Lesson 7 of 25 · 7 min

Why a sample maximum keeps growing with the sample

Ten years of data implies a worst day of ten percent and a century implies twenty-two, from one identical process.

The largest decline in a record feels like a property of the market. It is mostly a property of how long the record is, and the rate at which it grows is computable from the tail exponent.

The level expected to be exceeded once in n days is the level where n multiplied by its probability equals 1. Under the power law from the previous lesson, with 312 of 25,200 days worse than 3 percent, the probability of a 3 percent day is 0.012381 and the once-in-n level is 3 × (n × 0.012381)^(1 ÷ 2.85).

Run it twice. Over ten years, 2,520 days, that gives 3 × 31.2^0.3509 = 10.03 percent. Over a century, 25,200 days, it gives 3 × 312^0.3509 = 22.50 percent. Ten times the data implies a worst day 2.243 times larger, because 10^(1 ÷ 2.85) = 2.243. Nothing about the market changed between the two calculations. Only the length of the window did.

Compare the same exercise under a normal distribution with a 1.0 percent daily standard deviation. The expected maximum over 2,520 days is 3.378 percent and over 25,200 days it is 3.964 percent, a factor of 1.173. Under a bell curve the record converges on a wall: ten times the observations buy you seventeen percent more extreme an extreme. Under a power law there is no wall, and the worst observation simply keeps travelling as you watch longer.

Calibrating a buffer to the worst thing in a ten-year window is calibrating to 10 percent when the same process over a century implies 22.5. The gap is not a forecasting failure. It is the window length, showing up as a number.

Key terms

Sample maximum: The largest observation in a record. A statistic of the sample rather than a bound on the process, and it grows predictably with sample size.
Return level: The size of move expected to be exceeded once in a stated number of observations. The natural way to state a tail without quoting a probability.
Window dependence: The fact that a risk figure derived from the worst observation changes with the length of the observation period, even when nothing about the underlying behaviour has changed.

Worked example

The worst day, as a function of how long you looked

Power law with exponent 2.85 anchored at a 3 percent daily decline occurring 312 times in 25,200 days. Normal comparison at 1.0 percent daily.

Power law, 2,520 days: 10.03%
Power law, 25,200 days: 22.50%
Growth from ten times the data: 2.243×
Normal, 2,520 days: 3.378%
Normal, 25,200 days: 3.964%
What changed in either case: Only the window

Under one shape the worst day more than doubles when you look ten times as long. Under the other it moves by a sixth.

Check yourself

Two analysts look at the same market. One uses ten years and concludes the worst plausible day is about 10 percent; the other uses a century and says 22.5 percent. Who is wrong?

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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.