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FRM Exam Part I · Measures of Financial Risk

Expected Shortfall (ES) Formula and How to Calculate It

Updated 11 October 2026 · Fact-checked

Expected shortfall (ES) is the average loss in the worst (1 − α) share of outcomes, that is, the expected loss given that loss is at or beyond VaR. For discrete or historical data, average the tail losses. For a normal loss, use ES = μ + σ × φ(z) ÷ (1 − α).

Understand Expected Shortfall (ES)

Value at Risk (VaR) gives you a cut-off. At 95% confidence, it is the loss you expect to exceed only 5% of the time. It says nothing about how bad things get once you cross that line.

Expected shortfall (ES) fills that gap. It is the average loss in the tail, given that the loss is at or beyond the VaR threshold. Some texts call it conditional VaR (CVaR), average VaR or expected tail loss. In FRM Part I, treat these as the same idea: the mean of the losses in the worst (1 − α) fraction of outcomes.

Because ES averages the whole tail, it is always at least as large as VaR at the same confidence level. It also reacts to fat tails and extreme losses, which VaR ignores. ES is a coherent risk measure, so it satisfies subadditivity: the ES of a combined portfolio is never more than the sum of the separate ES figures. VaR can break this rule.

For a discrete or historical sample, you pick the tail, then take a probability-weighted average of the tail losses. For a normal loss distribution there is a closed form that uses the standard normal density at the VaR point. Work in losses as positive numbers, and the arithmetic stays simple.

Key formulas to remember

Definition of ES
ES(α) = E[L | L ≥ VaR(α)]
L is loss, shown as a positive number. α is the confidence level, such as 95% or 99%. The tail has probability 1 − α.
Discrete or historical ES
ES(α) = Σ (tail loss × its probability) ÷ (1 − α)
For N equally likely historical observations, ES is the simple average of the worst N × (1 − α) losses. Dividing by the tail probability is essential.
Normal ES (losses)
ES(α) = μ + σ × φ(z_α) ÷ (1 − α)
z_α = Φ⁻¹(α) and φ is the standard normal density, φ(z) = (1 ÷ √(2π)) × e^(−z²÷2). μ and σ are the mean and standard deviation of the loss.
Normal VaR for comparison
VaR(α) = μ + σ × z_α
Common z values: 1.645 at 95% and 2.326 at 99%.
Normal ES multipliers (μ = 0)
95%: ES ≈ 2.063σ; 99%: ES ≈ 2.67σ
Memorise these. They come from φ(1.645) ≈ 0.1031 and φ(2.326) ≈ 0.0267. Add μ to get the general case.
Ordering of measures
ES(α) ≥ VaR(α)
Holds at the same confidence level for the same loss distribution.

How to solve Expected Shortfall (ES) questions

Use this method for any ES question, whether the data are a table of probabilities, a list of historical returns or a normal distribution.

  1. 1Express everything as losses. Convert returns or P&L to positive losses, so a loss of 3 is +3 and a gain is negative.
  2. 2Find the tail probability: 1 − α. At 95% it is 5%. At 99% it is 1%.
  3. 3Identify the tail. For a discrete table, sort losses from largest down and collect outcomes until their probabilities add to 1 − α. For historical data, count N × (1 − α) of the worst observations.
  4. 4For discrete or historical data, compute the probability-weighted sum of the tail losses and divide by the tail probability. For equally likely data, this is the simple average.
  5. 5For a normal distribution, find z_α, compute φ(z_α), then ES = μ + σ × φ(z_α) ÷ (1 − α).
  6. 6If a tail outcome only partly fits inside the tail probability, include only the part you need and divide by 1 − α.
  7. 7Check: ES must be at least VaR at the same confidence level, and it should lie among the largest losses. Attach the unit and horizon.

Quickest way: Memorised multipliers and tail-average shortcut

When to use it: Use it when the exam gives a normal distribution at 95% or 99%, or a short list of equally likely losses.

  1. Normal case: use ES = μ + 2.063σ at 95% or ES = μ + 2.67σ at 99%. Do not recompute φ unless another confidence level is asked.
  2. Scale for the horizon first if needed. Multiply σ by √t for t days, and the mean by t.
  3. Historical case: count N × (1 − α) tail observations, add them up and divide by that count.
  4. Sanity check against VaR: the 95% normal VaR multiplier is 1.645 and the ES multiplier is 2.063, so ES should be about 25% above VaR. At 99%, it is about 15% above (2.67 versus 2.326).
  5. Eliminate answer options smaller than VaR immediately.

Common mistakes in Expected Shortfall (ES)

  • Forgetting to divide by the tail probability (1 − α).

    You sum probability-weighted tail losses and stop, which gives a number that is too small.

    Fix: The weighted sum covers only part of the distribution. Always divide by 1 − α so the tail probabilities act as weights that sum to 1.

  • Using z_α instead of φ(z_α) ÷ (1 − α) in the normal formula.

    The VaR formula uses z and the ES formula looks similar, so candidates reuse the multiplier.

    Fix: VaR multiplier is z_α. ES multiplier is φ(z_α) ÷ (1 − α). Memorise 2.063 and 2.67 for 95% and 99%.

  • Averaging the wrong number of historical observations.

    Candidates confuse 95% confidence with 95 observations, or include the VaR observation inconsistently.

    Fix: Tail count = N × (1 − α). With 200 observations at 99%, average the worst 2. Follow the wording of the question if it says 'beyond VaR'.

  • Mixing returns and losses, giving a negative ES or the wrong sign on μ.

    Data are given as returns, where bad outcomes are negative.

    Fix: Convert to losses first. If the mean return is positive, the mean loss is negative, so μ in the loss formula is negative.

  • Believing ES is the same as VaR at a higher confidence level.

    Both numbers move into the tail as confidence rises.

    Fix: ES averages the whole tail beyond VaR. It depends on the tail shape, not just one quantile. Only for special cases do the two numbers coincide.

  • Saying VaR is coherent but ES is not, or the reverse.

    The subadditivity point is easy to mix up.

    Fix: ES is coherent (monotonic, translation invariant, homogeneous, subadditive). VaR is not always subadditive.

Worked examples

Example 1

A portfolio's one-day loss distribution (USD million) has these worst outcomes: loss 10 with probability 2%, loss 6 with probability 3%, and all other outcomes are losses below 6 or gains. Calculate the 95% one-day ES.

Show the solution
  1. Tail probability = 1 − 0.95 = 0.05.
  2. The two worst outcomes have total probability 0.02 + 0.03 = 0.05, so they fill the tail exactly.
  3. Weighted tail losses = 10 × 0.02 + 6 × 0.03 = 0.20 + 0.18 = 0.38.
  4. ES = 0.38 ÷ 0.05 = 7.6.

Answer: The 95% one-day ES is USD 7.6 million.

Example 2

A trading book has normally distributed one-day losses with mean 0 and standard deviation USD 1.5 million. Calculate the 99% one-day VaR and ES.

Show the solution
  1. Find z at 99%: z = 2.326.
  2. VaR = μ + σ × z = 0 + 1.5 × 2.326 = 3.489, about USD 3.49 million.
  3. Find the density: φ(2.326) = (1 ÷ √(2π)) × e^(−2.326² ÷ 2) ≈ 0.3989 × e^(−2.705) ≈ 0.3989 × 0.0669 ≈ 0.0267.
  4. ES multiplier = φ ÷ (1 − α) = 0.0267 ÷ 0.01 = 2.67.
  5. ES = 0 + 1.5 × 2.67 = 4.005, about USD 4.0 million.
  6. Check: ES (4.0) is above VaR (3.49), as it must be.

Answer: 99% one-day VaR is about USD 3.49 million and 99% one-day ES is about USD 4.0 million.

Exam tips

  • Expect the question to ask you to compare VaR and ES. The safe statements: ES ≥ VaR, ES captures tail severity, and ES is coherent while VaR can fail subadditivity.
  • Memorise the two normal multipliers, 2.063 at 95% and 2.67 at 99%. They save time on every normal-distribution question.
  • For historical data, always compute the tail count as N × (1 − α) before touching a calculator.
  • Read whether the data are returns or losses. Convert before you average.
  • A calculator helps only for φ at unusual confidence levels. Otherwise, do tail averages by hand and save time for harder questions.

Practice questions from Measures of Financial Risk

Expected Shortfall (ES) 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 (ES): frequently asked questions

What is the difference between VaR and expected shortfall?

VaR is a single cut-off: the loss exceeded with probability 1 − α. ES is the average loss given that you are beyond that cut-off. ES is never smaller than VaR at the same confidence level and it is sensitive to the shape of the tail.

Is conditional VaR the same as expected shortfall?

In the FRM context, yes. CVaR, expected tail loss and average VaR all describe the mean loss in the tail beyond VaR. For continuous distributions they are identical. For discrete data, read the question wording on how the tail is defined.

How do I calculate expected shortfall from historical data?

Sort the losses from worst to best. Compute N × (1 − α) to get the number of tail observations, then average those worst losses. For example, with 500 observations at 99%, you average the worst 5.

What is the expected shortfall formula for a normal distribution?

ES = μ + σ × φ(z_α) ÷ (1 − α), where φ is the standard normal density and z_α is the α quantile. With μ = 0, the multiplier is about 2.063 at 95% and 2.67 at 99%.