Expected shortfall averages the tail that value at risk only locates. On a normal model the switch changes almost nothing; on a real history it changes the answer.
Expected shortfall is the average loss on the days worse than the value at risk. On a normal distribution it has a closed form, net assets × daily volatility × φ(z) / (1 − confidence): on an 840M credit fund that gives a one-day 97.5 per cent expected shortfall of 7,278,117, only 1.005 times the 99 per cent VaR of 7,242,456. On a fat-tailed 500-day history the same measure is the average of the worst 12.5 days, 10.13 million, and 1.113 times the historical VaR.
Worked in full in Financial Risk Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The fund is Halverton, the open-ended credit fund that runs through Financial Risk Management: 840,000,000 of net assets in four credit sleeves, with an annual portfolio volatility of 5.883 per cent from the reported correlation matrix and 252 trading days a year. The 500-day loss history used in the second half is added for this article and is illustrative.
| Input | Value |
|---|---|
| Net assets | 840,000,000 |
| Annual volatility | 5.883% |
| Daily volatility (annual / √252) | 0.3706% |
| One daily standard deviation, in money | 3,113,230 |
| VaR confidence | 99% |
| Expected shortfall confidence | 97.5% |
| Observations in the history | 500 |
Value at risk answers where the tail starts. Expected shortfall answers how bad it is once you are in it: the mean of the loss distribution beyond the quantile. For a normal distribution with zero mean, the mean beyond the quantile z is the density at z divided by the tail probability. At 97.5 per cent, z is 1.9600 and the density there is 0.0584, so the multiplier on one standard deviation is 0.0584 / 0.025 = 2.3378.
ESα = net assets × σdaily × φ(zα) / (1 − α)
= 3,113,230 × 2.3378 = 7,278,117, or 0.866 per cent of net assets
VaR99% = 3,113,230 × 2.3263 = 7,242,456
Excel: =NAV*SD*NORM.S.DIST(NORM.S.INV(0.975),FALSE)/(1-0.975)
The two figures differ by 35,662, half of one per cent. That is not a coincidence. When the Basel Committee moved the internal models approach for market risk from 99 per cent VaR to 97.5 per cent expected shortfall in its Fundamental Review of the Trading Book, the confidence level was chosen so that the two measures broadly agree on a normal distribution. The change was meant to capture the shape of the tail, not to raise the number for a well-behaved book. For comparison, expected shortfall at the same 99 per cent level would be 8,297,424, 1.146 times the VaR.
The parametric figure inherits the normal tail, so it cannot show what expected shortfall is for. The historical version can. Take 500 daily profit and loss observations, sort the losses, and average the worst 2.5 per cent of them. 2.5 per cent of 500 is 12.5 days, so the worst twelve count in full and the thirteenth counts at half weight.
| Rank | Loss | Weight in ES | Rank | Loss | Weight in ES |
|---|---|---|---|---|---|
| 1 | 21.4 | 1 | 8 | 7.5 | 1 |
| 2 | 16.8 | 1 | 9 | 7.2 | 1 |
| 3 | 13.5 | 1 | 10 | 7.0 | 1 |
| 4 | 11.2 | 1 | 11 | 6.8 | 1 |
| 5 | 9.1 | 1 | 12 | 6.6 | 1 |
| 6 | 8.4 | 1 | 13 | 6.4 | 0.5 |
| 7 | 7.9 | 1 |
Historical ES97.5% = (sum of the worst 12 + 0.5 × the 13th) / 12.5
= (123.4 + 3.2) / 12.5 = 126.6 / 12.5 = 10.13 million, or 1.206 per cent of net assets
Historical VaR99%, read as the fifth-worst loss = 9.1 million
Excel, losses as positive numbers in a range L: =(SUMPRODUCT(LARGE(L,ROW(1:12)))+0.5*LARGE(L,13))/12.5
On this history the expected shortfall is 1.113 times the VaR, against 1.005 on the normal. The difference comes from the first three rows: 21.4, 16.8 and 13.5 million, losses a normal distribution with this volatility would almost never produce. VaR does not see them at all, because the fifth-worst day is the same whether the worst day was 10 million or 40. Expected shortfall averages them in. Taking the fifth-worst loss as the 99 per cent VaR is one convention among several (some use the sixth, some interpolate); state the one you use.
The ratio of 97.5 per cent expected shortfall to 99 per cent VaR is a compact measure of tail shape. Holding the daily volatility at 0.3706 per cent and replacing the normal with a Student t scaled to the same variance:
| Distribution | VaR 99%, in sd | ES 97.5%, in sd | ES / VaR | ES 97.5%, Halverton |
|---|---|---|---|---|
| Normal | 2.326 | 2.338 | 1.005 | 7,278,117 |
| Student t, 8 degrees of freedom | 2.508 | 2.572 | 1.025 | 8,007,273 |
| Student t, 5 | 2.606 | 2.728 | 1.047 | 8,492,275 |
| Student t, 4 | 2.649 | 2.824 | 1.066 | 8,791,360 |
| Student t, 3 | 2.622 | 2.910 | 1.110 | 9,058,268 |
The illustrative history, at 1.113, behaves like a t with three degrees of freedom. A fund whose expected shortfall comes out within a per cent of its VaR is either genuinely normal or being measured with a normal model; the ratio tells you which question to ask.
Three errors recur. The first is comparing measures at the same confidence: 99 per cent expected shortfall against 99 per cent VaR shows 1.146 even on a normal distribution, which reads as fat tails that are not there. The second is miscounting the tail. Averaging the worst twelve days gives 10.28 million and the worst thirteen 9.98; neither is 12.5 days, and on a short history the half observation matters. The third is believing the switch itself adds prudence. With a parametric normal engine, moving from VaR to expected shortfall changes the number by half a per cent. The improvement only arrives when the model has a tail worth averaging: a history, a fat-tailed distribution, or a stressed calibration window.
Expected shortfall is still a one-day, hold-everything number. It says nothing about what selling costs or how long the book takes to exit. Scale it with the same care as VaR, see how to scale a one-day VaR to ten days, and keep a stress loss beside it.
The parametric construction, with expected shortfall beside the VaR, is live in the free workbook for this case; the site's VaR and expected shortfall model runs both on any portfolio you type in.
Because on a normal distribution the two are almost equal, so the switch keeps the level while making the measure sensitive to the shape of the tail. The 97.5 per cent ES multiplier is 2.3378 standard deviations against 2.3263 for 99 per cent VaR, a ratio of 1.005. On a fat-tailed history the ratio rises, here to 1.113, and that is the information VaR discards.
In practice, yes. Conditional value at risk, expected tail loss and expected shortfall all name the average loss beyond the VaR quantile, and for continuous distributions they coincide. On a discrete history the definitions differ only in how the boundary observation is weighted: with 500 days at 97.5 per cent, the twelve worst count in full and the thirteenth at half weight, giving 12.5 days.
More than for VaR, because the estimate rests on very few days. With 500 observations, 97.5 per cent expected shortfall averages only 12.5 losses, and in the illustrative history here the worst single day, 21.4 million, contributes 1.71 million of the 10.13 million result. Drop or add one extreme day and the figure moves materially, which is why a stressed window is often used.
Expected shortfall is set beside the value at risk in the workbook for chapters 4, 6 and 7 of Financial Risk Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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