Expected Shortfall and Coherent Risk Measures
Expected shortfall averages losses in the worst tail at a chosen confidence level. Unlike VaR, it measures severity beyond the cutoff. Its quantile-integral definition handles discrete losses and supports the diversification property called subadditivity.
The method at a glance
Illustrative comparison · million · bars start at zero
- A · 95% VaR2
- B · 95% VaR2
- A · 95% ES4
- B · 95% ES8
100 equally likely scenarios; nearest-rank VaR. ES averages the five worst losses in the case below.
| Concept | How it works | Watch for |
|---|---|---|
| VaR versus ES | VaR locates a quantile; ES averages the specified tail. | Reporting the cutoff as the average tail loss. |
| Coherence | Check monotonicity, cash translation, positive homogeneity and subadditivity. | Assuming coherence makes an estimated model accurate. |
| Discrete samples | Average the exact tail probability, including fractional mass at a boundary if necessary. | Dropping all observations tied at the VaR threshold. |
01VaR versus ES
How it works: VaR locates a quantile; ES averages the specified tail.
Watch for: Reporting the cutoff as the average tail loss.
02Coherence
How it works: Check monotonicity, cash translation, positive homogeneity and subadditivity.
Watch for: Assuming coherence makes an estimated model accurate.
03Discrete samples
How it works: Average the exact tail probability, including fractional mass at a boundary if necessary.
Watch for: Dropping all observations tied at the VaR threshold.
Original worked example
Compute a 95% tail average
Use 100 equally likely scenarios. The 95th ordered loss is 2 million for both portfolios. The five worst losses are A: 3, 3, 4, 4, 6 and B: 3, 5, 7, 10, 15 million.
- 1
The worst 5% contains five full observations in this example.
- 2
A: 20/5 = 4 million; B: 40/5 = 8 million.
- 3
Both nearest-rank VaRs are 2 million despite their different tail averages.
Expected shortfall reveals twice the average tail loss for B. The shared VaR cutoff conceals this difference.
Illustrative assumptions and calculations for learning. Not observed market data or an official exam item.
Apply the method and check its limits
Before you start: Loss quantiles, probability weights and the convention that larger positive numbers are worse losses.
Use the full tail probability, including boundary mass
ESα = (1 / (1 - α)) × ∫α¹ VaRu du
For a discrete example, suppose loss is 0 with probability 96% and 100 with probability 4%. At 95% confidence, VaR is 0, but ES is (4% × 100 + 1% × 0)/5% = 80. Averaging only losses strictly above VaR gives 100 and ignores the needed boundary probability. Use a tail of exactly the required probability mass.
Check all four coherence properties
Monotonicity respects scenario-by-scenario loss ordering. Translation invariance adds c to measured risk when a certain loss c is added. Positive homogeneity scales risk with a nonnegative position multiplier. Subadditivity requires combined risk to be no greater than the sum of separate risks. ES has these properties under the appropriate definition; VaR can fail subadditivity.
Why coherence does not remove estimation risk
Subadditivity means the risk of combined losses is no greater than the sum of their separate risks under the same coherent measure. That is a mathematical property, not a promise about the quality of tail data. ES can be especially sensitive to a few extreme observations. Check the sample, horizon, dependence and estimation uncertainty.
Work through the topic
Calculate expected shortfall with probability weights, understand the four coherence properties, and distinguish a mathematical property from evidence that a model is adequate.
Average the correct amount of tail probability
Expected shortfall at confidence level α measures the mean loss in the worst 1 - α share of the distribution, with appropriate treatment of probability mass at the cutoff. For a continuous loss distribution, this is often written as the conditional mean beyond VaR. Discrete distributions need more care because several outcomes can sit exactly at the quantile. Simply averaging every observation greater than or equal to VaR can include too much probability.
Consider an original loss distribution: a loss of 0 with probability 90%, a loss of 10 with probability 7%, and a loss of 50 with probability 3%. Under the lower-quantile definition, 95% VaR is 10. The worst 5% includes all of the 3% probability at 50 and only 2 percentage points of the probability at 10. Therefore 95% ES equals (0.03 × 50 + 0.02 × 10)/0.05 = 34.
Averaging all outcomes with loss at least 10 would instead use 10% probability and give 22. Averaging only losses strictly above 10 would give 50. Neither calculation selects the required worst 5%. Writing a probability-allocation table prevents both mistakes. With equally weighted observations, the same principle determines whether a boundary observation contributes fully or fractionally under the chosen empirical convention.
| Loss outcome | Total probability | Probability included in ES |
|---|---|---|
| 50 | 3% | All 3% |
| 10 | 7% | 2% of total distribution |
| 0 | 90% | None |
| Tail average | (1.50 + 0.20) / 0.05 | 95% expected shortfall = 34 |
Original discrete distribution: constructing the worst 5%
- 50
- Total probability: 3%
- Probability included in ES: All 3%
- 10
- Total probability: 7%
- Probability included in ES: 2% of total distribution
- 0
- Total probability: 90%
- Probability included in ES: None
- Tail average
- Total probability: (1.50 + 0.20) / 0.05
- Probability included in ES: 95% expected shortfall = 34
Interpret coherence under a positive-loss convention
Monotonicity means that a position producing no smaller losses in every state should not receive a smaller risk measure. Translation invariance means adding a certain loss c increases measured risk by c. Positive homogeneity means scaling a position by a nonnegative amount scales its measured risk by the same amount. These statements require consistent signs and a specified loss model; mixing profit and loss conventions reverses the interpretation of a cash adjustment.
Subadditivity says that the risk of a combined position is no greater than the sum of its separate measured risks. It gives a mathematical expression to the possibility of diversification. Properly defined expected shortfall satisfies the coherence properties under the relevant integrability conditions. VaR does not satisfy subadditivity for every distribution, although it can do so within particular distribution families.
The axioms describe the risk functional applied to a distribution. They do not certify the distribution supplied to it. An ES estimate based on missing exposures, incorrect prices or unrealistic dependence assumptions can be coherent and still misleading. Likewise, positive homogeneity within a model does not mean real trading costs remain proportional when the position becomes large enough to affect market liquidity.
Work through a VaR diversification counterexample
Take two independent loans. Each loses 100 on default with probability 4% and otherwise loses zero. For either loan, 95% VaR is zero because 96% of the distribution has no loss. The sum of the two standalone VaRs is therefore zero. This result does not mean the loans are safe; the selected quantile falls inside the large mass of zero-loss outcomes.
For the combined portfolio, no default has probability 0.96² = 92.16%. Exactly one default has probability 2 × 0.04 × 0.96 = 7.68%, and both default with probability 0.04² = 0.16%. The combined 95% VaR is 100. Thus portfolio VaR exceeds the sum of standalone VaRs in this example. The calculation demonstrates failure of a general axiom, not failure of economic diversification.
The corresponding standalone 95% ES is 80 for each loan: the worst 5% includes a 4% chance of losing 100 and a 1% chance of losing zero. Portfolio ES is (0.0016 × 200 + 0.0484 × 100)/0.05 = 103.2, below the standalone sum of 160. Dependence assumptions matter: replacing independence with a shared default event changes the portfolio distribution and the diversification benefit.
Does ES tell me the maximum possible loss?
No. It is an average over a selected tail. Individual outcomes can exceed it, sometimes substantially. A bounded maximum, where one exists, is a different feature of the distribution and must be established separately.
Why can ES be difficult to estimate?
It uses tail-loss magnitudes, where observations may be scarce and unstable. A model can improve the amount of usable information but introduces assumptions about the tail. Report the estimation window, weighting, confidence level and sensitivity to alternative specifications.
Try a changed assumption
Two independent loans each lose 100 with probability 4%. What happens to 95% VaR when combined?
Show answer and reasoning
Each loan has VaR 0. Together, probability of no loss is 0.96² = 92.16%, and probability of loss at most 100 is 99.84%. Combined VaR is therefore 100, exceeding 0 + 0. This illustrates VaR’s failure of subadditivity.
Original learning check. No email or account needed.
Is ES always the mean of observations strictly above VaR?
For a continuous distribution that shortcut is appropriate. With discrete probability masses, use the worst tail of the required size and allocate boundary mass correctly.
Sources and scope
Links checked 2026-10-10. GARP establishes exam scope; the technical references support the methods identified below. These are selected study notes and original examples, not complete curriculum coverage or official exam questions.
- GARP FRM study materials and 2026 learning objectives
Exam-domain scope and the official route to the annual learning objectives. This page does not certify coverage of every reading.
- Acerbi and Tasche: On the coherence of Expected Shortfall
Expected shortfall, coherence and the treatment of discrete loss distributions.
- Basel Committee: minimum capital requirements for market risk (2019)
Regulatory context for expected shortfall and market-risk model assessment. Local implementation can differ.
Prepared by MiloPrep editorial team. MiloPrep is independent of GARP. References do not imply endorsement or independent review of these notes.