Expected Shortfall Calculator
Calculate the average loss in the tail beyond value at risk, which is what VaR by construction never tells you.
Expected shortfall answers the question value at risk leaves open. Value at risk marks the point where the worst slice of outcomes begins; expected shortfall averages what happens inside that slice. It is also called conditional value at risk, or CVaR, and this expected shortfall calculator reports it beside the value at risk it is measured against, computed from the same inputs by the same arithmetic.
Formula: expected shortfall = mean of losses beyond the value at risk threshold
Your figures
Parametric works from a volatility you supply. Historical reads the tail out of a series of returns you supply.
The position or portfolio the loss is measured against. Every cash figure scales directly with this.
How much of the distribution sits above the threshold. 95 leaves the worst 5% in the tail. Between 50% and 99.99%.
Parametric method only. Always your own measured figure annualised, whatever frequency you pick below — a monthly volatility of 5.77% is entered as 20. The frequency sets the clock for the holding period, not for this field.
Parametric method only. Counted in the same periods as the frequency below, so 10 with 252 periods a year means ten trading days.
Parametric method only. The clock the holding period is counted on, and what the annualised volatility is scaled down to. The volatility itself is always annualised.
Only used when the frequency is set to Custom.
Results
Expected shortfall
US$2,598.77
average loss across the worst 5.00% of periods
- Shortfall as a share of the positionThe average loss in the tail, before it is applied to the position value.
- 2.5988%
- Value at risk at the same confidence2.0723% of the position. The point where the tail begins, rather than the average inside it.
- US$2,072.32
- Shortfall divided by value at riskNever below one, by construction. Close to one says the tail falls away quickly past the threshold; larger says it does not.
- 1.2540
- z-score at this confidenceHow many standard deviations into the left tail of a normal distribution the threshold sits.
- 1.6449
- Volatility over the holding periodThe quoted volatility multiplied by the square root of holding periods divided by periods in a year.
- 1.2599%
- Position value
- US$100,000.00
What this means
Value at risk marks where the worst 5.00% of the distribution begins, at 2.0723% of the position. Expected shortfall takes the mean of that slice instead of its edge: the normal density at the threshold, divided by the tail probability, scales the 1.2599% horizon volatility up to 2.5988%, or US$2,598.77 on US$100,000.00. The ratio between the two is fixed by the normal assumption rather than by anything in a record: at 95.00% confidence it is 1.2540, whatever volatility is entered.
- Expected shortfall is never smaller than value at risk. It is the mean of a set of losses none of which is smaller than the threshold, so the two can meet but cannot cross.
- The parametric estimate assumes returns are normally distributed and describes the whole shape with one number. Real return distributions have fatter tails than a normal does, so it understates the size of the worst outcomes — in the direction that matters.
- A portfolio is treated as a single position with a single volatility. Offsetting holdings are not netted and no correlation between them is modelled, so the figure describes the position value you entered rather than the parts inside it.
- Scaling a volatility across time uses the square-root-of-time rule, which holds when returns are independent from one period to the next. Real series trend and mean-revert, so a market that grinds one way understates the figure and one that whips back and forth overstates it.
Expected shortfall: US$2,598.77
The link reopens this calculator with the same numbers. Nothing you enter is sent to a server.
Continue with these figures
About the Expected Shortfall calculation
The distinction matters because two positions can share a threshold and diverge completely beyond it. If both lose more than 3% on one period in twenty, but one then loses 3.5% and the other loses 25%, value at risk records them as identical. Expected shortfall does not, because the depth of the tail is the entire thing it measures rather than an afterthought to where the tail starts.
It is never smaller than value at risk, and that is structural rather than empirical. It averages a set of losses none of which is smaller than the threshold, so the two can meet — where the tail holds a single observation sitting exactly on the line — but they cannot cross. The ratio between them is reported for that reason: it compresses how quickly the tail falls away into one number.
Expected Shortfall formula
parametric:
φ(z) = e^(−z² ÷ 2) ÷ √(2π)
ES% = σ over the horizon × φ(z) ÷ (1 − c)
historical:
threshold = the return at the (1 − c) percentile
ES% = −( mean of every return at or below the threshold )
both:
ES = position value × ES% ÷ 100
ratio = ES% ÷ VaR%- φ(z)
- The standard normal density at the threshold — the height of the bell curve where the tail starts, not the area beyond it.
- 1 − c
- The tail probability. Dividing by it renormalises that sliver of distribution back into a distribution of its own, so its mean can be taken.
- threshold
- The value at risk return at the same confidence, produced by calling the value at risk calculation itself so the two figures cannot drift apart.
- ratio
- Expected shortfall divided by value at risk. Never below one; close to one says the tail ends quickly past the threshold, larger says it does not.
The parametric form is the mean of a normal distribution's own left tail, and it has a closed form because the normal density integrates back to itself up to a sign. The height of the curve at the threshold, divided by the probability mass beyond it, is exactly the average distance into the tail. Nothing about your data enters that step — only the confidence level does.
That has a consequence worth stating plainly: under the parametric method the ratio between expected shortfall and value at risk depends on the confidence level and on nothing else. At 95% it is 2.0627 ÷ 1.6449 = 1.2540, whatever volatility is entered and whatever the position is worth. At 99% it is 2.6652 ÷ 2.3263 = 1.1457. The ratio tightens as confidence rises because the slice being averaged gets narrower, not because the position became safer.
The historical form assumes nothing and averages the losses that actually landed. The tail is defined by value rather than by count, so ties are all included — three identical worst days count three times, which is the honest reading of a record in which that loss happened three times. Its weakness is arithmetic rather than conceptual: at 95% confidence a record needs about a hundred observations before five of them fall in the tail at all.
That is why the number of returns averaged is reported and flagged whenever it falls below five. A mean taken over one or two observations is not an average in any useful sense; it is a single bad period carrying an enormous standard error, and it will move sharply the first time a worse one joins the record. Lengthening the series or lowering the confidence level are the only two things that put more observations behind the figure.
Expected Shortfall: a worked example
Six monthly returns — +2.4%, −1.8%, +3.1%, −4.2%, +1.6%, −0.9% — on a $100,000 position at 95% confidence, read by the historical method.
- Sorted, worst first−4.2, −1.8, −0.9, +1.6, +2.4, +3.1
- Position of the 5th percentileh = (6 − 1) × 0.05 = 0.25
- Return at that percentile−4.2 + 0.25 × (−1.8 − −4.2) = −3.6000%
- Value at risk3.6000% × $100,000 = $3,600.00
- Returns at or below −3.6000%one — the −4.2% month
- Expected shortfall−(−4.2000%) = 4.2000%, or $4,200.00
- Shortfall ÷ value at risk4.2000 ÷ 3.6000 = 1.1667
$4,200 against a value at risk of $3,600. But the average behind that $4,200 is taken over a single observation, because only one of the six months sits at or below the threshold — and the calculator flags exactly that. The figure is the honest answer to the question asked; it is also one bad month with a decimal point after it, and it will move sharply the first time a worse one is recorded.
How to use the Expected Shortfall calculator
- Choose the method. The inputs are identical to the value at risk calculator, which is deliberate — the two answer consecutive questions about the same tail and are meant to be read together.
- Enter the position value and the confidence level. The slice being averaged is everything past that level, so 95 averages the worst 5% of periods and 99 averages the worst 1%.
- For the parametric method, supply the volatility annualised — as on the value at risk calculator, the frequency below sets the clock for the holding period, not for the volatility — and count the holding period in those periods.
- For the historical method, enter your own periodic returns in percent, oldest first, then read the 'returns averaged in the tail' row before anything else on the page.
- Compare the shortfall against the value at risk shown beside it. The ratio is the part worth carrying away, because it says how much worse the tail is than its own threshold.
- If the tail holds fewer than five observations the calculator says so in a note. No tail risk calculator can average losses a record does not contain, so read the number as an illustration rather than an estimate when that appears.
Limitations of the Expected Shortfall calculator
Every calculation makes assumptions. These are the ones that matter here.
- The historical estimate averages whatever the tail happens to contain. Six returns at 95% confidence leave one observation in it, and calling a single number an average does not make it one.
- The parametric estimate inherits the normal assumption whole. Real tails are fatter than a normal's, so the closed form understates the average loss beyond the threshold just as it understates the threshold itself.
- The fixed ratio between shortfall and value at risk under the parametric method is a property of the normal curve rather than of your position. Reading it as a measurement of the tail's shape reverses the logic entirely.
- A portfolio is treated as a single position. No correlation between holdings is modelled, and correlations that look comfortable in ordinary conditions have a habit of rising towards one in the conditions this figure is asked about.
- Nothing here estimates how often the tail arrives. Expected shortfall describes the average size of the bad outcomes, not the chance that the process generating them changes character.
- Over a short record any tail statistic is dominated by luck. Whether the worst month in a two-year sample was −4% or −14% is largely an accident of which two years the record happens to cover.
Frequently asked questions about Expected Shortfall
Is expected shortfall the same thing as conditional value at risk?
Yes. Expected shortfall, conditional value at risk, CVaR and average value at risk all name the same quantity: the average loss beyond VaR, which is to say the mean loss given that the value at risk threshold was breached. The names come from different literatures rather than different formulas, so a CVaR calculator and an expected shortfall calculator are the same page under two labels, and the arithmetic here is what all four refer to.
Why is expected shortfall always larger than value at risk?
Because it is the mean of a set of losses none of which is smaller than the threshold. Every number being averaged is at least as large as value at risk, so their average must be too. The two are equal only in the degenerate case where the tail holds one observation sitting exactly on the threshold, which is why the ratio has a floor of one rather than a range around it.
Why does the ratio not change when I change the volatility?
Under the parametric method both figures are the same horizon volatility multiplied by a constant that depends only on the confidence level — z for value at risk, φ(z) ÷ (1 − c) for expected shortfall. The volatility cancels in the division, so the ratio is 1.2540 at 95% and 1.1457 at 99% regardless of what you enter. Under the historical method the ratio does move, because it then reflects the actual shape of your record.
Only one return fell in my tail. Is the answer wrong?
It is arithmetically right and statistically fragile, which is why it is flagged. At 95% confidence the tail holds roughly 5% of the record, so a six-return series leaves one observation and a thirty-return series leaves one or two. The figure describes those observations faithfully; it simply carries a standard error large enough that it should not be compared with another record's figure or tracked over time.
Which method suits a record like mine?
It depends mostly on length. Under about a hundred observations the historical tail is too thin to average meaningfully, and the parametric form at least produces a stable number from a stable volatility. With a long record the historical form captures skew and fat tails the normal assumption throws away. Running both and comparing them is more informative than either alone, since the gap measures how un-normal the record actually is.
Why does raising the confidence level make the ratio smaller?
Because the slice being averaged shrinks towards the threshold. At 95% the tail spans everything beyond 1.6449 standard deviations, and its mean sits well inside at 2.0627. At 99% the tail starts further out at 2.3263 and its mean is 2.6652 — a smaller proportional gap, because the normal density is falling steeply by then. Under a fatter-tailed distribution the ratio would shrink far more slowly, which is one way of seeing what the normal assumption costs.
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