Lesson 10 of 25 · 7 min

Expected shortfall, and the question it answers instead

Averaging the losses past the line separates two distributions that share an identical threshold to the dollar.

Expected shortfall asks the question the previous lesson could not: given that the threshold has been passed, how large is the loss on average? It is the mean of everything beyond the value-at-risk level rather than the level itself.

Under a normal distribution the arithmetic is a ratio of the density to the tail probability. At the 99 percent level that is 0.0266521 ÷ 0.01 = 2.6652 standard deviations, against 2.3263 for the threshold itself. On the same $1,840,000 position at 1.9 percent daily, the expected shortfall is 2.6652 × 1.9 percent = 5.0639 percent, which is $93,176 against a value at risk of $81,329 — a ratio of 1.1457.

Now calibrate a power law with exponent 2.85 so that its 99 percent threshold is identical, $81,329 to the dollar. Its expected shortfall is the threshold multiplied by 2.85 ÷ 1.85 = 1.5405, which is $125,291. Same value at risk, expected shortfall 34.5 percent higher. The two distributions are indistinguishable on the first statistic and 34.5 percent apart on the second, which is the entire reason the second exists.

The 1.5405 multiple is not new. It is the mean excess from lesson 5, appearing again because expected shortfall is exactly a conditional mean. Under a normal tail that multiple converges toward 1 as the confidence level rises; under a power law it is fixed by the exponent and never moves.

Whether a threshold is breached determines that a demand arrives. How far past it the collateral went determines the size of the shortfall and therefore the size of the forced sale. The second question is the one expected shortfall answers.

Key terms

Expected shortfall: The average loss on the occasions when the value-at-risk threshold is exceeded. A statement about the far side of the line rather than the line.
Tail ratio: Expected shortfall divided by value at risk. About 1.15 under a normal tail at 99 percent and 1.54 under an exponent of 2.85.
Conditional mean: An average taken over a restricted set of outcomes. Here, over the worst one percent, which is why it is sensitive to a shape that value at risk ignores.

Worked example

Identical threshold, different average beyond it

Position $1,840,000, daily standard deviation 1.9 percent. Both distributions calibrated to the same 99 percent threshold.

Value at risk, both cases: $81,329
Normal shortfall multiple: 2.6652 sigma
Expected shortfall, normal: $93,176
Power-law shortfall multiple: 1.5405×
Expected shortfall, power law: $125,291
Difference: $32,115

Value at risk could not tell the two apart. Expected shortfall put $32,115 between them, and that gap is what a forced sale would be sized against.

Check yourself

Two positions report the same $81,329 value at risk. The first has an expected shortfall of $93,176 and the second $125,291. What does the second number tell you that the first did not?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.