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FRM Exam Part I · Calculating and Applying VaR

Expected Shortfall and Coherent Risk Measures for FRM Part I

Updated 11 October 2026 · Fact-checked

Expected shortfall (ES, also called CVaR) is the average loss in the tail beyond the VaR level, given that VaR is exceeded. A coherent risk measure satisfies monotonicity, subadditivity, positive homogeneity and translation invariance. ES is coherent. VaR is not, because it can fail subadditivity.

Understand Expected Shortfall and Coherent Risk Measures

Value at Risk (VaR) gives a loss threshold. At 95% confidence it says: losses should exceed this number only 5% of the time. It says nothing about how bad losses are once you pass that threshold.

Expected shortfall (ES) fills that gap. It is the expected loss given that the loss is at or beyond the VaR level. At 95% confidence, ES is the average of the worst 5% of outcomes. Because it averages the tail, ES is always at least as large as VaR at the same confidence level.

A coherent risk measure is one that behaves sensibly. It must satisfy four properties: monotonicity (a portfolio that always loses more has higher risk), subadditivity (the risk of a combined portfolio is no more than the sum of the separate risks), positive homogeneity (doubling the position doubles the risk) and translation invariance (adding risk-free cash of amount c lowers risk by c).

Subadditivity is the property that matters most in exams. It captures diversification: merging portfolios should never create more risk. VaR can violate it, especially with discrete or non-elliptical loss distributions, such as two separate bonds with small default probabilities. ES always satisfies it, so ES is coherent. For normally distributed losses, VaR is subadditive, so the failure shows up in non-normal cases.

In practice, regulators have moved toward ES for market risk capital for this reason. VaR still has strengths: it is simple, widely used and easier to backtest.

Key formulas to remember

Expected shortfall definition
ES(α) = E[ L | L ≥ VaR(α) ]
L is the loss. α is the confidence level, e.g. 95% or 99%. It is the average loss in the tail beyond VaR.
Normal VaR (loss as positive number)
VaR(α) = μ + σ × z(α)
z(95%) = 1.645, z(99%) = 2.326. Use μ = 0 if the question assumes zero mean.
Normal ES
ES(α) = μ + σ × φ(z(α)) ÷ (1 − α)
φ is the standard normal density. φ(1.645) ≈ 0.1031 and φ(2.326) ≈ 0.0267. So ES factor = 2.063 at 95% and 2.665 at 99%.
Discrete ES
ES = (sum of the worst losses in the tail) ÷ (number of tail outcomes)
With n equally likely scenarios, the tail has n × (1 − α) outcomes. Average them.
Coherence axioms
Monotonicity; Subadditivity: ρ(X + Y) ≤ ρ(X) + ρ(Y); Positive homogeneity: ρ(λX) = λρ(X) for λ > 0; Translation invariance: ρ(X + c) = ρ(X) − c
Here X and Y are returns or P&L, with cash c added. Learn the intent of each, not only the symbols.

How to solve Expected Shortfall and Coherent Risk Measures questions

Exam questions on this topic are either calculations (ES from a normal distribution or a list of scenarios) or conceptual (which property does a measure fail).

  1. 1Identify whether the question asks for VaR, ES or a property test.
  2. 2Write down the confidence level and whether losses or returns are used.
  3. 3For a normal distribution, find z(α), then compute VaR = μ + σz. For ES, use μ + σ × φ(z) ÷ (1 − α).
  4. 4For a discrete list of scenarios, sort losses from worst to best. Count the tail size as n × (1 − α).
  5. 5Average the tail losses to get ES. Compare it with VaR: ES must be at least VaR.
  6. 6For a coherence question, test each of the four properties in turn. Subadditivity is the usual failing property for VaR.
  7. 7Scale for time or position size if asked. Positive homogeneity allows scaling by position size; for time, use the square root of time under standard assumptions.

Quickest way: Memorised normal ES factors

When to use it: Use when the question says losses are normal and asks for ES at 95% or 99%.

  1. Remember the factors: 95% VaR = 1.645σ and ES = 2.063σ; 99% VaR = 2.326σ and ES = 2.665σ.
  2. Multiply the factor by σ (in currency terms, σ times portfolio value) and add the mean loss if non-zero.
  3. Sanity check that ES is larger than VaR.
  4. For discrete scenarios, skip the formula: take the worst n × (1 − α) outcomes and average them.

Common mistakes in Expected Shortfall and Coherent Risk Measures

  • Saying ES is smaller than or equal to VaR.

    Students confuse ES with a conditional average of everything inside the VaR.

    Fix: ES averages losses beyond VaR, so it is always at least as large as VaR at the same confidence level.

  • Claiming VaR is never subadditive.

    Students overstate the rule.

    Fix: VaR is not always subadditive. It fails in some cases, such as non-elliptical distributions, but it is subadditive for normal distributions.

  • Using the wrong tail size in discrete ES.

    Students mix up confidence level and tail probability.

    Fix: Tail size = n × (1 − α). With 100 scenarios at 95%, average the worst 5.

  • Using the VaR z-value in the ES formula.

    Both use z(α), so students stop there.

    Fix: ES needs φ(z) ÷ (1 − α). Use the memorised factors 2.063 and 2.665.

  • Mixing up translation invariance and positive homogeneity.

    Both sound like scaling.

    Fix: Translation is adding cash, which lowers risk by that amount. Homogeneity is scaling the position, which scales risk proportionally.

Worked examples

Example 1

A portfolio's daily loss is normally distributed with mean 0 and standard deviation $2 million. Calculate the 99% one-day VaR and ES.

Show the solution
  1. VaR = 2.326 × 2 = $4.652 million.
  2. ES = σ × φ(z) ÷ (1 − α) = 2 × 0.0267 ÷ 0.01 = 2 × 2.665 = $5.33 million (approx., using φ(2.326) ≈ 0.0267).
  3. Check: ES (5.33) is greater than VaR (4.65).

Answer: VaR ≈ $4.65 million; ES ≈ $5.33 million.

Example 2

A portfolio has 20 equally likely loss scenarios. The five worst losses are $12m, $9m, $8m, $7m and $6m. Calculate the 90% ES.

Show the solution
  1. Tail probability = 1 − 0.90 = 10%.
  2. Tail size = 20 × 0.10 = 2 scenarios.
  3. The two worst losses are $12m and $9m.
  4. ES = (12 + 9) ÷ 2 = $10.5 million.

Answer: ES = $10.5 million.

Exam tips

  • Memorise the four coherence properties and be ready to say which one VaR can fail: subadditivity.
  • Memorise the normal ES factors for 95% and 99%. They save time.
  • Always check that ES ≥ VaR in your answer.
  • Read the question for losses versus returns. A VaR quoted as a positive loss is standard.
  • In discrete questions, count the tail size carefully before averaging.

Practice questions from Calculating and Applying VaR

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 at a given confidence level. ES is the average loss beyond that threshold. ES therefore captures tail severity and is always at least as large as VaR.

Is VaR a coherent risk measure?

No. VaR satisfies monotonicity, positive homogeneity and translation invariance, but it can fail subadditivity. ES satisfies all four properties, so it is coherent.

How do you calculate expected shortfall for a normal distribution?

Use ES = μ + σ × φ(z) ÷ (1 − α). For zero mean, ES is about 2.063σ at 95% and 2.665σ at 99%.

Why does subadditivity matter?

It reflects diversification: combining portfolios should not increase total risk. If a measure violates it, the risk of a merged book could be reported as higher than the sum of its parts.