Lesson 18 of 25 · 6 min

From a volatility figure to a one-day loss estimate

A percentile of an assumed distribution times a position gives a dollar figure, and the assumption is the weakest link in it.

Track 4 lesson 5 ran the conversion from a move to a number of standard deviations. This lesson runs it the other way: from an annualised volatility to a dollar loss over one session at a stated confidence. It is the building block every haircut is assembled from.

There are three steps and no subtlety in any of them. Convert the annual figure to a daily one by dividing by 15.8745. Multiply by the number of standard deviations corresponding to the confidence level wanted. Multiply by the position value.

one-day loss = position value × confidence multiple × (annual volatility ÷ 15.8745)

one-day loss: The loss that would be exceeded only with the stated probability, over a single session.
position value: Market value of the holding, in currency.
confidence multiple: Standard deviations corresponding to the confidence level: about 1.28 for 90 percent, 1.65 for 95 percent, 2.33 for 99 percent.
annual volatility: The annualised standard deviation of returns, as a percentage.
15.8745: The square root of 252 trading sessions, which converts an annual figure to a daily one.

Run it on an illustrative $742,000 position with 29.4 percent annualised volatility. The daily figure is 1.8520 percent. At 99 percent confidence the move is 2.33 × 1.8520 = 4.3152 percent and the loss is $32,018.94. At 95 percent it is 3.0558 percent and $22,674.35. At 90 percent, 2.3706 percent and $17,589.80.

What that figure is, precisely, is a quantile of an assumed distribution — and Track 4 lessons 3 and 6 established that the assumption fails in exactly this region. Suppose the worst one percent of sessions on this name actually average 6.4 percent rather than the 4.3152 percent the normal assumption produces. The loss is then $47,488, which is 1.483 times the modelled figure. The arithmetic is exact; the input distribution is the part that is wrong.

This is the number a haircut is made of, one session at a time. A discount covering an exit of several sessions is this calculation with its horizon stretched, and lesson 19 does the stretching.

Key terms

Confidence multiple: The number of standard deviations corresponding to a stated confidence level. Raising the level widens the loss estimate without changing the volatility input.
One-day loss estimate: A loss that would be exceeded only with a stated probability over a single session. A statement about an assumed distribution, not about history.
Tail understatement: The amount by which a normal-based loss estimate falls short of what the worst sessions actually deliver. Here, a factor of about 1.5.

Worked example

One position, three confidence levels, and the fat-tail correction

An illustrative $742,000 position with 29.4 percent annualised volatility, so a daily volatility of 1.8520 percent.

90 percent confidence: 2.3706%, or $17,589.80
95 percent confidence: 3.0558%, or $22,674.35
99 percent confidence: 4.3152%, or $32,018.94
If the worst 1% of sessions average 6.4% instead: $47,488.00
Understatement of the modelled figure: 1.483 times

Moving from 90 to 99 percent confidence nearly doubled the figure. Replacing the distribution with something more honest added half again on top.

Check yourself

A $1,315,000 holding has 41.8 percent annualised volatility. What is the one-day loss at 99 percent confidence, and what exactly does that number claim?

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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.