Value at Risk & Expected Shortfall

Artzner et al.\ (1999) · Coherent risk measurement · Basel III

Family 03 Risk Parametric / Historical / MC

Value at Risk (VaR) is the $\alpha$-quantile of the loss distribution: the number $x$ such that losses exceed $x$ with probability at most $1 - \alpha$. Expected Shortfall (ES, a.k.a.\ CVaR) is the mean loss in the worst $(1-\alpha)$ tail. Artzner, Delbaen, Eber & Heath (1999) showed that ES is coherent (subadditive, so diversification always helps); VaR is not. In 2019 the Basel Committee on Banking Supervision (BCBS) replaced 99% VaR with 97.5% Expected Shortfall in its market-risk capital rule.

Educational Purpose

Educational demonstration of risk measurement. Not investment or regulatory advice. Parametric, historical, and Monte Carlo VaR/ES estimates can disagree substantially — backtesting, model risk, and liquidity horizon all matter in practice.

The definitions

VaR, ES, coherence

OR family: Risk measurement Solver class: Sample statistics / Gaussian / simulation Measure: Physical $\mathbb{P}$ Realism: ★★★ Exact (under each method's assumptions)

Definitions

Symbol	Meaning
$L$	Loss random variable (positive = loss, negative = gain) over a given horizon
$F_L(x)$	CDF of $L$ under the physical measure $\mathbb{P}$
$\alpha$	Confidence level, e.g.\ 0.95, 0.975, 0.99
$\text{VaR}_\alpha$	$\alpha$-quantile of the loss distribution: $\inf\{x : F_L(x) \ge \alpha\}$
$\text{ES}_\alpha$	Expected loss conditional on $L \ge \text{VaR}_\alpha$ (a.k.a.\ CVaR)

Value at Risk $$ \text{VaR}_\alpha(L) \;=\; \inf\bigl\{ x \in \mathbb{R} \;:\; \mathbb{P}(L \le x) \ge \alpha \bigr\} $$

“We are $\alpha$-confident that losses will not exceed VaR over the horizon.” A single number summarising the loss distribution at a confidence level.

Expected Shortfall (Conditional VaR) $$ \text{ES}_\alpha(L) \;=\; \mathbb{E}\bigl[\, L \;\big|\; L \ge \text{VaR}_\alpha(L) \,\bigr] \;=\; \frac{1}{1-\alpha}\int_{\alpha}^{1} \text{VaR}_u(L)\, du $$

Tail mean; the integral form is always well-defined (even when $F_L$ has flat regions at the $\alpha$-quantile). $\text{ES}_\alpha \ge \text{VaR}_\alpha$ always.

Three estimation methods

(a) Parametric (variance-covariance). Assume $L \sim \mathcal{N}(\mu_L, \sigma_L^2)$; then

$$ \text{VaR}_\alpha = \mu_L + \sigma_L \, \Phi^{-1}(\alpha), \qquad \text{ES}_\alpha = \mu_L + \sigma_L \, \frac{\phi(\Phi^{-1}(\alpha))}{1-\alpha} $$

Fast, closed-form; breaks down for fat-tailed returns. Standard Student-$t$ extension scales the Gaussian formulas by $\nu$-dependent factors.

(b) Historical simulation. Use the empirical $\alpha$-quantile of observed losses. Non-parametric; reflects actual fat tails if the sample is long enough but is limited to historical extremes.

(c) Monte Carlo simulation. Sample $M$ scenarios from a calibrated model (GARCH, copula, multivariate $t$, etc.), compute portfolio losses, and take sample quantile / tail mean. Error is $O(1/\sqrt{M})$.

Coherence & the regulatory shift to ES

Artzner et al.\ (1999) axiomatised what it means to be a coherent risk measure: monotonicity, translation invariance, positive homogeneity, and subadditivity — $\rho(X+Y) \le \rho(X) + \rho(Y)$ for any portfolios. ES satisfies all four. VaR satisfies the first three but can violate subadditivity on discrete or heavy-tailed distributions: merging two portfolios can increase VaR, a property that penalises diversification. Basel III (BCBS d457, 2019) moved banks’ market-risk capital onto $\text{ES}_{0.975}$ explicitly for this reason.

Interactive estimator

Parametric, historical, and MC estimates side by side

VaR / ES estimator

★★☆ Synthetic Student-$t$ sample

Sample size $M$

2000

Confidence $\alpha$

0.975

Tail heaviness (Student-$t$ $\nu$)

Volatility $\sigma$

15.00%

Seed

Parametric VaR

—

Historical VaR

—

Parametric ES

—

Historical ES

—

ES/VaR ratio

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Histogram of losses from synthetic Student-$t$ sample. Gold dashed: parametric VaR. Red dashed: historical VaR. Hatched tail: mass beyond VaR, whose mean is ES.

Backtesting & pitfalls

Where VaR goes wrong

Practitioner pitfalls

Fat tails. Parametric Gaussian VaR systematically understates tail risk when returns are leptokurtic. Switching to Student-$t$ or generalised hyperbolic distributions improves fit but requires $\nu$ estimation.
Time-varying volatility. GARCH and stochastic-volatility models make the marginal VaR depend on recent market state; ignoring this produces static VaR that misses vol clustering.
Liquidity horizon. A 1-day VaR assumes positions can be liquidated in one day. For illiquid holdings (private equity, distressed debt), 10-day or longer horizons (and scaling $\sqrt{10}$) are required; Basel uses horizon-dependent liquidity adjustments.
Backtesting. Kupiec (1995), Christoffersen (1998) tests check whether the observed frequency of VaR breaches matches the $1-\alpha$ target. Over-/under-rejection drives regulatory traffic-light systems.
ES is not elicitable. Gneiting (2011) proved ES does not admit a strictly-consistent scoring function, making direct forecast comparison harder than VaR; workarounds exist via joint VaR-ES scoring (Fissler & Ziegel 2016).
Model risk. VaR/ES are model-dependent. Changing the distributional assumption (Gaussian vs $t$) can shift ES by 30-100%.

Key references

Foundational papers and regulation

Artzner, P., Delbaen, F., Eber, J.-M. & Heath, D. (1999).

Coherent Measures of Risk.

Mathematical Finance, 9(3), 203–228. doi:10.1111/1467-9965.00068

Rockafellar, R. T. & Uryasev, S. (2000).

Optimization of Conditional Value-at-Risk.

Journal of Risk, 2(3), 21–41. doi:10.21314/JOR.2000.038

Kupiec, P. H. (1995).

Techniques for Verifying the Accuracy of Risk Measurement Models.

Journal of Derivatives, 3(2), 73–84. doi:10.3905/jod.1995.407942

Christoffersen, P. F. (1998).

Evaluating Interval Forecasts.

International Economic Review, 39(4), 841–862. doi:10.2307/2527341

Gneiting, T. (2011).

Making and Evaluating Point Forecasts.

Journal of the American Statistical Association, 106(494), 746–762. doi:10.1198/jasa.2011.r10138

Fissler, T. & Ziegel, J. F. (2016).

Higher Order Elicitability and Osband's Principle.

The Annals of Statistics, 44(4), 1680–1707. doi:10.1214/16-AOS1439

McNeil, A. J., Frey, R. & Embrechts, P. (2015).

Quantitative Risk Management: Concepts, Techniques and Tools, 2nd ed.

Princeton University Press. ISBN 978-0-691-16627-9.

Föllmer, H. & Schied, A. (2016).

Stochastic Finance: An Introduction in Discrete Time, 4th ed.

de Gruyter. ISBN 978-3-11-046344-9.

Basel Committee on Banking Supervision (2019).

Minimum Capital Requirements for Market Risk (FRTB, d457).

bis.org/bcbs/publ/d457 (Expected Shortfall at 97.5% adopted.)