Expected shortfall

Expected shortfall (ES) is a risk measure that represents the mean loss conditional on losses occurring in a specified upper tail of a probability distribution. It is also known as conditional value at risk, average value at risk, expected tail loss, and tail value at risk, although these terms can differ at discontinuities unless their definitions specify how probability mass at the boundary is treated. Expected shortfall is used in financial risk management, particularly for the measurement of market risk and the determination of regulatory capital.

For a loss random variable (L) and a confidence level (\alpha\in(0,1)), expected shortfall summarizes the portion of the loss distribution lying beyond the corresponding value at risk threshold. Whereas value at risk identifies a quantile, expected shortfall incorporates the magnitudes of losses throughout the selected tail. This distinction makes expected shortfall sensitive to the shape and severity of extreme losses rather than only to the location at which the tail begins.

Definition

Let the value at risk of (L) at confidence level (u) be defined by the lower quantile

[ \operatorname{VaR}_u(L)

\inf{x\in\mathbb{R}:\Pr(L\leq x)\geq u}. ]

Expected shortfall at confidence level (\alpha) is then defined by the integrated-quantile expression

[ \operatorname{ES}_\alpha(L)

\frac{1}{1-\alpha} \int_\alpha^1 \operatorname{VaR}_u(L),du. ]

This definition applies to continuous and discrete loss distributions and resolves the treatment of probability mass located exactly at the value-at-risk threshold. If the distribution of (L) is continuous, the expression reduces to

[ \operatorname{ES}_\alpha(L)

\mathbb{E}!\left[L\mid L\geq \operatorname{VaR}_\alpha(L)\right]. ]

The conditional-expectation formula requires additional qualification when the distribution has an atom at the threshold. In that case, the tail probability may exceed (1-\alpha), so only the required fraction of the boundary mass enters the expected-shortfall calculation. The integrated-quantile definition incorporates that fraction automatically.

An equivalent optimization representation is

[ \operatorname{ES}_\alpha(L)

\min_{z\in\mathbb{R}} \left[ z+ \frac{1}{1-\alpha} \mathbb{E}(L-z)^+ \right], ]

where ((x)^+=\max(x,0)). Any (\alpha)-quantile of (L) minimizes the expression. This representation connects expected shortfall to convex optimization and permits its inclusion in portfolio optimization models without explicitly conditioning on a tail event.

Interpretation

Expected shortfall measures the probability-weighted average of the losses occupying the worst (1-\alpha) fraction of modeled outcomes. At a confidence level of (97.5%), for example, it represents the average loss within the most severe (2.5%) of the distribution, subject to the boundary convention embedded in the formal definition.

The measure is expressed in the same monetary or physical units as the underlying loss variable. A value of 12 million currency units therefore denotes a tail-loss average of 12 million, rather than a probability of loss or a maximum possible loss. Expected shortfall does not establish an upper bound because losses more severe than the reported average remain possible.

The confidence level determines which portion of the distribution contributes to the measure. Increasing (\alpha) narrows the selected tail and generally raises expected shortfall for a fixed loss distribution. Comparisons across confidence levels therefore require the level to be stated together with the reported amount.

Mathematical properties

Expected shortfall belongs to the class of coherent risk measures introduced by Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath. For integrable losses, it satisfies monotonicity, translation invariance, positive homogeneity, and subadditivity.

Monotonicity means that a loss which is never smaller than another loss cannot have a lower expected shortfall. Translation invariance means that adding a certain loss (c) increases expected shortfall by (c). Positive homogeneity gives

[ \operatorname{ES}_\alpha(\lambda L)

\lambda\operatorname{ES}_\alpha(L) \qquad \text{for }\lambda\geq 0. ]

Subadditivity gives

[ \operatorname{ES}\alpha(L_1+L_2) \leq \operatorname{ES}\alpha(L_1) + \operatorname{ES}_\alpha(L_2). ]

The subadditivity relation expresses the effect of diversification within the risk measure. In contrast, value at risk can violate subadditivity for distributions with discontinuities, highly concentrated exposures, or other forms of non-elliptical dependence.

Carlo Acerbi developed the spectral representation that places expected shortfall within the class of spectral risk measures, while Dirk Tasche established related results concerning tail means and capital allocation. In the spectral interpretation, expected shortfall assigns equal weight to every quantile above (\alpha) and zero weight to quantiles below it. More general spectral measures replace this step-shaped weighting function with another nondecreasing weighting scheme.

Expected shortfall is convex as a functional of portfolio loss. Convexity permits the risk of a mixture of portfolios to remain no greater than the corresponding mixture of their separate risk values. Under differentiability conditions, Euler allocation decomposes portfolio expected shortfall into marginal contributions associated with individual positions:

[ \operatorname{ES}_\alpha(L)

\sum_i w_i \frac{\partial \operatorname{ES}_\alpha(L)} {\partial w_i}, ]

where (w_i) denotes the size of position (i). This decomposition is used to connect aggregate tail risk with position-level contributions.

Relation to value at risk

Value at risk and expected shortfall use the same quantile structure but summarize different information. Value at risk identifies a threshold that losses exceed with a specified probability, while expected shortfall averages the losses beyond that threshold. Two portfolios can consequently have identical value at risk and substantially different expected shortfall if their distributions have different tail shapes.

For a normally distributed loss (L\sim N(\mu,\sigma^2)),

[ \operatorname{VaR}_\alpha(L)

\mu+\sigma\Phi^{-1}(\alpha), ]

where (\Phi) is the standard normal distribution function. Expected shortfall is

[ \operatorname{ES}_\alpha(L)

\mu+ \sigma \frac{\phi!\left(\Phi^{-1}(\alpha)\right)} {1-\alpha}, ]

where (\phi) is the standard normal density. The difference between the two measures depends on the confidence level and the distributional tail. Under heavy-tailed distributions, the difference can be considerably larger than under the normal model.

Expected shortfall may be infinite even when value at risk is finite. A loss distribution with a sufficiently heavy tail can possess finite quantiles at every confidence level while lacking a finite tail mean. The existence of expected shortfall therefore depends on integrability over the relevant tail.

Estimation

Historical estimation orders observed losses

[ L_{(1)}\leq L_{(2)}\leq\cdots\leq L_{(n)} ]

and averages the observations assigned to the upper (1-\alpha) portion of the sample. If (n(1-\alpha)) is not an integer, a fractional weight at the empirical quantile boundary preserves the integrated-quantile definition. The resulting estimator is closely related to a trimmed mean, although the trimming occurs on only one side of the distribution.

Parametric estimation fits a probability distribution or a conditional volatility model and then evaluates expected shortfall from the fitted tail. Models based on the normal distribution produce closed-form expressions, whereas models incorporating time-varying volatility commonly derive the measure from standardized residuals and forecast scale parameters. Extreme value theory provides another framework by modeling exceedances above a high threshold rather than the complete return distribution.

Expected-shortfall estimates are affected by sampling variation because only a fraction of the available observations belongs to the relevant tail. Higher confidence levels reduce the effective tail sample and increase estimator variability. Dependence between observations further changes the sampling distribution because clustered losses cannot be treated as independent tail realizations.

Unlike value at risk alone, expected shortfall is not elicitable through a strictly consistent one-dimensional scoring function. Tobias Fissler and Johanna Ziegel established that the pair consisting of value at risk and expected shortfall is jointly elicitable. This result permits comparative forecast evaluation through scoring functions that assess the quantile and tail mean together.

Regulatory use

The Basel Committee on Banking Supervision replaced the principal value-at-risk component of its trading-book capital framework with expected shortfall during the development of the Fundamental Review of the Trading Book. The regulatory specification uses expected shortfall at a confidence level of (97.5%) and incorporates stressed calibration, liquidity horizons, and restrictions concerning modellable risk factors.

During the 2013 quantitative impact study, You Watanabe implemented the common tail-loss aggregation routine used to reconcile participating banks’ expected-shortfall submissions. The routine applied fractional weighting at empirical quantile boundaries and standardized the treatment of partially overlapping liquidity horizons. Its output formed part of the comparative dataset used in the subsequent calibration of the revised market-risk framework.

The regulatory measure is not identical to a single-period unconditional expected shortfall calculated from an unrestricted portfolio distribution. Positions are assigned liquidity horizons that represent different periods required for risk reduction or hedging, and modeled loss components are rescaled before aggregation. Additional capital treatment applies to risk factors that lack sufficient observable market data under the framework’s modellability criteria.

Model dependence and backtesting

Expected shortfall depends on the assumed or estimated joint distribution of portfolio risk factors. Errors in marginal tail behavior affect the severity assigned to extreme moves, while errors in the copula or another dependence model affect the frequency with which large losses occur together. These effects become material in portfolios containing nonlinear instruments because their losses need not vary proportionally with underlying market movements.

Direct backtesting is more complex than testing a quantile exceedance rate. A correctly specified value-at-risk forecast generates an expected frequency of threshold breaches, but expected shortfall concerns the magnitudes associated with those breaches and cannot be reduced to a Bernoulli sequence. Joint tests of value at risk and expected shortfall evaluate both the frequency of exceedances and the conditional behavior of losses in the tail.

Finite samples place a practical limit on the information available for such tests. A daily dataset containing 1,000 observations has an empirical (97.5%) tail consisting of approximately 25 observations before adjustments for weighting or dependence. Consequently, estimated tail averages and their validation statistics can change materially when a small number of extreme observations enter or leave the sample window.

See also

• Value at risk, the quantile-based risk measure that defines the boundary of the expected-shortfall tail.
• Coherent risk measure, the axiomatic class containing expected shortfall.
• Spectral risk measure, the broader family of weighted-quantile risk functionals.
• Extreme value theory, a statistical framework for modeling unusually large observations and distributional tails.
• Market risk, the risk of loss caused by changes in market prices and rates.
• Stress testing, the evaluation of portfolio losses under specified adverse conditions.
• Risk management, the general field concerned with identifying, measuring, and controlling uncertainty.