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FRM Exam Part I · Measures of Financial Risk

Spectral Risk Measures and Distortion Risk Measures Explained

Updated 11 October 2026 · Fact-checked

A spectral risk measure is a weighted average of loss quantiles. The weighting function must be non-negative, integrate to one and be non-decreasing, which makes the measure coherent. ES puts equal weight on the tail. VaR is a single quantile, a degenerate limiting case rather than a proper spectral measure, and is not coherent in general.

Understand Spectral and Distortion Risk Measures

Start with quantiles. VaR at 95% is one quantile of the loss distribution. Expected shortfall (ES) at 95% is the average of all quantiles from 95% to 100%. Both are weighted averages of quantiles. They differ only in the weights.

A spectral risk measure generalises this. You pick a weighting function φ(p) that tells you how much weight to give the loss quantile at each probability level p. The risk measure is M = ∫ φ(p) q(p) dp, where q(p) is the loss quantile at level p, integrated from 0 to 1.

The weighting function is the risk aversion function. For the measure to be coherent, φ must satisfy three conditions: it is non-negative, it integrates to 1 over 0 to 1, and it is non-decreasing in p. The last condition is the key one. It says a worse outcome never gets less weight than a better one. That is what risk aversion means here. A weighted average of quantiles is a coherent spectral measure if and only if φ is non-negative, integrates to 1 and is non-decreasing. In that case the measure satisfies subadditivity. When you work with discrete weights wᵢ instead of a continuous φ, the weights sum to 1.

Two special cases to know. For ES at confidence level α, φ(p) = 0 for p < α and φ(p) = 1 ÷ (1 − α) for p ≥ α. This integrates to 1, because the weight 1 ÷ (1 − α) is spread over a tail of width (1 − α). It is flat in the tail, so it is non-decreasing and ES is coherent. In a discrete version with n equally likely tail points, each tail point gets weight 1 ÷ n. For VaR, all weight sits on a single quantile, a spike. This is a limiting, degenerate weighting rather than a proper non-decreasing φ, since the weight drops to zero beyond α. So VaR fails the condition and is not coherent in general.

A common choice that shows risk aversion directly is the exponential weighting function, φ(p) = e^(−(1 − p)/γ) ÷ (γ(1 − e^(−1/γ))), where a smaller γ means greater risk aversion. The denominator is the normalising constant that makes φ integrate to 1. You do not need to memorise the exact constant. You need to know that a more risk-averse user puts more weight on the worst outcomes, and that the weights must still integrate to one.

Distortion risk measures are a related family. They reshape the probability distribution of losses with a distortion function, so that tail outcomes get more weight. A spectral measure is a distortion measure whose distortion function is concave. A concave distortion function corresponds to a non-decreasing weighting function φ, which is why those distortion measures are coherent.

Key formulas to remember

Spectral risk measure
M = ∫₀¹ φ(p) q(p) dp
q(p) is the loss quantile at probability p. φ(p) is the risk aversion weighting function.
Conditions for a coherent spectral measure
φ(p) ≥ 0; ∫₀¹ φ(p) dp = 1; φ is non-decreasing in p
Non-decreasing weights mean worse losses never get less weight. This gives subadditivity.
Expected shortfall as a spectral measure
φ(p) = 1 ÷ (1 − α) for p ≥ α; φ(p) = 0 for p < α
Equal weights on the tail quantiles. Coherent.
VaR as a limiting case
φ(p) = spike at p = α (all weight on q(α))
Not non-decreasing, so VaR is not coherent in general.
Discrete approximation
M ≈ Σ wᵢ qᵢ, with wᵢ ≥ 0, Σ wᵢ = 1, w₁ ≤ w₂ ≤ … ≤ wₙ
Quantiles qᵢ are ordered from best to worst. Weights rise with the loss.

How to solve Spectral and Distortion Risk Measures questions

Exam questions ask you to compute a weighted average of quantiles, test whether a weighting function gives a coherent measure, or relate VaR and ES to the spectral form. Use this order.

  1. 1Identify what is given: a list of loss quantiles with weights, a weighting function, or a verbal description of the measure.
  2. 2Order the quantiles from smallest loss to largest loss, and line the weights up with them.
  3. 3Check the weights. They must be non-negative and sum to 1. If they sum to something else, rescale or note the measure is not properly normalised.
  4. 4Check monotonicity. Weights must be non-decreasing as the loss gets worse. If not, the measure is not guaranteed to be coherent.
  5. 5Compute M = Σ wᵢ qᵢ, multiplying each quantile by its weight and adding.
  6. 6Interpret: a higher weight on worse losses means more risk aversion. Compare with ES (equal weights in the tail) or VaR (one quantile).
  7. 7State the answer with units, such as USD millions of loss.

Quickest way: Weights check, then weighted sum

When to use it: Use this for multiple-choice questions with a short table of quantiles and weights, or a true/false test of coherence.

  1. Scan the weights. Do they sum to 1? Are they non-decreasing with loss? If a question asks about coherence, these two checks decide it.
  2. If the weights are equal over the tail and zero elsewhere, it is ES. Just average the tail quantiles.
  3. If all weight is on one quantile, it is VaR. Read off that quantile.
  4. Otherwise multiply and add. Do the arithmetic with the largest weights first to catch obvious wrong options.
  5. Eliminate any option that gives a result outside the range of the quantiles. A weighted average cannot be below the smallest or above the largest quantile.

Common mistakes in Spectral and Distortion Risk Measures

  • Saying VaR is a spectral risk measure that is coherent.

    VaR is a weighted average of quantiles in the trivial sense, so students assume it qualifies.

    Fix: VaR puts all weight on one quantile, a spike followed by zero weight. That is a degenerate limiting weighting, not a proper non-decreasing spectral φ. So VaR is not a proper spectral risk measure, and it fails subadditivity in general.

  • Allowing weights that fall as losses get worse.

    Students think any weights that sum to 1 are fine.

    Fix: Coherence needs non-decreasing weights. Falling weights mean the measure under-weights extreme losses, which breaks subadditivity.

  • Forgetting to normalise the weights.

    Weights are given as raw numbers such as 1, 2, 3.

    Fix: Divide each by the total so the weights sum to 1 before computing the weighted average.

  • Mixing up the direction of risk aversion.

    Students think a more risk-averse user spreads weight evenly.

    Fix: More risk aversion means steeper weights that favour the worst outcomes. Constant weights across the whole distribution (φ = 1) are non-decreasing and give the expected loss. That is the risk-neutral boundary case of a coherent spectral measure, with no extra weight on bad outcomes.

  • Treating ES weights as 1 ÷ N over all quantiles.

    Students forget ES only averages the tail.

    Fix: ES at α gives weight 1 ÷ (1 − α) to quantiles beyond α and zero below. Only the tail counts.

Worked examples

Example 1

A loss distribution has a 95% quantile of USD 10 million, which marks the boundary of the tail. The tail beyond 95% is approximated by five equally likely points, with loss quantiles of USD 12, 14, 17, 21 and 30 million. Compare ES at 95%, the average of these five tail points, with a spectral measure that gives weights 0.1, 0.1, 0.2, 0.2, 0.4 to the same five points and zero weight to every quantile below the tail. The USD 10 million boundary quantile is not one of the five tail points, so it is not used in either calculation.

Show the solution
  1. ES averages the tail, here the five points that approximate the tail beyond 95%. Each gets weight 1 ÷ 5 = 0.2, and everything below the tail gets zero.
  2. ES = 0.2 × (12 + 14 + 17 + 21 + 30) = 0.2 × 94 = 18.8.
  3. Check the spectral weights. They are zero below the tail, then 0.1, 0.1, 0.2, 0.2, 0.4 across the five tail points. They sum to 1.0 and are non-negative.
  4. The weights are non-decreasing as the loss gets worse, because they stay at zero below the tail and then never fall. So the measure is coherent.
  5. Compute: 0.1 × 12 = 1.2; 0.1 × 14 = 1.4; 0.2 × 17 = 3.4; 0.2 × 21 = 4.2; 0.4 × 30 = 12.0.
  6. Add: 1.2 + 1.4 + 3.4 + 4.2 + 12.0 = 22.2.
  7. The spectral value is higher because it puts more weight on the worst outcome.

Answer: ES = USD 18.8 million. The spectral measure = USD 22.2 million, which reflects greater risk aversion.

Exam tips

  • Memorise the three conditions for a coherent spectral measure: non-negative, sum to 1, non-decreasing. Many questions test only these.
  • Remember that ES and VaR are both special cases, but only ES is coherent.
  • Always check that your answer lies between the smallest and largest quantile. A weighted average cannot fall outside that range.
  • Questions on risk aversion usually ask which weighting gives the larger measure. The one that puts more weight on the worst outcomes does.

Practice questions from Measures of Financial Risk

Spectral and Distortion 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.

Spectral and Distortion Risk Measures: frequently asked questions

What is a spectral risk measure in simple terms?

It is a weighted average of the quantiles of the loss distribution. The weights express how much you care about each part of the distribution. If the weights never fall as losses get worse, the measure is coherent.

How is a spectral risk measure different from expected shortfall?

ES is one particular spectral measure with equal weights on all tail quantiles beyond the confidence level. A general spectral measure can use any non-decreasing weights, so it can weight the worst losses more heavily than ES does.

How do I choose the weighting function?

The choice reflects the user's risk aversion. Steeper, more tail-heavy weights suit a more risk-averse user. The function must be non-negative, integrate to 1 and be non-decreasing in the loss, otherwise coherence is lost.

Is VaR a spectral risk measure?

VaR can be written as a weighting that puts all weight on a single quantile. That weighting is not non-decreasing across the tail, so VaR is not coherent in general.