02 — Risk Measures

Expected Shortfall (CVaR)#

Tail AverageStatistical1 min read
◆ The PatternAverage loss in the worst α % of scenarios

Expected Shortfall (ES), also called Conditional VaR, is the mean of all losses exceeding the VaR cutoff. It is sub-additive — the ES of a combined portfolio is never more than the sum of the parts’ ES — so regulators prefer it to VaR.

ESα = E[ L | L > VaRα ]
For a normal distribution ES has a closed-form: ES = μ + σ · φ(zα) / (1 − α). For fat-tailed distributions Monte Carlo or historical methods are used.
// Interactive — tail threshold and expected shortfall region
Tail α %
Basel III. Banks must now report Expected Shortfall at 97.5 % under the Fundamental Review of the Trading Book (FRTB).
Pattern bridge: Expectation and conditioning connect to The Toolkit — Distribution Shape.
Worked example

10,000 days of fat-tailed returns with 1% daily volatility. At 97.5% the VaR is 1.83% — below the 1.96% a normal distribution would give. Expected Shortfall, the average loss beyond the VaR, is 2.92%, against 2.34% for the normal. Fat tails can make VaR look safer while the losses past it are bigger; ES sees what VaR misses.

Where it misleads
  • ES averages very few observations — 2.5% of the sample — so it is noisier than VaR and needs a long history or a model of the tail.
  • It is harder to backtest than VaR, because there is no simple count of breaches to check.
  • Like VaR it is only as good as the data behind it: a sample without a crisis gives an ES without one.
In code
rng = np.random.default_rng(4)
r = rng.standard_t(3, 10_000) * 0.01 / np.sqrt(3)          # fat-tailed daily returns, 1% volatility
var = -np.quantile(r, 0.025)                               # VaR at 97.5%
es = -r[r <= -var].mean()                                  # the average loss beyond it
es_normal = 0.01 * np.exp(-1.96**2 / 2) / np.sqrt(2 * np.pi) / 0.025   # ES of a normal with the same volatility
Python, in your browser — edit the code and run it again

Assumes import numpy as np and import pandas as pd. Each snippet carries its own example numbers; the comments say which are made up or simulated.

Sources
  • P. Artzner, F. Delbaen, J.-M. Eber & D. Heath, “Coherent Measures of Risk”, Mathematical Finance 9(3), 1999
  • C. Acerbi & D. Tasche, “On the Coherence of Expected Shortfall”, Journal of Banking & Finance 26(7), 2002
  • Basel Committee on Banking Supervision, Minimum Capital Requirements for Market Risk, 2019 — ES at 97.5% for the trading book

Last reviewed October 2026 · How this site checks its content · Report a mistake

Check yourself

1. What is the 97.5% expected shortfall?

Show answer

B. It looks beyond the VaR threshold, so it reflects how bad the tail is, not just where it starts.

2. How does expected shortfall compare with VaR at the same confidence level?

Show answer

A. It averages losses that are all beyond the VaR, so it cannot be smaller.

3. What did the Basel Committee’s market-risk framework (FRTB) do with VaR?

Show answer

A. For a normal distribution the two are similar in size, but expected shortfall reacts to the shape of the tail.

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