric weighted-quantile forecasting, ES-based counterparty limit metrics, mixtures of ES with the mean, distortion-based generalized ES, and adjusted ES defined through an ES profile constraint (Dimitriadis et al., 2017, Bartl et al., 2024, Storti et al., 2020, Kenyon et al., 2017, Papayiannis et al., 17 Jul 2025, Gong et al., 13 Jul 2025, Burzoni et al., 2020). Several of these papers do not use the label as a formal term; this suggests that FES is best understood as an umbrella concept for flexible, tail-sensitive extensions of standard Expected Shortfall rather than as a single universally fixed formalism.

1. Definitions, conventions, and scope

Expected Shortfall is used in the cited literature with both lower-tail and upper-tail conventions. For a financial loss random variable $X$, one definition is

$$\mathrm{ES}_\alpha(X) := \frac{1}{\alpha}\int_{1-\alpha}^1 \mathrm{VaR}_u(X)\,du,$$

with

$$\mathrm{VaR}_u(X):=\inf\{t\in\mathbb{R}: \mathbb{P}(X\le t)\ge u\},$$

whereas for lower-tail returns $Y$, another paper uses

$$es_\alpha = \frac{1}{\alpha}\int_0^\alpha F^{-1}(u)\,du.$$

An upper-tail regression paper instead defines

$$v_{[Y]}(\tau)=E\!\left[Y \mid Y\ge q_{[Y]}(\tau)\right], \qquad v_{[Y\mid X]}(\tau,x)=x^T\beta.$$

These sign and tail conventions differ, but all of them retain the same structural idea: ES averages outcomes beyond a quantile threshold (Bartl et al., 2024, Zwingmann et al., 2016, Li et al., 21 Feb 2026).

Within this broad scope, flexibility enters through different mechanisms. In some formulations, ES is made flexible by allowing dependence on covariates; in others, by changing how tail quantiles are aggregated; in others still, by replacing the expectation in the ES construction, or by constraining the entire ES profile against a benchmark.

Formulation	Defining idea	Representative source
Joint quantile–ES regression	$Q_\alpha(Y\mid X)$ and $\ES_\alpha(Y\mid X)$ depend on covariates	(Dimitriadis et al., 2017)
Robust block-based ES estimator	Full-sample plug-in ES is clipped by blockwise robust quantiles	(Bartl et al., 2024)
Same-level FES mixture	FES is a mixture of ES and the mean with parameter $\theta$	(Papayiannis et al., 17 Jul 2025)
Distortion-based generalized ES	Ordinary expectation in ES is replaced by a distortion risk measure	(Gong et al., 13 Jul 2025)
Adjusted ES	Acceptability is defined by $\mathrm{ES}_p(X)\le g(p)$ for all $$\mathrm{ES}_\alpha(X) := \frac{1}{\alpha}\int_{1-\alpha}^1 \mathrm{VaR}_u(X)\,du,$$0	(Burzoni et al., 2020)

2. Joint quantile–Expected Shortfall regression

A central FES-like formulation is the joint regression framework in which, at a fixed level $$\mathrm{ES}_\alpha(X) := \frac{1}{\alpha}\int_{1-\alpha}^1 \mathrm{VaR}_u(X)\,du,$$1, the conditional quantile and conditional ES are both linear in covariates: $$\mathrm{ES}_\alpha(X) := \frac{1}{\alpha}\int_{1-\alpha}^1 \mathrm{VaR}_u(X)\,du,$$2 Equivalently, the paper writes

$$\mathrm{ES}_\alpha(X) := \frac{1}{\alpha}\int_{1-\alpha}^1 \mathrm{VaR}_u(X)\,du,$$3

with

$$\mathrm{ES}_\alpha(X) := \frac{1}{\alpha}\int_{1-\alpha}^1 \mathrm{VaR}_u(X)\,du,$$4

The paper explicitly characterizes this as a regression specification for ES with covariates that is feasible only because ES is estimated jointly with the corresponding quantile. The theoretical reason is equally explicit: ES is not elicitable on its own, whereas the pair $$\mathrm{ES}_\alpha(X) := \frac{1}{\alpha}\int_{1-\alpha}^1 \mathrm{VaR}_u(X)\,du,$$5 is jointly elicitable, which permits strictly consistent estimation through a joint loss function (Dimitriadis et al., 2017).

The corresponding loss depends on two specification functions, $$\mathrm{ES}_\alpha(X) := \frac{1}{\alpha}\int_{1-\alpha}^1 \mathrm{VaR}_u(X)\,du,$$6 and $$\mathrm{ES}_\alpha(X) := \frac{1}{\alpha}\int_{1-\alpha}^1 \mathrm{VaR}_u(X)\,du,$$7, and has a quantile component plus an ES interaction component that cannot be separated into a quantile-only term and an ES-only term. Differentiation yields a Z-type estimating function, but the paper finds the Z-estimator numerically unstable because the estimating equations are redescending to zero for many attractive choices of $$\mathrm{ES}_\alpha(X) := \frac{1}{\alpha}\int_{1-\alpha}^1 \mathrm{VaR}_u(X)\,du,$$8. For that reason, the paper recommends M-estimation with global optimization, using starting values from two quantile regressions, Nelder–Mead simplex optimization, iterated local search with random perturbations, and repeated re-optimization until improvement stops (Dimitriadis et al., 2017).

The asymptotic theory underpinning this approach was strengthened by work deriving the joint asymptotic distribution of empirical quantiles and expected shortfalls under general conditions. A key result is that the quantile can have a non-standard rate and non-Gaussian limit when the distribution is irregular at the target quantile, but the expected shortfall estimator remains $$\mathrm{ES}_\alpha(X) := \frac{1}{\alpha}\int_{1-\alpha}^1 \mathrm{VaR}_u(X)\,du,$$9-asymptotically normal. That paper uses the Fissler–Ziegel bivariate scoring function for $$\mathrm{VaR}_u(X):=\inf\{t\in\mathbb{R}: \mathbb{P}(X\le t)\ge u\},$$0, and shows that the score minimizer is essentially the empirical quantile together with the empirical ES up to an $$\mathrm{VaR}_u(X):=\inf\{t\in\mathbb{R}: \mathbb{P}(X\le t)\ge u\},$$1 term (Zwingmann et al., 2016).

Subsequent work on ES regression with many regressors replaces direct minimization of the non-differentiable, non-convex Fissler–Ziegel loss by a two-step, Neyman-orthogonal procedure. The first step is ordinary quantile regression,

$$\mathrm{VaR}_u(X):=\inf\{t\in\mathbb{R}: \mathbb{P}(X\le t)\ge u\},$$2

and the second step estimates the ES coefficients through an orthogonal-score loss, yielding the closed-form estimator

$$\mathrm{VaR}_u(X):=\inf\{t\in\mathbb{R}: \mathbb{P}(X\le t)\ge u\},$$3

The paper emphasizes that the impact of first-stage quantile error enters only at higher order because of Neyman orthogonality (He et al., 2022).

A distinct optimization-based contribution argues that superquantile regression from the operations research literature does not coincide with expected shortfall regression in general. It proposes the i-Rock loss

$$\mathrm{VaR}_u(X):=\inf\{t\in\mathbb{R}: \mathbb{P}(X\le t)\ge u\},$$4

and proves that under the linear ES model and a positive-definiteness condition, the unique minimizer is the ES regression coefficient $$\mathrm{VaR}_u(X):=\inf\{t\in\mathbb{R}: \mathbb{P}(X\le t)\ge u\},$$5. This paper therefore separates expected shortfall regression from superficially related superquantile formulations (Li et al., 21 Feb 2026).

3. Estimation, robustness, and inference

A nonparametric robustness line of work begins from the classical plug-in estimator

$$\mathrm{VaR}_u(X):=\inf\{t\in\mathbb{R}: \mathbb{P}(X\le t)\ge u\},$$6

and shows that its asymptotic normality can be misleading in finite samples. The paper states that the plug-in estimator suffers from poor statistical performance for heavy-tailed distributions, lacks exponential concentration in general, exhibits an additional $$\mathrm{VaR}_u(X):=\inf\{t\in\mathbb{R}: \mathbb{P}(X\le t)\ge u\},$$7 second-order effect when the quantile function is Lipschitz, and is extremely sensitive to outliers because changing even one observation can distort the empirical tail dramatically (Bartl et al., 2024).

The proposed alternative is a robust block-based estimator. After partitioning the sample into blocks and computing block plug-in ES estimates $$\mathrm{VaR}_u(X):=\inf\{t\in\mathbb{R}: \mathbb{P}(X\le t)\ge u\},$$8, the estimator is defined by

$$\mathrm{VaR}_u(X):=\inf\{t\in\mathbb{R}: \mathbb{P}(X\le t)\ge u\},$$9

The full-sample plug-in estimate is retained when it is typical and clipped between robust block-based quantiles when it is dragged by outliers. Under finite variance of ES and a Lipschitz condition on the quantile function, the paper proves an exponential deviation inequality of the form

$Y$0

It also gives an adversarial contamination guarantee and states that the dependence on the number of corrupted points is minimax-optimal (Bartl et al., 2024).

In regression settings, robustness to skewness and heavy tails is handled differently. The many-regressor ES regression paper replaces second-stage least squares with adaptive Huber loss,

$Y$1

and estimates

$Y$2

The resulting estimator admits finite-sample deviation bounds, a Bahadur representation, and Gaussian approximation results, while the confidence parameter enters through $Y$3 rather than $Y$4, which the paper interprets as markedly better tail behavior than the unrobust estimator (He et al., 2022).

The optimization-based ES regression approach offers a different route to feasible inference. For discrete covariates, the estimator reduces to a quantile regression on pseudo-responses built from empirical ES estimators on bins. For continuous covariates, binning and local linear ES estimation lead to an asymptotic linearization in which the final estimator behaves like a weighted least squares regression of initial ES estimates on the covariates, with implicit weights

$Y$5

The paper interprets these as heterogeneity-adaptive weights and attributes frequent efficiency gains over existing linear ES regression approaches to this feature (Li et al., 21 Feb 2026).

4. Forecasting architectures and time-series implementations

A semi-parametric ES forecasting framework reconstructs ES from a grid of conditional quantile forecasts rather than specifying a separate dynamic law for ES itself. Using the identity

$Y$6

the method first estimates a set of tail quantiles through CAViaR regressions and then forms

$Y$7

The weighting scheme is parameterized by a Beta weight function,

$Y$8

and the parameters are estimated by minimizing a strictly consistent joint VaR–ES Fissler–Ziegel score specialized to the Asymmetric Laplace log score. The paper stresses that this approach imposes no additional dynamic assumption on ES itself and that even very small grids, notably $Y$9, can perform strongly in forecasting experiments (Storti et al., 2020).

A different semi-parametric line models ES jointly with expectiles and realized measures. In the ES-CARE setup, the expectile recursion

$$es_\alpha = \frac{1}{\alpha}\int_0^\alpha F^{-1}(u)\,du.$$0

is paired with an ES recursion obtained by scaling: $$es_\alpha = \frac{1}{\alpha}\int_0^\alpha F^{-1}(u)\,du.$$1 The realized version replaces lagged returns by realized measures $$es_\alpha = \frac{1}{\alpha}\int_0^\alpha F^{-1}(u)\,du.$$2, and a threshold extension allows regime-dependent coefficients when the previous return crosses a threshold. Estimation is carried out by adaptive Bayesian MCMC with a joint likelihood combining the ES/expectile part and a Gaussian measurement equation for the realized measure. The paper reports that realized and threshold extensions improve tail-risk forecasting across seven stock indices (Wang et al., 2019).

The joint quantile–ES regression framework also has a direct forecasting interpretation. One empirical example forecasts VaR and ES of IBM returns using realized volatility: $$es_\alpha = \frac{1}{\alpha}\int_0^\alpha F^{-1}(u)\,du.$$3 Forecasts are evaluated with strictly consistent scoring rules for the pair $$es_\alpha = \frac{1}{\alpha}\int_0^\alpha F^{-1}(u)\,du.$$4 and Murphy diagrams; the paper reports dominance over historical simulation and AR-GARCH-$$es_\alpha = \frac{1}{\alpha}\int_0^\alpha F^{-1}(u)\,du.$$5 in most threshold regions, with competitiveness relative to the HAR model (Dimitriadis et al., 2017).

5. Generalized FES as a risk-measure family

One paper gives an explicit formal definition of FES as a same-level mixture of ES and the mean: $$es_\alpha = \frac{1}{\alpha}\int_0^\alpha F^{-1}(u)\,du.$$6 It states the bounds

$$es_\alpha = \frac{1}{\alpha}\int_0^\alpha F^{-1}(u)\,du.$$7

together with the limits $$es_\alpha = \frac{1}{\alpha}\int_0^\alpha F^{-1}(u)\,du.$$8 and $$es_\alpha = \frac{1}{\alpha}\int_0^\alpha F^{-1}(u)\,du.$$9, and uses this mixture to build a same-level coherent representation of VaR. For $$v_{[Y]}(\tau)=E\!\left[Y \mid Y\ge q_{[Y]}(\tau)\right], \qquad v_{[Y\mid X]}(\tau,x)=x^T\beta.$$0, the paper defines the matching flexibility parameter

$$v_{[Y]}(\tau)=E\!\left[Y \mid Y\ge q_{[Y]}(\tau)\right], \qquad v_{[Y\mid X]}(\tau,x)=x^T\beta.$$1

and then sets

$$v_{[Y]}(\tau)=E\!\left[Y \mid Y\ge q_{[Y]}(\tau)\right], \qquad v_{[Y\mid X]}(\tau,x)=x^T\beta.$$2

The associated $$v_{[Y]}(\tau)=E\!\left[Y \mid Y\ge q_{[Y]}(\tau)\right], \qquad v_{[Y\mid X]}(\tau,x)=x^T\beta.$$3-index is interpreted as a normalized tail-risk or flexibility index, and the paper also develops Euler-based capital allocation formulas for FES and PELVaR (Papayiannis et al., 17 Jul 2025).

A more general construction replaces the ordinary expectation in the optimization representation of ES by a distortion risk measure. With $$v_{[Y]}(\tau)=E\!\left[Y \mid Y\ge q_{[Y]}(\tau)\right], \qquad v_{[Y\mid X]}(\tau,x)=x^T\beta.$$4, the generalized ES is

$$v_{[Y]}(\tau)=E\!\left[Y \mid Y\ge q_{[Y]}(\tau)\right], \qquad v_{[Y\mid X]}(\tau,x)=x^T\beta.$$5

For a vector $$v_{[Y]}(\tau)=E\!\left[Y \mid Y\ge q_{[Y]}(\tau)\right], \qquad v_{[Y\mid X]}(\tau,x)=x^T\beta.$$6, this induces a scaled generalized-ES norm

$$v_{[Y]}(\tau)=E\!\left[Y \mid Y\ge q_{[Y]}(\tau)\right], \qquad v_{[Y\mid X]}(\tau,x)=x^T\beta.$$7

where $$v_{[Y]}(\tau)=E\!\left[Y \mid Y\ge q_{[Y]}(\tau)\right], \qquad v_{[Y\mid X]}(\tau,x)=x^T\beta.$$8. The paper proves that if $$v_{[Y]}(\tau)=E\!\left[Y \mid Y\ge q_{[Y]}(\tau)\right], \qquad v_{[Y\mid X]}(\tau,x)=x^T\beta.$$9 is strictly increasing and convex, the resulting functional is a norm for all $Q_\alpha(Y\mid X)$0, derives a dual norm via the ordered weighted $Q_\alpha(Y\mid X)$1 norm structure, and applies the construction to projection problems and anomaly detection in financial time series (Gong et al., 13 Jul 2025).

Adjusted Expected Shortfall constitutes another flexible generalization. Given an increasing benchmark profile $Q_\alpha(Y\mid X)$2, the acceptance set is

$Q_\alpha(Y\mid X)$3

and the corresponding monetary risk measure is

$Q_\alpha(Y\mid X)$4

A central proposition rewrites this as

$Q_\alpha(Y\mid X)$5

Standard ES is recovered as a special case for a suitable step-function $Q_\alpha(Y\mid X)$6. The paper proves monotonicity, cash additivity, convexity, law invariance, and SSD-consistency, but it also proves that positive homogeneity holds if and only if the adjusted ES collapses to an ordinary ES. This is an important qualification: flexibility in this profile-based sense generally yields a convex monetary risk measure rather than a coherent one (Burzoni et al., 2020).

A further structural result concerns distributional mixtures. For a finite mixture random variable $Q_\alpha(Y\mid X)$7 with cdf $Q_\alpha(Y\mid X)$8, the concavity theorem states

$Q_\alpha(Y\mid X)$9

The paper extends this immediately to spectral risk measures

$\ES_\alpha(Y\mid X)$0

This mixture-concavity property is relevant to flexible ES constructions that are built by mixing ES levels or distributions, because it shows that ES-based risk need not decrease under probability mixing (Tselishchev, 2019).

6. Applications, misconceptions, and adjacent concepts

In counterparty risk management, ES flexibility appears as a move from Potential Future Exposure (PFE), a high quantile of exposure, to Potential Future Loss (PFL), an ES-based tail-loss metric that includes Loss Given Default. The paper defines

$\ES_\alpha(Y\mid X)$1

with $\ES_\alpha(Y\mid X)$2 the corresponding quantile threshold, and extends this to Adjusted PFL and Protected Adjusted PFL by subtracting incurred CVA and existing credit protection. The framework is explicitly collateral-aware and accommodates initial margin, collateral timing, flow netting, seniority differences, and existing hedges. The stated motivation is that PFE can become close to zero under widespread regulatory initial margin or flow netting while still ignoring the tail beyond the reference percentile, whereas PFL remains tail-sensitive (Kenyon et al., 2017).

Several recurrent misconceptions are addressed by the literature. First, FES is not equivalent to standalone ES regression: in the joint regression framework, ES is estimable only by modeling it together with the quantile because ES is not elicitable on its own (Dimitriadis et al., 2017). Second, FES is not synonymous with superquantile regression: the optimization-based ES regression paper gives a counterexample showing that Rockafellar-style superquantile regression does not generally recover the ES regression coefficient (Li et al., 21 Feb 2026). Third, FES is not uniformly coherent across formulations: the same-level mixture FES is presented as coherent, and distortion-based generalized ES can produce norms under convexity conditions, but adjusted ES is generally not positively homogeneous except in the ordinary ES case (Papayiannis et al., 17 Jul 2025, Gong et al., 13 Jul 2025, Burzoni et al., 2020). Fourth, FES is not merely VaR relabeled at another confidence level: one paper’s point is precisely that VaR can be represented at the same level $\ES_\alpha(Y\mid X)$3 through FES by selecting the unique $\ES_\alpha(Y\mid X)$4 satisfying $\ES_\alpha(Y\mid X)$5 (Papayiannis et al., 17 Jul 2025).

Taken together, these works position FES as a technically heterogeneous but conceptually coherent research area organized around flexible tail averaging. The common objectives are to let ES depend on covariates, to stabilize estimation under heavy tails and contamination, to forecast ES without rigid parametric tail dynamics, to compare or represent VaR within coherent or profile-based frameworks, and to embed ES into broader structures such as counterparty limits, generalized norms, spectral measures, and backtesting systems (Dimitriadis et al., 2017, Bartl et al., 2024, Gong et al., 13 Jul 2025).