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Subadditivity means a risk measure must not assign more risk to a combined position than the sum it assigns to its parts. Value at risk can fail this test because a loss that is too rare to appear in either stand-alone quantile may become frequent enough to appear in the combined quantile. Expected shortfall averages losses in the tail and satisfies subadditivity under its standard definition.
What coherence requires
A coherent measure satisfies four checks. Monotonicity means a position with no worse payoff in any state cannot be judged riskier. Positive homogeneity means scaling a position scales its measured risk. Translation invariance describes how a certain cash amount changes measured risk. Subadditivity constrains the risk of combining positions. For losses, the last property can be written as follows.
GARP’s FRM study materials place risk measurement in the exam programme. A paper hosted by GARP notes both that value at risk may fail subadditivity and that expected shortfall is coherent. The distinction matters when interpreting an aggregated risk number: a larger combined value at risk does not, by itself, prove that combining exposures created an economic loss.
An illustrative quantile counterexample
Take the invented joint loss outcomes below, in £ millions. Use a 95% confidence level. Define value at risk as the smallest loss whose cumulative probability reaches at least 95%. For this discrete example, define expected shortfall as the average loss in the worst 5% of probability, taking a fraction of an outcome if the boundary cuts through it. The numbers are illustrative, not market observations.
| Neither loss | 92% | 0 | 0 | 0 |
| A only | 4% | 100 | 0 | 100 |
| B only | 4% | 0 | 100 | 100 |
For A, the probability of zero loss is 96%, so its 95% value at risk is zero. B is identical. For the combined position, zero loss has only 92% probability; the 95% quantile is therefore £100 million. Combined value at risk is £100 million against a sum of zero for the two positions, violating subadditivity under the stated convention.
Now average the worst 5% of each distribution. For A, that tail contains 4% probability at £100 million and 1% at zero, giving expected shortfall of £80 million. B also has £80 million. The combined position loses £100 million throughout its worst 5%, so its expected shortfall is £100 million, below the £160 million sum of the separate measures.
The exam mistake to avoid
Do not infer that diversification has failed merely because a combined value-at-risk number exceeds the sum of the stand-alone numbers. The quantile can jump when probabilities move across its cutoff. State the confidence level and quantile convention before reading a discrete table; then calculate the joint loss in each outcome, rather than adding separate quantiles outcome by outcome. Expected shortfall needs the full specified tail, including any fractional boundary outcome.
The FRM programme page places this topic in the wider course, while the methodology page explains the study approach.
Sources
- FRM® Study Material, Guide, Books, Practice Exams | GARP · garp.org · Retrieved
- Estimation of Tail Risk Measures for Heteroskedastic Financial Time Series: A Extreme Value Approach With Covariates · garp.org · Retrieved
