Lesson 11 of 25 · 7 min

Why a 99th percentile is not a worst case

A level crossed two and a half times a year on average is a design specification, not a limit.

A 99 percent level sounds like an outer bound and is nothing of the sort. It is a level chosen to be crossed one day in a hundred, and a market year contains about 252 of those days.

So the expected number of crossings in a year is 252 × 0.01 = 2.52. The chance of at least one is 1 − 0.99^252 = 92.06 percent. The chance of a whole year passing without one is 7.94 percent. The chance of five or more in a single year is 11.05 percent. Over three years the expectation is 7.56 crossings and the chance of at least one is 99.95 percent.

Read those figures the right way round. A crossing is not evidence that the model failed; it is the model working to specification. A long stretch with no crossing at all is the more informative observation, and it is the one nobody investigates. Lesson 24 shows how little either observation can establish on a sample of realistic length.

Each crossing also carries a size, and that is where the previous lesson returns. Under a normal tail the average crossing overshoots the level by 14.57 percent. Under an exponent of 2.85 it overshoots by 54.05 percent. Two and a half crossings a year with a 15 percent overshoot and two and a half with a 54 percent overshoot are very different years, and the percentile figure is identical in both.

A buffer built from a 99 percent daily move is not a promise that the buffer holds. It is an arrangement under which the buffer is expected to be tested about two and a half times a year, and the open question is only how far past it each test goes.

Key terms

Exceedance: An occasion on which the stated level is passed. Under a correctly specified model these arrive on schedule rather than signalling failure.
Overshoot: How far beyond the level an exceedance travels. Fifteen percent on average under a normal tail, fifty-four under an exponent of 2.85.
Design specification: The frequency a threshold was built to permit. Confusing it with a bound is the most common misreading of a percentile.

Worked example

What a one-in-a-hundred level does over a year and over three

252 trading days a year, a level with a 1 percent daily exceedance probability.

Expected exceedances in a year: 2.52
Chance of at least one: 92.06%
Chance of none at all: 7.94%
Chance of five or more: 11.05%
Expected exceedances over three years: 7.56
Average overshoot beyond the level: 14.57% or 54.05%

The level is crossed about two and a half times a year by design. The only genuinely open question is the size of each crossing.

Check yourself

A buffer is set at the 99th percentile of daily moves. Over the next year it is breached three times. Is that evidence the buffer was set wrongly?

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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.