FRM Exam Part I · Measures of Financial Risk

Coherent Risk Measures and Subadditivity Explained

Updated 11 October 2026 · Fact-checked

A coherent risk measure satisfies four properties: monotonicity, subadditivity, positive homogeneity and translation invariance. Subadditivity means the risk of a combined portfolio is no more than the sum of the separate risks. VaR can break it, so VaR is not coherent. Expected shortfall satisfies all four, so it is coherent.

Understand Coherent Risk Measures and Subadditivity

A risk measure turns a portfolio's loss distribution into one number, such as a capital requirement. Not every such number behaves sensibly. Coherence is a set of four common-sense rules that a good measure should obey.

The four properties, with L as loss and ρ as the risk measure:

Monotonicity: if portfolio X always has losses at least as large as portfolio Y, then X has at least as much risk as Y.
Subadditivity: ρ(X + Y) ≤ ρ(X) + ρ(Y). Merging portfolios cannot create risk. This is the mathematical form of diversification.
Positive homogeneity: scaling a position by a factor k > 0 scales risk by k. Doubling the position doubles the risk.
Translation invariance: adding a sure amount of cash reduces risk by exactly that amount (in loss terms, adding a sure gain of c lowers the measure by c).

Subadditivity is the property that matters most in exams. If a measure fails it, a firm could cut its reported risk by splitting itself into separate entities. A regulator who sets capital by such a measure can be gamed.

VaR is the loss threshold that is not exceeded with a given confidence level. It ignores how bad losses are beyond the threshold. Because of this, VaR can rise when two portfolios are combined. This happens typically with discrete, skewed or fat-tailed payoffs, such as two bonds with small default probabilities. For elliptical distributions such as the normal, VaR is subadditive. VaR does satisfy monotonicity, homogeneity and translation invariance. Only subadditivity can fail.

Expected shortfall (ES) is the average loss in the tail beyond the VaR level. It is coherent, which is the main reason regulators have moved toward it. Standard deviation is subadditive and homogeneous but not monotonic, so it is not coherent either.

Key formulas to remember

Monotonicity: If X ≤ Y in every state (X has lower losses), then ρ(X) ≤ ρ(Y); Check the direction in terms of losses. Worse outcomes mean higher risk.
Subadditivity: ρ(X + Y) ≤ ρ(X) + ρ(Y); The key property. VaR can violate it. ES cannot.
Positive homogeneity: ρ(kX) = k × ρ(X), for k > 0; Ignores liquidity effects, which is a known criticism.
Translation invariance: ρ(X + c) = ρ(X) − c, where c is a sure gain (cash added); If written with losses, a sure loss of c increases ρ by c.
Expected shortfall: ES = average of losses that are greater than or equal to VaR at the same confidence level; Coherent. ES is always at least as large as VaR at the same confidence level.
Coherent measure: Coherent = monotonicity + subadditivity + positive homogeneity + translation invariance; All four must hold.

How to solve Coherent Risk Measures and Subadditivity questions

Use this method for conceptual and numerical questions on coherence and subadditivity.

1Identify what the question asks: name a property, test a property, or classify a measure as coherent or not.
2Write the four properties in your own words, using ρ for risk and X, Y for portfolios.
3If numbers are given, compute the risk measure for X, for Y and for the combined portfolio X + Y separately.
4Compare ρ(X + Y) with ρ(X) + ρ(Y). If the combined value is larger, subadditivity fails.
5For homogeneity, scale the position by k and check whether risk scales by exactly k. For translation invariance, add cash and check the risk falls by exactly that cash.
6Classify the measure: VaR fails subadditivity only; ES is coherent; standard deviation fails monotonicity.
7Pick the answer that matches, watching for options that mix up the property names.

Quickest way: Property-matching shortcut

When to use it: For definition-style questions or when a numerical VaR comparison is given.

Match keywords: 'merging' or 'diversification' means subadditivity; 'double the position' means homogeneity; 'add cash' means translation invariance; 'worse payoff in every state' means monotonicity.
For VaR, remember it fails only subadditivity, and only for non-elliptical distributions such as discrete or skewed losses.
For a numerical test, compute VaR of each position and of the combination, then compare combined VaR with the sum.
Remember ES is coherent and standard deviation is not.

Common mistakes in Coherent Risk Measures and Subadditivity

Saying VaR violates all four properties or is never coherent for any distribution.
Students memorise 'VaR is not coherent' without the detail.
Fix: VaR fails only subadditivity, and only in some cases. For normal (elliptical) distributions it is subadditive.
Confusing homogeneity with subadditivity.
Both involve sums and scaling.
Fix: Homogeneity scales one portfolio by k. Subadditivity combines two different portfolios.
Getting the sign wrong in translation invariance.
Mixing gains-based and loss-based notation.
Fix: Adding sure cash c reduces risk by exactly c. Think of it as capital that absorbs losses.
Calling standard deviation coherent.
It is subadditive and homogeneous, so it looks fine.
Fix: It fails monotonicity and translation invariance, because adding cash does not change it.
Assuming a higher combined VaR means an arithmetic error.
Diversification intuition says combining must lower risk.
Fix: With discrete payoffs the combined VaR can exceed the sum. This is the standard counterexample.
Thinking ES can be lower than VaR at the same confidence level.
Confusing ES with a body-of-distribution average.
Fix: ES averages losses at or beyond VaR, so it is at least VaR.

Worked examples

Example 1

Two independent bonds each have a 4% chance of default within a year, with a loss of ₹10,00,000 on default and zero otherwise. Find the 95% VaR of each bond alone and of the portfolio of both. Is VaR subadditive here?

Show the solution

Single bond: P(loss = 0) = 96%, which exceeds 95%. So the 95% VaR is 0.
Portfolio: P(no default) = 0.96 × 0.96 = 0.9216. P(exactly one default) = 2 × 0.04 × 0.96 = 0.0768. P(two defaults) = 0.04 × 0.04 = 0.0016.
Cumulative probability of loss ≤ 0 is 0.9216, which is below 95%. Cumulative probability of loss ≤ ₹10,00,000 is 0.9216 + 0.0768 = 0.9984, which is above 95%.
So the 95% VaR of the portfolio is ₹10,00,000.
Sum of individual VaRs = 0 + 0 = 0. Portfolio VaR of ₹10,00,000 is greater than 0.

Answer: Portfolio VaR ₹10,00,000 exceeds the sum of individual VaRs (₹0), so VaR is not subadditive in this case.

Example 2

A portfolio has a risk measure ρ = ₹8,00,000. The manager doubles every position and then adds ₹3,00,000 of sure cash. If the measure is coherent, what is the new risk?

Show the solution

Positive homogeneity: doubling all positions gives ρ = 2 × 8,00,000 = ₹16,00,000.
Translation invariance: adding sure cash of ₹3,00,000 reduces the measure by that amount.
New risk = 16,00,000 − 3,00,000 = ₹13,00,000.

Answer: ₹13,00,000

Exam tips

Expect at least one question asking which property VaR violates. The answer is subadditivity.
Read the wording: 'combining portfolios' signals subadditivity, while 'scaling' signals homogeneity.
For the discrete-bond VaR example, always build the cumulative distribution and compare it with the confidence level.
Remember ES is coherent but harder to estimate and to backtest than VaR.
Know that VaR is subadditive for normal distributions, so delta-normal VaR does not show the problem.

Practice questions from Measures of Financial Risk

Coherent Risk Measures and Subadditivity: frequently asked questions

What are the four properties of a coherent risk measure?

They are monotonicity, subadditivity, positive homogeneity and translation invariance. A measure must satisfy all four to be coherent.

Why is VaR not subadditive?

VaR only looks at one quantile and ignores losses beyond it. With discrete or skewed losses, combining two positions can push the tail loss over the confidence threshold, so combined VaR exceeds the sum.

Is expected shortfall coherent?

Yes. ES satisfies all four properties, including subadditivity. It also captures the size of losses beyond VaR.

Is VaR ever subadditive?

Yes. For elliptical distributions such as the normal, VaR is subadditive. The failure appears with non-elliptical distributions, such as discrete defaults or heavy-tailed losses.