Risk / tail measures

Expected shortfall

VaR tells you where the tail starts; expected shortfall tells you how deep it is. Basel moved to a 97.5% ES because it is sub-additive — two books cannot have less risk together than apart — and because it responds to the shape of the tail rather than a single quantile.

Horizon vol1.134%ES multiplier2.338RegimeModerate

ES at 97.5%

$265,082

2.65% of exposure over 1 day

VaR at 99%

$263,783

2.64% of exposure

ES ÷ VaR

1.005

0.5% above the 99% VaR

Tail shape

Normal

thin tails

Tail measures side by side

Same book, two measures, two distributions — the differences are the point.

$265,082

Tail measures side by side
Measure	Amount	Share of exposure
Volatility over the horizon	1.134%	σ√(h/252)
VaR at 95%	$186,509	1.87%
VaR at 99%	$263,783	2.64%
ES at 97.5%, normal	$265,082	2.65%
ES at 97.5%, selected	$265,082	2.65%
ES at 99%, selected	$302,207	3.02%
Quantile at the ES confidence	1.960	standard deviations
ES ÷ VaR 99%	1.005	the tail multiplier

Expected shortfall is the average loss in the tail beyond the quantile, so it is always larger than the VaR at the same confidence. It is also sub-additive, which is the property that made Basel replace VaR with ES.

ES and VaR against confidence

ES diverges from VaR as the confidence rises — the tail is where the two measures disagree.

Normal against fat tails

The same volatility, two distributions. The t case costs more capital for the same headline risk.

Canonical equations

The exact relationships implemented by this engine.

μ + σ·φ(z_α)/(1 − α) · z at the confidence, φ the standard normal density.

μ + σ√((ν−2)/ν)·(f_ν(t_α)/(1−α))·(ν + t_α²)/(ν−1) · Scaled to unit variance so σ means the same thing.

σ_h = σ√(h/252) · Square root of time, zero drift assumption.

ES₉₇.₅ vs VaR₉₉ · FRTB replaced the quantile measure with the tail average.

Show model notation and computationOpen the symbolic form, the variable definitions, and the step-by-step derivation when you need the detail.

A clear path from inputs to results.

Step-by-step methodWorked exampleAssumptions explained

The formula

\mathrm{ES}_\alpha = \mu + \sigma\frac{\varphi(z_\alpha)}{1-\alpha}

Reference: Basel Committee, Minimum capital requirements for market risk (d457, 2019), §3 for the 97.5% ES; McNeil, Frey & Embrechts, Quantitative Risk Management, ch. 2 for the closed forms.

Worked example

$10m book at 18% volatility, one-day horizon

Inputs

$10,000,000 of exposure.
18% annual volatility, zero drift.
97.5% confidence, one trading day, normal distribution.

Calculation path

1
Daily volatility is 18% / √252 = 1.134%.
2
The 97.5% quantile is 1.960 standard deviations, where the normal density is 0.0584.
3
The ES multiplier is 0.0584 / 0.025 = 2.338, so ES is $10m × 2.338 × 1.134% = $265,082.
4
The 99% VaR is 2.326 × 1.134% × $10m = $263,783.
5
ES is only 0.5% above the VaR on a normal distribution, because the tail is thin.
6
Switching to a Student-t with five degrees of freedom raises ES to $309,304 — 17% more for the same volatility.

Result

ES $265,082 · VaR $263,783 · ratio 1.005 · 17% more on fat tails.

Assumptions & limitations

Parametric ES assumes a known, stable distribution and constant volatility over the horizon.

Volatility scales with the square root of time, with zero drift over short horizons.

The Student-t is scaled to unit variance so the volatility input means the same thing in both cases.

No skew, no jumps and no regime change are modelled.

Basel pairs ES with a non-modellable-risk add-on and a stressed calibration, neither of which is applied here.

Frequently asked questions

Why did Basel replace VaR with expected shortfall?

Two reasons. VaR is not sub-additive, so splitting a book can appear to reduce risk; and VaR says nothing about how bad the losses beyond the quantile are. ES fixes both, at the cost of needing a full distribution rather than a single quantile.

How sensitive is ES to the degrees of freedom?

Very. Moving from a normal to a t with five degrees of freedom adds about 17% at this volatility; three degrees of freedom adds far more. That is why tail calibration, not the confidence level, is where the modelling risk sits.

Does ES add across desks?

Not linearly, because diversification still applies. But it is sub-additive, which means the ES of a combined book is never greater than the sum of the parts — the property VaR lacks and the reason it was replaced.

Why 97.5% and not 99%?

A 97.5% ES is roughly equivalent to a 99% VaR for a normal distribution, so the calibration was chosen to keep capital broadly comparable while changing the measure. It also reduces the sample size needed to estimate the tail.

Can ES be estimated from historical data?

Yes — average the losses beyond the quantile. The estimate is noisy because it uses only the tail observations, so a 250-day window gives you roughly six data points at 97.5%, which is not enough for a stable number.