Marginal Expected Shortfall (MES)

• MES is defined as the conditional expected loss of an institution or asset when the market experiences extreme distress.
• Different formulations, including bivariate thresholds, multivariate regular variation, and dynamic extensions, allow for nuanced systemic risk decomposition.
• Advanced estimation techniques use extreme-value theory, tail extrapolation, and dependence uncertainty to robustly quantify tail risk.

Marginal Expected Shortfall (MES) is a systemic-risk measure that quantifies the expected loss of an institution, asset, or component conditional on a market or system-wide distress event. In the bivariate threshold form, it is written as $\mathrm{MES}(u)=E[X\mid Y>u]$, where $X$ is the institution-level loss or return and $Y$ is a market portfolio or systemic factor; in aggregate-loss formulations, it appears as $\theta_j(\tau)=E\!\left(X_j\mid R>Q_R(\tau)\right)$ or $$\mathrm{MES}_p(X_j,S)=\mathbb{E}[X_j\mid S>\mathrm{VaR}_p(S)]$$, where $S$ or $R$ is the system loss. Across these formulations, MES measures the conditional severity of component-level losses in extreme market states and serves as a building block for decomposing system Expected Shortfall into institution-level contributions (Liu et al., 7 May 2025, Padoan et al., 2023, Chen et al., 28 Apr 2025).

1. Core definitions and conceptual scope

The classical systemic-risk interpretation of MES is the expected loss of one entity when another entity, or the market as a whole, is already in distress. In the Acharya-style formulation used in several papers, MES is the conditional expected loss of component $i$ given that aggregate system loss exceeds a systemic tail threshold. In the bivariate rare-event formulation, MES is the conditional mean of $X$ given that $Y$ exceeds a high threshold. These formulations differ in notation and aggregation structure, but they share the same tail-conditioning principle: firm-level vulnerability is measured relative to an extreme state of the system rather than in isolation (Liu et al., 7 May 2025, Teruzzi, 2023, Pu et al., 2024).

A central distinction is between MES and Expected Shortfall (ES). ES measures the tail mean of the system itself, whereas MES measures the conditional mean of a constituent or related variable during systemic distress. In multivariate systemic-risk work, this distinction is made explicit by the identity $X$0, where $X$1 is system ES and $X$2 is institution $X$3’s MES contribution (Padoan et al., 2023).

Measure	Definition	Role
MES	$X$4	Conditional vulnerability in market distress
QMES	$X$5	Quantile-threshold extreme MES
XMES	$X$6	Expectile-threshold MES
TMES	$X$7	Lagged extension of MES
JMES	$X$8	Joint-tail extension of MES

The table reflects the main line of development in the recent literature. QMES and XMES adapt the distress threshold; TMES introduces temporal lag structure; JMES strengthens the conditioning event by requiring both trigger and affected variable to lie in their respective tail regions (Teruzzi, 2023, Liu et al., 7 May 2025, Pu et al., 2024).

2. Extreme-value formulations and inference in the bivariate setting

A substantial part of the MES literature studies the extreme regime $X$9 or $Y$0, where direct empirical conditioning becomes data-sparse. In the asymptotically independent but positively associated setting, MES is written as

$Y$1

or equivalently $Y$2. The key modeling device is the tail dependence coefficient $Y$3, together with heavy-tail regular variation for $Y$4. Under the paper’s conditions, MES scales as

$Y$5

showing that MES can remain materially large even under asymptotic independence, but with an order determined jointly by the marginal tail index $Y$6 and the dependence coefficient $Y$7 (Cai et al., 2017).

This asymptotic result motivates an extrapolation estimator based on an intermediate threshold: $Y$8 The same work proves asymptotic normality with rate

$Y$9

which is slower than $\theta_j(\tau)=E\!\left(X_j\mid R>Q_R(\tau)\right)$0 because the asymptotic independence regime weakens the effective extremal signal (Cai et al., 2017).

A related tail-oriented line replaces quantile thresholds by expectile thresholds. In the cryptocurrency study, standard MES is embedded in two extreme conditional variants,

$\theta_j(\tau)=E\!\left(X_j\mid R>Q_R(\tau)\right)$1

with the second using an expectile threshold. The paper’s rationale is that expectiles are coherent and elicitable, and it constructs extrapolating MES estimators for dependent heavy-tailed data using intermediate thresholds, Hill-type tail index estimation, and either LAWS or QB expectile estimators. In its empirical application to Bitcoin, Ethereum, and Litecoin, Ethereum has the highest MES (Teruzzi, 2023).

These bivariate extreme-value approaches establish an important point: MES inference is fundamentally a tail-extrapolation problem when systemic events are rare. Direct empirical conditioning is typically viable only at intermediate levels; extreme MES therefore depends on regular variation assumptions, dependence characterization, and threshold selection.

3. Multivariate regular variation and systemic ES decomposition

Bivariate MES analysis isolates one institution and one market variable. A different line of work argues that this is structurally limited because systemic events are generated by the joint extremal dependence of many institutions. In that framework, the market return is the sum of institution-level losses or negative returns,

$\theta_j(\tau)=E\!\left(X_j\mid R>Q_R(\tau)\right)$2

and institution $\theta_j(\tau)=E\!\left(X_j\mid R>Q_R(\tau)\right)$3’s MES is

$\theta_j(\tau)=E\!\left(X_j\mid R>Q_R(\tau)\right)$4

The system ES then decomposes additively as $\theta_j(\tau)=E\!\left(X_j\mid R>Q_R(\tau)\right)$5 (Padoan et al., 2023).

Under multivariate regular variation, extreme MES admits the approximation

$\theta_j(\tau)=E\!\left(X_j\mid R>Q_R(\tau)\right)$6

where $\theta_j(\tau)=E\!\left(X_j\mid R>Q_R(\tau)\right)$7 is the tail index of the radial component and $\theta_j(\tau)=E\!\left(X_j\mid R>Q_R(\tau)\right)$8 is the angular mean of institution $\theta_j(\tau)=E\!\left(X_j\mid R>Q_R(\tau)\right)$9 under the limiting spectral measure. This representation separates three ingredients: market-tail severity through $$\mathrm{MES}_p(X_j,S)=\mathbb{E}[X_j\mid S>\mathrm{VaR}_p(S)]$$0, tail-thickness scaling through $$\mathrm{MES}_p(X_j,S)=\mathbb{E}[X_j\mid S>\mathrm{VaR}_p(S)]$$1, and systemic allocation through $$\mathrm{MES}_p(X_j,S)=\mathbb{E}[X_j\mid S>\mathrm{VaR}_p(S)]$$2 (Padoan et al., 2023).

The corresponding estimator combines a Hill estimator for $$\mathrm{MES}_p(X_j,S)=\mathbb{E}[X_j\mid S>\mathrm{VaR}_p(S)]$$3, a Weissman estimator for $$\mathrm{MES}_p(X_j,S)=\mathbb{E}[X_j\mid S>\mathrm{VaR}_p(S)]$$4, and an empirical estimator of the angular mean: $$\mathrm{MES}_p(X_j,S)=\mathbb{E}[X_j\mid S>\mathrm{VaR}_p(S)]$$5 The same paper proposes a bias-corrected version and proves asymptotic normality for both the i.i.d. and strictly stationary $$\mathrm{MES}_p(X_j,S)=\mathbb{E}[X_j\mid S>\mathrm{VaR}_p(S)]$$6-mixing cases. In simulations, the new estimators have smaller squared bias and lower MSE than existing bivariate competitors, with especially pronounced gains in higher dimensions. In an application to weekly returns of 23 major North American banks from 2010–2022, the largest MES contributors are JPMorgan Chase, Bank of America, Wells Fargo, and Morgan Stanley (Padoan et al., 2023).

This multivariate perspective recasts MES as a tail allocation functional rather than a purely pairwise dependence measure. A plausible implication is that bivariate MES can be informative for a single exposure channel, but it may underrepresent the market-wide extremal structure when systemic risk is generated by many simultaneously stressed institutions.

4. Conditional, dynamic, and lagged MES

Static MES summarizes a contemporaneous tail relation. Several papers generalize it to conditional and dynamic settings. For time-varying systemic risk forecasting, one paper considers a bivariate process $$\mathrm{MES}_p(X_j,S)=\mathbb{E}[X_j\mid S>\mathrm{VaR}_p(S)]$$7 and defines the conditional MES forecast

$$\mathrm{MES}_p(X_j,S)=\mathbb{E}[X_j\mid S>\mathrm{VaR}_p(S)]$$8

Under a multiplicative volatility model

$$\mathrm{MES}_p(X_j,S)=\mathbb{E}[X_j\mid S>\mathrm{VaR}_p(S)]$$9

the forecast decomposes as

$S$0

This yields a two-stage procedure: de-volatilize the data, estimate innovation-level MES by EVT, and rescale by the next-period volatility forecast. The proposed forecast is $S$1, and the paper establishes asymptotic normality and forecast intervals. In the empirical application to the 8 US global systemically important banks, MES forecasts and their interval widths vary strongly over time, with intervals becoming much wider during crisis periods such as 2007–2009 and March 2020 (Hoga, 2023).

A more structural dynamic extension is time-lagged marginal expected shortfall (TMES). For a strictly stationary bivariate time series $S$2, TMES at lag $S$3 is defined by

$S$4

where the event $S$5 represents an extreme systemic state $S$6 periods earlier. When $S$7 and $S$8, $S$9 matches the asymptotic MES. The centered version

$R$0

is zero under independence and admits the representation

$R$1

which makes TMES a lagged extremal dependence measure with a direct conditional-mean interpretation (Liu et al., 7 May 2025).

TMES is shown to exist under partial regular variation, allowing the systemic component to be heavy-tailed while $R$2 may be light-tailed. The paper proposes the empirical estimator

$R$3

proves consistency and a pre-asymptotic CLT, and uses the stationary bootstrap to construct asymptotically valid confidence bands. Applications to gold volatility versus U.S. Economic Policy Uncertainty and to water level versus precipitation show that the lagged formulation can reveal delayed propagation patterns that static MES omits (Liu et al., 7 May 2025).

5. Dependence uncertainty, robust bounds, and factor restrictions

Another branch of the literature does not estimate MES under a fixed dependence model, but instead studies the range of possible MES values when the marginals are known and the dependence structure is unknown. Let

$R$4

Under pure dependence uncertainty, the worst-case and best-case bounds are

$R$5

The upper bound is exactly the marginal expected shortfall,

$R$6

attained under comonotonicity. The general lower bound is

$R$7

where $R$8 is the left expected shortfall. The lower bound is not always sharp; the paper identifies exact tail-separation conditions under which equality holds (Chen et al., 28 Apr 2025).

The same paper shows that these unconstrained bounds can be very wide, and therefore potentially uninformative, when dependence is left unrestricted. It then introduces partial dependence information through additive, minimum-based, and multiplicative factor models. In the additive background risk model,

$R$9

the factor information can materially tighten MES bounds. In the multiplicative background risk model,

$i$0

the dependence restriction does not necessarily reduce the uncertainty spread. In the minimum-based model,

$i$1

the range can also tighten, with the size of the gain depending strongly on distributional parameters (Chen et al., 28 Apr 2025).

A further restriction assumes a linear conditional expectation relation $i$2, consistent with the Capital Asset Pricing Model in finance and the Weighted Insurance Pricing Model in insurance. In the bivariate normal case this yields the explicit formula

$i$3

together with corresponding upper and lower bounds as $i$4 varies in $i$5 (Chen et al., 28 Apr 2025).

A recurrent source of confusion is terminological. In work on robust spectral risk measures under unknown dependence, the acronym “MES” denotes Maximum Expected Shortfall,

$i$6

not Marginal Expected Shortfall. That object is a worst-case ES over all couplings with fixed marginals, and it is conceptually distinct from the conditional systemic-risk measure discussed above (Ghossoub et al., 2020).

6. Generalizations, properties, and statistical fragility

MES has generated several nearby constructs designed to capture spillovers more precisely. Joint Marginal Expected Shortfall (JMES) is defined as

$i$7

It reduces to MES when $i$8 and $i$9 with $X$0, and to ES when $X$1. JMES is therefore a two-threshold extension that conditions on a joint-tail event rather than on distress in the trigger variable alone (Pu et al., 2024).

Theoretical properties established for JMES include monotonicity in the affected-variable stress level $X$2, comonotonic additivity, and comparison results based on copula structure and stochastic orders. The same paper also proves a negative result: JMES is neither identifiable nor elicitable on classes containing all bivariate normals and their finite mixtures. That conclusion is specific to JMES rather than to classical MES, but it illustrates a broader issue in systemic tail-risk measurement: extending MES to richer conditional structures can increase interpretive power while complicating inference and backtesting (Pu et al., 2024).

A broader statistical warning arises from work on Expected Shortfall estimation error in high-dimensional portfolios. That paper does not discuss MES explicitly. However, it argues that tail-based risk measures can become unstable or even infeasible when the effective information content of the tail sample is too small relative to problem dimension, and it emphasizes regularization as the natural remedy. This suggests that MES and related conditional tail expectations may also be vulnerable to severe estimation error, false precision, and concentration on a small set of extreme observations when systemic events are rare (Kondor, 2014).

Taken together, the recent literature portrays MES as a family of conditional tail functionals rather than a single fixed statistic. The common core is a conditional expectation under systemic distress; the main axes of variation are the definition of the distress event, the treatment of tail dependence, the dimensionality of the system, the inclusion of temporal propagation, and the robustness of the measure to dependence uncertainty. Within that family, classical MES remains the canonical one-sided measure of firm-level exposure to market collapse, while multivariate, lagged, expectile-based, and robust-bound formulations extend it to settings where contagion, delayed transmission, or model ambiguity are central.