Lesson 4 of 25 · 7 min

Kurtosis, computed rather than described

One day out of twenty supplies ninety-eight percent of the fourth moment, which is what the statistic is actually measuring.

Track 4 lesson 3 named excess kurtosis as the measure of fat tails and left it there. Computing one changes what the number means, because the computation reveals where it comes from.

Twenty invented daily returns, in percent, one of which is a bad day:

0.42, −0.31, 0.88, −1.24, 0.15, 0.63, −0.47, 1.07, −0.92, 0.28
−7.60, 0.51, −1.83, 2.14, −0.36, 0.74, −0.19, 1.42, −0.68, 0.33

The sum is −5.03, so the mean is −5.03 ÷ 20 = −0.2515. The average squared deviation is 3.663953, giving a standard deviation of 1.9141. The average fourth power of the deviations is 148.5491. Kurtosis is the second of those divided by the square of the first: 148.5491 ÷ 3.663953² = 148.5491 ÷ 13.4245 = 11.07. A normal distribution scores 3, so the excess kurtosis is 8.07.

kurtosis = m₄ ÷ m₂², excess = kurtosis − 3

m₄: The average of the fourth powers of the deviations from the mean. Here 148.5491.
m₂: The average of the squared deviations — the variance. Here 3.663953.
3: The kurtosis of a normal distribution, whatever its width. Subtracting it gives the excess.

Now find out what produced the 11.07. The −7.60 percent day contributes 98.15 percent of the sum of fourth powers on its own. Delete that single observation and recompute from the remaining nineteen: the standard deviation falls to 0.9301 and the kurtosis falls to 2.85, an excess of −0.15, which is very slightly thinner than normal.

So excess kurtosis is not a property distributed through a sample. It is a measurement of how much one or two observations dominated. That makes it an honest statistic and an unstable one: a window containing the bad day reports 11.07 and the identical process observed one day later reports 2.85. For a lender whose risk parameters come from a trailing window, this is the difference between collateral that looks explosive and collateral that looks tame.

Key terms

Kurtosis: The average fourth power of the deviations divided by the square of the variance. Three for a normal distribution, higher when extremes dominate.
Excess kurtosis: Kurtosis minus three. Positive means more weight in the extremes than a bell curve of the same width would carry.
Moment dominance: The situation where a single observation supplies most of a statistic's value. It makes the statistic true for the sample and unreliable as an estimate.

Worked example

One number, computed twice

The twenty invented daily returns above, then the same nineteen days with the −7.60 percent day removed.

Mean: −0.2515%
Average squared deviation: 3.663953
Average fourth power of deviations: 148.5491
Kurtosis: 11.07
Share of the fourth-moment sum from one day: 98.15%
Kurtosis of the other nineteen days: 2.85

The same twenty days produce 11.07 or 2.85 depending on whether one observation is included. The statistic is measuring that observation.

Check yourself

A twenty-day sample has excess kurtosis of 8.07. Removing the single worst day takes it to −0.15. Has the fat tail gone away?

Saved in this browser only. No account, nothing sent anywhere.

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.