Expected shortfall
Expected shortfall (ES), also known as conditional value at risk (CVaR) or tail conditional expectation (TCE), is a coherent risk measure used in financial risk management to quantify the expected loss of a portfolio over a specified time horizon, conditional on the loss exceeding the value at risk (VaR) threshold at a high confidence level, typically 97.5% or 99%.[1][2] Formally, for a loss random variable X (positive values indicate losses) and confidence level \alpha \in (0,1), ES is defined as \mathrm{ES}_\alpha(X) = \frac{1}{1-\alpha} \int_\alpha^1 \mathrm{VaR}_p(X) \, dp, where \mathrm{VaR}_p(X) is the p-quantile of the loss distribution, providing an average of the losses exceeding \mathrm{VaR}_\alpha(X) (i.e., the worst $1-\alpha tail).[3] ES addresses key limitations of VaR by incorporating the magnitude of extreme losses beyond the quantile threshold, rather than merely identifying a cutoff point.[1] It satisfies the four axioms of coherence—monotonicity (greater losses imply greater or equal risk), subadditivity (risk of combined positions does not exceed the sum of individual risks), positive homogeneity (risk scales linearly with position size), and translation invariance (adding cash reduces risk by the same amount)—making it suitable for diversification and capital allocation in portfolios.[4] These properties ensure ES promotes prudent risk-taking without incentivizing excessive concentration, unlike non-subadditive measures such as VaR.[4][1] The concept of coherent risk measures, including ES as TCE, was formalized by Artzner et al. in 1999 to evaluate risks in incomplete markets.[4] ES gained prominence in regulatory practice through the Basel III framework, where it replaced VaR in the internal models approach for market risk capital requirements, calibrated at a 97.5% confidence level over a 10-day horizon (adjusted for liquidity), to better capture tail risks during stressed market conditions.[2] This adoption reflects its robustness in estimating capital needs for banks' trading books, with calculations incorporating both modellable and non-modellable risk factors.[2]Fundamentals
Definition
Expected shortfall (ES), also known as conditional value at risk or tail value at risk, is a risk measure that quantifies the expected loss of a portfolio in the worst (1-α) portion of cases, where α is the confidence level (typically 0.95 or 0.99). For a loss random variable X (where positive values indicate losses), the formal definition is given by the average of the value at risk (VaR) over the tail: \text{ES}_\alpha(X) = \frac{1}{1-\alpha} \int_\alpha^1 \text{VaR}_u(X) \, du, where VaR_u(X) is the value at risk at level u, defined as the u-quantile of the loss distribution, i.e., the smallest value such that the probability of exceeding it is 1-u. This integral form captures the severity of extreme losses beyond the VaR threshold by averaging the quantiles in the upper tail.[5] For continuous loss distributions, expected shortfall admits an alternative expression as the tail conditional expectation: \text{ES}_\alpha(X) = \mathbb{E}[X \mid X > \text{VaR}_\alpha(X)]. This represents the expected loss given that the loss exceeds the α-VaR threshold. In financial contexts, the convention treats X as the loss (positive for adverse outcomes), distinguishing it from profit-and-loss frameworks where negative values denote losses; adjustments for sign conventions ensure consistency in risk assessment.[1] Expected shortfall was introduced as part of the framework for coherent risk measures by Artzner et al. (1999), who proposed axioms such as subadditivity and positive homogeneity to evaluate risk metrics, with ES emerging as a prominent example satisfying these properties. This development addressed limitations of VaR by providing a more comprehensive tail risk perspective.Relation to Value at Risk
Value at Risk (VaR) at confidence level α is defined as the quantile threshold VaR_α(X) = inf{x | P(X > x) ≤ 1-α}, where X represents portfolio losses, providing a measure of the maximum expected loss exceeded with probability no greater than 1-α.[1] However, VaR has significant limitations as a risk measure: it is not subadditive, meaning the risk of a combined portfolio may exceed the sum of individual risks, potentially discouraging diversification.[6] Additionally, VaR ignores the severity of losses beyond the quantile threshold, offering no information on the magnitude of extreme tail events.[1] Expected Shortfall (ES) addresses these shortcomings by calculating the average loss conditional on exceeding the VaR threshold, thereby capturing the expected severity of tail losses rather than just the boundary.[1] Unlike VaR, ES is a coherent risk measure, satisfying subadditivity, monotonicity, positive homogeneity, and translation invariance, which ensures it promotes diversification and provides consistent risk assessments across portfolios.[6][1] This makes ES particularly suitable for managing extreme events, as it integrates the full tail distribution for a more robust evaluation of potential downside risks. The development of ES gained prominence following critiques of VaR's incoherence, as highlighted in foundational work on coherent risk measures, leading to its proposal as a superior alternative in financial regulation.[6] In response to the limitations exposed during the 2008 financial crisis, Basel III incorporated ES through the Fundamental Review of the Trading Book, replacing VaR for market risk capital requirements to better account for tail risks.[7] For instance, in fat-tailed distributions common in financial returns, ES yields more conservative risk estimates than VaR by incorporating the heavier probabilities and magnitudes of extreme losses beyond the quantile.[8]Properties
Mathematical Properties
Expected shortfall (ES) satisfies the axioms of coherent risk measures, including monotonicity, subadditivity, positive homogeneity, and translation invariance.[6] Monotonicity implies that if one portfolio's outcomes are always less than or equal to another's, then the ES of the first is less than or equal to that of the second.[9] Subadditivity ensures that the risk of a combined portfolio does not exceed the sum of individual risks, promoting diversification.[9] Positive homogeneity means that scaling a portfolio by a positive factor scales its ES by the same factor, while translation invariance states that adding a constant to all outcomes reduces the ES by that constant.[6] ES is continuous with respect to the confidence level α, providing stability under small perturbations in α.[9] More broadly, as a monetary risk measure, ES is Lipschitz continuous with respect to the L¹ norm on the space of random variables, ensuring robustness to changes in the underlying distribution.[10] Among coherent risk measures, ES is the smallest one that dominates value at risk (VaR) at the same confidence level α and depends only on the distribution of the loss variable.[9] This property positions ES as a tight upper bound on VaR while maintaining coherence. ES exhibits greater sensitivity to heavy-tailed distributions compared to VaR, as it averages losses beyond the VaR threshold, thereby penalizing extreme tail events more severely.[8] For instance, under distributions with fatter tails, ES increases more than VaR, reflecting the higher expected severity of losses in the tail.[11] A specific limiting case occurs when α = 0, where ES reduces to the unconditional expectation of the loss variable -X, ES₀(X) = E[-X].[9]Axiomatic Foundations
Expected shortfall (ES) is classified as a coherent risk measure within the axiomatic framework introduced by Artzner et al., which defines desirable properties for assessing financial risks.[4] A coherent risk measure \rho must satisfy four axioms: monotonicity, subadditivity, positive homogeneity, and translation invariance. Monotonicity requires that if random variable X (representing portfolio returns) is less than or equal to Y almost surely, then \rho(X) \geq \rho(Y), ensuring worse outcomes incur higher risk assessments. Subadditivity states \rho([X + Y](/page/X+Y)) \leq \rho(X) + \rho(Y), promoting diversification by capping combined risk at the sum of individual risks. Positive homogeneity demands \rho(\lambda X) = \lambda \rho(X) for \lambda > 0, reflecting scalability of risks with position size. Translation invariance specifies \rho(X + c) = \rho(X) - c for constant c, adjusting risk linearly for cash additions.[4] Expected shortfall satisfies these axioms, as demonstrated through its integral representation: for a confidence level \alpha \in (0,1), \text{ES}_\alpha(X) = \frac{1}{1-\alpha} \int_\alpha^1 \text{VaR}_u(X) \, du, where \text{VaR}_u(X) = \inf \{ z \in \mathbb{R} : P(X \leq -z) \leq u \} is the value at risk at level u. Monotonicity follows directly: if X \leq Y almost surely, then \text{VaR}_u(X) \geq \text{VaR}_u(Y) for all u, implying \text{ES}_\alpha(X) \geq \text{ES}_\alpha(Y). For translation invariance, adding c > 0 shifts \text{VaR}_u(X + c) = \text{VaR}_u(X) - c, so \text{ES}_\alpha(X + c) = \text{ES}_\alpha(X) - c. Positive homogeneity holds because \text{VaR}_u(\lambda X) = \lambda \text{VaR}_u(X) for \lambda > 0, preserving the integral scaling. Subadditivity is the most involved: it arises from the dual representation of coherent measures as suprema over probability measures, where ES corresponds to the set of measures with density bounded by $1/(1-\alpha), ensuring \text{ES}_\alpha(X + Y) \leq \text{ES}_\alpha(X) + \text{ES}_\alpha(Y) via Hölder's inequality on the tail expectations.[12] In contrast, value at risk (VaR) fails subadditivity, rendering it non-coherent. A classic counterexample involves two identical bonds, each with a 4% default probability leading to a 100-unit loss and otherwise 0, under a 95% confidence level where \text{VaR}_{0.95} = 0 for each. Merging them yields a binomial default risk with 8% probability of at least one default (loss of 100), so \text{VaR}_{0.95} = 100 > 0 + 0, violating subadditivity and discouraging diversification.[4] Expected shortfall extends naturally to the broader class of convex risk measures, which relax subadditivity to convexity (\rho(\lambda X + (1-\lambda) Y) \leq \lambda \rho(X) + (1-\lambda) \rho(Y) for \lambda \in [0,1]) while retaining the other coherent axioms; coherent measures are precisely the positively homogeneous convex ones. ES is convex because its tail expectation form inherits convexity from the expectation operator, positioning it as a special case that fully satisfies coherence for \alpha < 1. Under law invariance and comonotonic additivity conditions, expected shortfall is the unique coherent risk measure for elliptical distributions, such as multivariate normal or Student's t, where risks exhibit symmetric dependence structures.[13]Examples
Numerical Illustrations
To illustrate the calculation of expected shortfall (ES), consider a simple discrete uniform distribution for losses L taking values \{-1, 0, 1, 10\}, each with probability $0.25. This example demonstrates how ES captures the magnitude of extreme losses in the tail.[3] First, compute the Value at Risk (VaR) at the 95% confidence level, denoted \mathrm{VaR}_{0.95}, which is the 0.95 quantile of the loss distribution: the smallest value q such that P(L \leq q) \geq 0.95. Sorting the outcomes in ascending order gives -1, 0, 1, 10. The cumulative probabilities are P(L \leq -1) = 0.25, P(L \leq 0) = 0.5, P(L \leq 1) = 0.75, and P(L \leq 10) = 1. Since P(L \leq 1) = 0.75 < 0.95 and P(L \leq 10) = 1 \geq 0.95, \mathrm{VaR}_{0.95} = 10.[3] Next, compute \mathrm{ES}_{0.95} using the formula \mathrm{ES}_p = \mathrm{VaR}_p + \frac{1}{1-p} E[(L - \mathrm{VaR}_p)^+], where (x)^+ = \max(x, 0). Here, p = 0.95, so $1-p = 0.05. The excess term E[(L - 10)^+] = \sum (l_i - 10)^+ P(L = l_i). Only l = 10 contributes: (10 - 10)^+ \cdot 0.25 = 0, and all others are negative, so E[(L - 10)^+] = 0. Thus, \mathrm{ES}_{0.95} = 10 + \frac{1}{0.05} \cdot 0 = 10. Alternatively, since the tail consists solely of the outcome 10 with probability 0.25 (exceeding the 0.05 tail probability), the conditional expectation E[L \mid L \geq 10] = 10, confirming the result. In this case, ES highlights the impact of the rare large loss of 10, pulling the risk measure to that extreme value despite its 25% probability, far beyond the distribution's mean of 2.5.[3]| Outcome l_i | Probability | Sorted Order |
|---|---|---|
| -1 | 0.25 | 1st |
| 0 | 0.25 | 2nd |
| 1 | 0.25 | 3rd |
| 10 | 0.25 | 4th |