Lesson 9 of 25 · 7 min

Value at risk: what it states, and what it is silent about

One number, one horizon, one probability — and nothing at all about how bad the bad days actually get.

Value at risk is a threshold, not a forecast and not a worst case. It states the loss level that a stated fraction of outcomes fall short of, over a stated horizon. Three pieces of information are required before the number means anything: the horizon, the confidence level, and the distribution used to produce it.

Compute one. A position worth $1,840,000 with a daily standard deviation of 1.9 percent, under a normal assumption, has a one-day 99 percent value at risk of 2.3263 × 1.9 percent = 4.4201 percent, which is $81,329. The statement that number makes is precise and narrow: on 99 of every 100 days the loss is smaller than $81,329.

What the figure does not contain, none of which is a defect so much as a boundary:

How large the loss is on the hundredth day. The threshold says nothing about the region beyond it.
Whether two such figures can be combined, which lesson 12 shows they cannot.
Which distribution produced it. Two very different tails can share an identical 99 percent level to the dollar.
Anything about a horizon other than the one stated. A one-day figure is not a ten-day figure divided by ten.

The last point is worth the arithmetic. Under a normal assumption the ten-day figure is the one-day figure multiplied by √10 = 3.1623, so $81,329 becomes $257,185. Under a fat-tailed process that scaling is wrong in a known direction, because the aggregation of fat-tailed draws does not converge as quickly as the square-root rule assumes.

A haircut is a value-at-risk statement about a liquidation window: enough withheld to survive the move the lender expects to be exceeded only rarely. The clause permitting the haircut to be revised is the lender's acknowledgement that the number is silent about the rest.

Key terms

Value at risk: The loss level that a stated proportion of outcomes over a stated horizon fall short of. A threshold on a distribution, quoted in currency.
Confidence level: The proportion of outcomes below the threshold. Ninety-nine percent means one day in a hundred goes past it, by an unstated amount.
Horizon: The period over which the loss is measured. A figure is meaningless without it, and converting between horizons requires an assumption about how risk accumulates.

Worked example

A 99 percent one-day figure, and its boundaries

Position $1,840,000, daily standard deviation 1.9 percent, normal assumption. Illustrative.

Sigma multiple for 99 percent: 2.3263
Move at that level: 4.4201%
Value at risk, one day: $81,329
Days per year past that level: 2.52
Ten-day figure under the square-root rule: $257,185
Average loss on the days past it: Not stated

The figure is exact about where the line sits and says nothing about what happens on the far side of it. The next lesson asks that question.

Check yourself

Two positions are each reported with a one-day 99 percent value at risk of $81,329. One holds a broad fund, the other a single volatile name. What have you learned about the difference between them?

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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.