Lesson 5 of 25 · 7 min

What "fat" means arithmetically in a fat tail

Under a bell curve a move twice as large is a million times rarer. Under a power law, seven times.

Track 4 lesson 4 described a fat tail by comparing a modelled frequency against a historical one. The arithmetic definition is cleaner and does not need any history: pick a level, double it, and take the ratio of the two probabilities.

Under a normal distribution that ratio collapses, and it collapses faster the further out you start. The chance of a four-sigma move divided by the chance of a two-sigma move is 0.001392, one in 718. The chance of a six-sigma move divided by the chance of a three-sigma move is 0.00000073086, one in 1,368,249. Doubling the level got 1,900 times more expensive simply by starting further out.

Under a power law with exponent 2.85 — the case the next lesson identifies — that ratio is 2 to the power of −2.85, which is 0.1387, one in 7.21, and it is the same at every level. Doubling costs a factor of 7.21 whether you start at three percent or at thirty. That constancy is what the word fat is pointing at.

The same fact stated as an average is more useful. Given that a normal variable has already exceeded three sigma, its expected value is 3.283, which is 1.094 times the level it just passed. Given that it has exceeded five sigma, the expectation is 1.037 times the level. Under a normal tail a large move is effectively at its ceiling. Under a power law with exponent 2.85, the conditional expectation is 2.85 ÷ 1.85 = 1.5405 times the level, at every level, forever.

A lender selling collateral is operating inside the conditional distribution, not the unconditional one. The question is never how likely a bad day was — it is how much further a bad day travels once it has started.

Key terms

Conditional tail ratio: The probability of a move twice as large divided by the probability of the move you already have. Near zero under a bell curve, constant under a power law.
Mean excess: The average size of a move given that it has already passed a level. Under a fat tail it stays proportional to the level rather than converging on it.
Scale invariance: The property that the tail looks the same whatever magnification you view it at. It is the defining feature of a power law and the opposite of a bell curve's behaviour.

Worked example

The cost of doubling, in two distributions

One-tailed probabilities. The power law uses an exponent of 2.85 throughout.

Normal, 2 sigma to 4 sigma: One in 718
Normal, 3 sigma to 6 sigma: One in 1,368,249
Power law, any level to twice it: One in 7.21
Normal mean excess at 3 sigma: 1.094× the level
Normal mean excess at 5 sigma: 1.037× the level
Power-law mean excess, any level: 1.5405× the level

Under one shape a large move is nearly as bad as it will get. Under the other it is, on average, half as far again from being over.

Check yourself

Collateral has just fallen 6 percent in a day. Under a normal tail, what is the expected size of the move given that it has passed that level, and how does the answer differ under a power law with exponent 2.85?

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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.