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

Spectral Risk Measures and Risk-Aversion Weights

Q: What is a spectral risk measure in simple terms?

It is a weighted average of loss quantiles. The weights show how much you care about each part of the loss distribution. Worse losses get at least as much weight as milder ones.

Q: Why are spectral risk measures coherent?

When the weights are non-negative, non-decreasing and sum to one, the measure satisfies monotonicity, sub-additivity, positive homogeneity and translation invariance. The non-decreasing weights are what give sub-additivity.

Q: Is expected shortfall a spectral risk measure?

Yes. It gives equal weight to every quantile beyond the confidence level and zero weight below it. That weight pattern is non-decreasing, so ES is coherent.

Q: What does exponential weighting do?

It gives weights that rise exponentially toward the worst losses. A larger risk-aversion coefficient pushes more weight onto extreme quantiles and raises the measure.

Updated 11 October 2026 · Fact-checked

A spectral risk measure is a weighted average of the quantiles of the loss distribution. The weights come from a risk-aversion function that must be non-negative, non-decreasing in the tail and sum to one. If these conditions hold, the measure is coherent. Expected shortfall is a special case; VaR is not.

Understand Spectral Risk Measures and Risk-Aversion Weights

Start with a loss distribution. VaR picks one quantile. Expected shortfall (ES) averages all quantiles beyond a cutoff, giving each equal weight. Both use a fixed rule for how much each part of the tail matters.

A spectral risk measure generalises this. It takes a weighted average of all quantiles of the loss distribution. The weights are given by a risk-aversion function φ(p), where p is the cumulative probability level of the loss quantile. You choose φ to reflect how much the user dislikes losses at each probability level.

The key condition is that φ must give more weight to worse losses. If a user is risk averse, a loss at the 99.9% quantile must weigh at least as much as a loss at the 95% quantile. Formally φ must be non-negative, non-decreasing in p, and integrate to one over 0 to 1. A measure built this way is coherent: it is monotonic, sub-additive, positively homogeneous and translation invariant. Sub-additivity is what VaR lacks.

Special cases show the idea. ES at confidence α sets φ(p) = 1 ÷ (1 − α) for p ≥ α and 0 below. That is flat weight in the tail and zero outside, so ES is a spectral measure. VaR is not a spectral measure. It puts all weight on a single quantile, a spike that is not an admissible non-decreasing weighting function. VaR is also not sub-additive in general, which is why it is not coherent.

A popular smooth choice is the exponential weighting function: φ(p) = γ · e^(−γ(1 − p)) ÷ (1 − e^(−γ)), with γ > 0. Here γ is the coefficient of absolute risk aversion. It controls how fast weight rises toward the tail. A larger γ means a more risk-averse user and heavier weight on extreme quantiles. In practice the integral is approximated by averaging many quantiles with their weights.

Key formulas to remember

Spectral risk measure: M_φ = ∫₀¹ φ(p) · q_p dp; q_p is the loss quantile at probability p. φ is the risk-aversion weighting function.
Admissibility conditions on φ: φ(p) ≥ 0; φ(p) non-decreasing in p; ∫₀¹ φ(p) dp = 1; Non-decreasing weights reflect risk aversion. If all three hold, the measure is coherent.
Expected shortfall as a spectral measure: φ(p) = 1 ÷ (1 − α) for p ≥ α; φ(p) = 0 for p < α; Equal weights on all tail quantiles beyond the confidence level α.
Discrete approximation: M_φ ≈ Σ wᵢ · qᵢ, with wᵢ ≥ 0, Σ wᵢ = 1; Weights must be non-decreasing as the quantile gets worse.
Exponential weighting idea: weight rises exponentially toward the worst tail; larger risk aversion → heavier tail weight; Know the behaviour, not a memorised constant.

How to solve Spectral Risk Measures and Risk-Aversion Weights questions

Use this routine for any spectral risk measure question, whether it asks for a number, a coherence test or an interpretation.

1Identify the loss quantiles given and the probability level each one sits at.
2Read the weights. Check that they are non-negative and sum to one. If not, normalise them only if the question says so.
3Check the ordering: weights must not fall as losses get worse. If a weight falls in the tail, the measure is not a valid spectral measure and may not be coherent.
4Multiply each quantile by its weight.
5Add the products to get the measure.
6Interpret: compare with VaR or ES at similar levels and say whether more risk aversion raises the number.
7State the coherence conclusion with the reason (non-decreasing weights give sub-additivity).

Quickest way: Weights check then weighted sum

When to use it: Use when the question gives a short list of quantiles and weights and asks for a value or a validity check.

Scan the weights first: do they rise toward the worst loss and sum to 1?
If yes, compute Σ weight × quantile directly.
If asked which option is more risk averse, pick the one with more weight on extreme quantiles; it gives the larger measure.
If asked about coherence, answer yes only when weights are non-negative and non-decreasing.

Common mistakes in Spectral Risk Measures and Risk-Aversion Weights

Treating VaR as a spectral measure that is coherent.
VaR is a weighted quantile in a loose sense, so it looks like a special case.
Fix: VaR is not a spectral measure and is not coherent because it is not sub-additive in general. It puts all weight on one quantile, which is not an admissible weighting function.
Forgetting that weights must sum to one.
Students focus on the shape of the weights and skip the normalisation.
Fix: Always add the weights. If they do not total 1, the number is not a proper spectral measure.
Allowing weights to decrease in the tail.
Students assume any positive weights are fine.
Fix: Risk aversion means worse losses get at least as much weight. A falling weight violates the rule and can break coherence.
Thinking higher risk aversion lowers the measure.
Confusion between risk aversion and risk tolerance.
Fix: More risk aversion puts more weight on worst losses, so the measure rises.
Confusing the weight with a probability of the loss.
Both are numbers between 0 and 1 that sum to one.
Fix: The weight reflects user attitude to that quantile. The probability level is the position of the quantile in the distribution.

Worked examples

Example 1

A risk manager approximates a spectral risk measure using four loss quantiles (in USD million): 10 at p = 0.90, 14 at p = 0.95, 20 at p = 0.99, 30 at p = 0.999. The weights are 0.10, 0.20, 0.30 and 0.40 respectively. Compute the measure and say whether the weights are admissible.

Show the solution

Check weights: 0.10 + 0.20 + 0.30 + 0.40 = 1.00, all non-negative.
Check ordering: weights rise as the loss gets worse, so non-decreasing.
Compute: 0.10 × 10 = 1.0
0.20 × 14 = 2.8
0.30 × 20 = 6.0
0.40 × 30 = 12.0
Sum: 1.0 + 2.8 + 6.0 + 12.0 = 21.8

Answer: The measure is USD 21.8 million. The weights are admissible, so the measure is coherent.

Example 2

Two users apply weights to the same three quantiles (USD million): 8, 15 and 25. User A uses weights 0.2, 0.3, 0.5. User B uses weights 0.1, 0.2, 0.7. Which user is more risk averse, and what are the two measures?

Show the solution

Both sets sum to 1 and rise toward the worst loss, so both are valid.
User A: 0.2 × 8 = 1.6; 0.3 × 15 = 4.5; 0.5 × 25 = 12.5.
Sum for A: 1.6 + 4.5 + 12.5 = 18.6
User B: 0.1 × 8 = 0.8; 0.2 × 15 = 3.0; 0.7 × 25 = 17.5.
Sum for B: 0.8 + 3.0 + 17.5 = 21.3
User B puts more weight on the worst quantile, so B is more risk averse.

Answer: User A's measure is USD 18.6 million and User B's is USD 21.3 million. User B is more risk averse.

Exam tips

Memorise the three conditions on the weighting function: non-negative, non-decreasing in the tail, sum to one.
Expect a question asking which measure is coherent. Spectral with non-decreasing weights and ES pass; VaR does not in general.
When comparing users, the more risk-averse one has more weight on extreme quantiles and a higher measure.
For exponential weighting, you need the behaviour: higher risk aversion means faster-rising tail weights. Constants are rarely needed.
Do the weights-sum check first. It is the fastest way to eliminate wrong options.

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