Expected shortfall vs Value at Risk: what the tail average adds for FRM candidates
Value at Risk (VaR) names a loss threshold. Expected shortfall (ES), also called conditional VaR or expected tail loss, asks the next question: given that we are past the threshold, how bad is the average loss? Both appear in FRM® Part I (Valuation and Risk Models) and Part II (Market Risk). If you are new to VaR itself, start with Value at Risk explained. The full list of guides is on the FRM page.
What each measure says
At confidence level c (say 99%):
- VaR is the loss that is exceeded with probability 1 - c. It is a quantile.
- ES is the expected loss, conditional on the loss being at least the VaR. It is the average over the whole tail.
Because it averages values that are all at or beyond the VaR, ES at a given level is never smaller than VaR at that level. For a continuous loss distribution, ES(c) is the average of VaR(u) for u between c and 1, which is why it responds to how fat the tail is, while VaR ignores everything past the cut-off. Two portfolios can have identical 99% VaR and very different ES.
For normally distributed losses with mean zero and standard deviation sigma:
VaR(c) = z x sigma, and ES(c) = sigma x pdf(z) / (1 - c)
where z is the standard normal quantile (2.326 at 99%) and pdf is the standard normal density.
Why ES is called coherent
A risk measure is called coherent (Artzner and co-authors) if it satisfies four properties: monotonicity (worse outcomes mean at least as much risk), translation invariance (adding cash reduces risk by that amount), positive homogeneity (doubling the position doubles the risk) and subadditivity: the risk of a combined portfolio is at most the sum of the separate risks.
Subadditivity is the diversification property. VaR can break it, so merging two books can show a higher VaR than the two apart. ES satisfies all four. Practitioners and regulators differ on how far this matters in practice, but it is a standard theoretical argument for ES, and the Basel Committee's market risk framework moved towards ES in its trading book review.
Worked example 1: a normal portfolio
An invented desk, Cormorant Rates Desk, has daily profit and loss that is normal with mean zero and standard deviation 1.5 million.
- 99% one-day VaR = 2.326 x 1.5 = 3.49 million.
- 99% one-day ES = 1.5 x 0.0267 / 0.01 = 4.00 million.
- At 95%, VaR = 1.645 x 1.5 = 2.47 million and ES = 3.09 million.
Notice that 97.5% ES (not shown) is close to 99% VaR in the normal case; under fat tails the two diverge, so the comparison is not a general rule.
Worked example 2: a discrete case where VaR fails subadditivity
Two independent bonds in an invented fund each lose 100 with probability 4% and nothing otherwise. Use 95% confidence.
- Each bond alone: the loss is 0 with probability 96%, so 95% VaR = 0. The sum of the two VaRs is 0.
- Together: the loss is 0 with probability 0.9216, 100 with 0.0768 and 200 with 0.0016. Since 0.9216 is below 0.95, the 95% VaR of the combined fund is 100.
So the combined VaR (100) exceeds the sum of the parts (0). VaR has penalised diversification.
Now ES at 95%, averaging over the worst 5% of probability:
- Each bond: 4% at 100 and 1% at 0, so ES = (0.04 x 100) / 0.05 = 80. The sum is 160.
- Combined: 0.16% at 200, then 4.84% at 100, so ES = (0.0016 x 200 + 0.0484 x 100) / 0.05 = 103.2.
103.2 is below 160, as subadditivity requires.
Usual mistakes
- Treating ES as the worst case. It is an average, still conditional on a confidence level.
- Comparing VaR and ES at the same c and calling it like for like. Many risk managers compare 99% VaR with 97.5% ES, since they coincide under normality.
- Averaging only the observations beyond VaR in a small historical sample without noting how noisy that is. ES needs more tail data than VaR.
- Saying VaR is never subadditive. It is subadditive for normal (elliptical) distributions; the failures come with skewed or discrete losses.
- Forgetting that ES is harder to backtest directly than VaR. See backtesting VaR and the two errors.
How the exam typically tests it
Typically you compute VaR and ES for a normal or a small discrete distribution, state which coherence property VaR violates, or explain why ES responds to tail shape. Thick tails are covered in fat tails, skewness and kurtosis. Charta's practice questions for this topic let you rehearse the arithmetic.
Common questions
Is expected shortfall always larger than VaR?
At the same confidence level, yes: it averages losses that are all at least the VaR. It can be equal only in degenerate cases.
Why is VaR not coherent?
It can fail subadditivity, as the two-bond example shows, so a combined portfolio can report more risk than its parts. ES does not fail in this way.
Is ES the same as conditional VaR?
Under the usual definitions for continuous distributions, yes. Names also include tail VaR and expected tail loss; check the definition when the distribution is discrete.
Does ES replace VaR?
Not entirely. VaR remains widely used and is easier to backtest, while some regulatory frameworks have moved to ES for capital.
Sources
- GARP, the FRM programme: what the exam is and its two parts.
- Basel Committee on Banking Supervision, the committee's website: its market risk standards discuss VaR and expected shortfall.
The exam and the syllabus can change. Check GARP's current details before you book.