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FRM Exam Part II · Estimating Market Risk Measures: An Introduction and Overview

Expected Shortfall and Coherent Risk Measures Explained

Updated 11 October 2026 · Fact-checked

Expected shortfall (ES) is the average loss in the worst (1 − α) share of outcomes, so it looks inside the tail that VaR ignores. A coherent risk measure is monotonic, subadditive, positively homogeneous and translation invariant. ES meets all four. VaR can fail subadditivity. To get ES from historical data, average the tail losses.

Understand Expected Shortfall and Coherent Risk Measures

Value at Risk (VaR) gives one cutoff: the loss you expect to exceed only (1 − α) of the time. It says nothing about how bad losses are once that cutoff is crossed. Two portfolios can share the same 99% VaR, yet one loses a little more beyond it and the other loses a catastrophic amount.

Expected shortfall (ES), also called conditional VaR or expected tail loss, fixes this. It is the average of all losses that are at or beyond the VaR level. At 97.5% confidence, ES is the average of the worst 2.5% of outcomes. For the same confidence level, ES is always at least as large as VaR.

A coherent risk measure satisfies four axioms from Artzner and others. Monotonicity: if portfolio X always loses at least as much as Y, then its risk is at least as high. Translation invariance: adding a risk-free amount of cash reduces risk by exactly that amount. Positive homogeneity: doubling a position doubles its risk. Subadditivity: the risk of a combined portfolio is no more than the sum of the separate risks.

Subadditivity is the key property because it expresses diversification. Merging portfolios should never raise measured risk. VaR can break this rule, especially with skewed or lumpy losses such as defaults. ES is coherent, so it always respects diversification. It also gives a sensible basis for allocating capital and setting limits.

This is why Basel's FRTB moved the internal models approach for market risk from VaR to ES. VaR is still easy to explain and backtest, so it remains widely used. For normal distributions, VaR is subadditive, so the failure shows up with non-normal, heavy-tailed or discrete losses.

Key formulas to remember

Expected shortfall (continuous)
ES(α) = E[L | L ≥ VaR(α)]
Average loss given that loss is at or beyond the VaR at confidence α. ES ≥ VaR at the same α.
ES from n equally likely historical losses
ES(α) = average of the worst n × (1 − α) losses
If n × (1 − α) is not a whole number, a common treatment is to weight the boundary observation fractionally, or follow the convention the question states.
ES for normal losses
ES(α) = μ + σ × φ(z_α) ÷ (1 − α)
φ is the standard normal density and z_α the α quantile. For α = 99%, φ(2.326) ÷ 0.01 ≈ 2.665; for 95%, ≈ 2.063.
Subadditivity
ρ(X + Y) ≤ ρ(X) + ρ(Y)
Diversification should not increase risk. VaR can violate this; ES cannot.
Positive homogeneity
ρ(λX) = λ × ρ(X), for λ > 0
Scaling a position scales risk by the same factor.
Translation invariance
ρ(X + c) = ρ(X) − c
Where X is a gain/P&L and c is a risk-free cash amount. In loss terms, adding cash c reduces the loss by c.
Monotonicity
If X ≤ Y in every state, then ρ(X) ≥ ρ(Y)
X is P&L. A worse outcome in every state means higher risk.

How to solve Expected Shortfall and Coherent Risk Measures questions

Use this method for any question on ES or coherence, whether it asks for a number or a concept.

  1. 1Identify whether the data are losses or P&L. Convert so that larger numbers mean bigger losses.
  2. 2Note the confidence level α and the number of observations n.
  3. 3Compute the tail size: n × (1 − α). This is how many worst observations are in the tail.
  4. 4Sort losses from largest to smallest and take the worst tail observations. For VaR, use the cutoff loss at the boundary.
  5. 5Average the tail observations to get ES. Check that ES ≥ VaR.
  6. 6For a coherence question, test each property: monotonicity, subadditivity, positive homogeneity, translation invariance.
  7. 7For a normal-distribution question, use μ + σ × φ(z) ÷ (1 − α) with the correct z and state the units.

Quickest way: Tail-average shortcut

When to use it: Use when a question gives a short list of losses or a simple discrete distribution and asks for ES or a comparison with VaR.

  1. Count the tail: n × (1 − α). Often it is a small whole number.
  2. Circle the worst losses in that count.
  3. Add and divide by the count.
  4. Sanity check: the answer must be at least the VaR cutoff.
  5. For coherence questions, remember only VaR fails subadditivity; ES and spectral measures pass.

Common mistakes in Expected Shortfall and Coherent Risk Measures

  • Saying ES is the loss at the 97.5% or 99% quantile.

    Students mix ES with VaR because both use a confidence level.

    Fix: VaR is a single quantile. ES is the average of all losses beyond that quantile.

  • Claiming VaR is never subadditive.

    Students over-learn the headline that VaR is not coherent.

    Fix: VaR can violate subadditivity. For normally distributed losses it is subadditive. State the condition.

  • Averaging the wrong end of the sorted data.

    P&L and losses are mixed up, so the best outcomes get averaged.

    Fix: Convert to losses first, or take the lowest P&L values. Always check ES ≥ VaR.

  • Using the wrong tail size, such as n × α.

    Confidence level and tail probability are confused.

    Fix: Tail size is n × (1 − α). At 95% with 100 observations, the tail is 5 observations.

  • Mixing up translation invariance and positive homogeneity.

    Both sound like scaling or shifting rules.

    Fix: Translation invariance is about adding cash. Positive homogeneity is about multiplying position size.

Worked examples

Example 1

A portfolio has 100 daily historical losses. The five largest are ₹9 lakh, ₹7 lakh, ₹6 lakh, ₹5 lakh and ₹3 lakh. All other losses are smaller than ₹3 lakh. Estimate the 95% ES.

Show the solution
  1. Tail size = 100 × (1 − 0.95) = 5 observations.
  2. The tail consists of the five largest losses: 9, 7, 6, 5 and 3 (₹ lakh).
  3. Sum = 9 + 7 + 6 + 5 + 3 = 30.
  4. ES = 30 ÷ 5 = 6.

Answer: 95% ES ≈ ₹6 lakh. It exceeds the VaR cutoff of about ₹3 lakh to ₹5 lakh, depending on the VaR convention used.

Example 2

Losses are normal with mean 0 and standard deviation $2 million. Using φ(2.326) ≈ 0.0267, compute the 99% ES. Also state which coherence property VaR can fail.

Show the solution
  1. z at 99% = 2.326, and 1 − α = 0.01.
  2. ES = μ + σ × φ(z) ÷ (1 − α) = 0 + 2 × 0.0267 ÷ 0.01.
  3. 0.0267 ÷ 0.01 = 2.67, so ES = 2 × 2.67 = 5.34.
  4. For comparison, 99% VaR = 2 × 2.326 = 4.652, so ES > VaR as expected.
  5. The property VaR can fail is subadditivity.

Answer: 99% ES ≈ $5.34 million (VaR ≈ $4.65 million). VaR can fail subadditivity.

Exam tips

  • Expect conceptual questions asking which property VaR violates. The answer is subadditivity.
  • When asked to compare ES and VaR at the same confidence level, ES is at least as large.
  • In historical ES questions, compute the tail count first. It saves time and avoids averaging errors.
  • Watch the wording on confidence level. 'Worst 2.5%' and '97.5% ES' describe the same tail.
  • Link ES to FRTB: the internal models approach uses ES rather than VaR for market risk capital.

Practice questions from Estimating Market Risk Measures: An Introduction and Overview

Expected Shortfall and Coherent Risk Measures in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Expected Shortfall and Coherent Risk Measures: frequently asked questions

What is the difference between expected shortfall and VaR?

VaR is the loss threshold that is exceeded with probability (1 − α). Expected shortfall is the average loss given that the threshold has been exceeded. ES therefore captures tail severity, which VaR ignores.

What is subadditivity in risk measures?

Subadditivity means the risk of two portfolios combined is no greater than the sum of their individual risks. It captures the idea that diversification should not increase risk. VaR can violate it; ES satisfies it.

What are the properties of a coherent risk measure?

The four properties are monotonicity, subadditivity, positive homogeneity and translation invariance. A measure must satisfy all four to be coherent. ES and spectral risk measures with suitable weights do.

How do I calculate expected shortfall from historical data?

Sort the losses from largest to smallest. Find the tail size as n × (1 − α). Average the losses in that tail. Check the result is not below VaR.