Lesson 8 of 25 · 7 min

Why a standard deviation understates a fat-tailed risk

The estimate is lowest exactly when the window has been calm, which is exactly before the day that needs it.

Track 4 lesson 7 separated volatility from risk as concepts. The quantitative complaint is narrower and sharper: a standard deviation is an estimate, and under a fat tail it is a badly behaved one.

Return to the twenty days from lesson 4. With the −7.60 percent day included, the standard deviation is 1.9141 percent. Without it, across the other nineteen days, it is 0.9301 percent. A single observation in twenty more than doubled the estimate, raising it by 105.8 percent, because that one day supplies 73.7 percent of the sum of squared deviations on its own.

Which means the same day gets two different descriptions. Measured against the standard deviation of the full sample it is a 3.84 sigma day. Measured against the standard deviation of the nineteen days that preceded it, it is an 8.32 sigma day. Both are arithmetic; they differ only in whether the event is allowed to set the scale it is being judged against.

There is a formal version of the problem. Under a normal distribution the standard error of a volatility estimate is the volatility divided by √(2n), so 23 percent from three years of daily data is pinned to within 0.59 points. That formula depends on the fourth moment being finite. When the tail exponent falls below 4 the fourth moment does not converge, and the estimator has no reliable error bar at all — the uncertainty on the volatility estimate is itself unestimable.

A haircut computed from a trailing window falls steadily while nothing happens and jumps after the event that made it necessary. The buffer is at its smallest on the day before it is needed, and that is a property of the estimator rather than bad luck.

Key terms

Estimator instability: The tendency of a statistic to swing on the inclusion of one observation. Standard deviation has it under fat tails; it is worse for kurtosis.
Standard error of volatility: How precisely a volatility figure is known. Equal to the volatility divided by the square root of twice the sample size, and only when the fourth moment is finite.
Trailing window: A fixed recent period used to compute a risk parameter. It makes the parameter a function of what happened to fall inside it.

Worked example

The same day, two scales

The twenty invented daily returns from lesson 4, with and without the −7.60 percent day.

Standard deviation, all twenty days: 1.9141%
Standard deviation, the other nineteen: 0.9301%
Share of squared deviations from one day: 73.7%
The bad day, scaled by the full sample: 3.84 sigma
The bad day, scaled by the prior nineteen: 8.32 sigma
Three standard deviations, prior nineteen: 2.79%

A buffer of three standard deviations built from the calm window would have covered 2.79 percent against a day that moved 7.60.

Check yourself

A trailing window gives a standard deviation of 0.9301 percent, and a buffer is set at three of them, 2.79 percent. The next day moves 7.60 percent. How many standard deviations was that, and which figure should the buffer have used?

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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.