Lesson 2 of 25 · 7 min

What the normal distribution says about a five-sigma day

Each extra standard deviation costs more likelihood than the one before it, and that acceleration is the whole mechanism.

Track 4 lesson 6 put a five-sigma day at about once in thirteen thousand years. The exact figure, at 252 trading days a year, is 13,843 years: the one-tailed probability is 0.00000028665, which is one day in 3,488,556. The interesting part is not that number but the shape that produces it.

Compute the ladder rather than a single rung. Moving from three sigma to four divides the probability by 42.6. From four to five, by 110.5. From five to six, by 290.5. From six to seven, by 770.9. The multiplier is not constant — it grows by roughly a factor of 2.6 with each additional standard deviation.

That acceleration comes straight from the formula. The normal density carries exp(−z² ÷ 2), so the logarithm of the probability falls with the square of the distance rather than in proportion to it. Every step further out costs more than the step before. A useful approximation shows the same thing: the tail probability is close to the density divided by z.

P(Z > z) ≈ φ(z) ÷ z, where φ(z) = exp(−z² ÷ 2) ÷ √(2π)

P(Z > z): The chance of a move beyond z standard deviations in one direction.
z: Distance from the centre, measured in standard deviations.
φ(z): The height of the bell curve at that distance. It contains the squared term that drives the acceleration.
π: The usual constant, 3.14159, appearing in the normalising factor.

Now land it. A buffer set at 2.3263 standard deviations is a one-day-in-a-hundred buffer. A buffer at five standard deviations is roughly twice as wide in dollars, and the model claims it is 34,886 times safer. No physical process gives a payoff that steep for a doubling. When a modest widening of a haircut appears to buy a factor of tens of thousands, the buffer is not doing the work — the assumption is.

Key terms

Survival probability: The chance of exceeding a stated level. The quantity that falls faster than exponentially as the level rises under a normal assumption.
Gaussian decay: The exp(−z² ÷ 2) shape of the normal tail. Because the exponent is squared, rarity accelerates rather than compounding at a fixed rate.
Mills approximation: Estimating a normal tail probability as the density divided by the sigma count. At five sigma it is 3.7 percent too high, which is close enough to reason with.

Worked example

The sigma ladder, and the multiplier between rungs

One-tailed normal probabilities, 252 trading days a year.

Beyond 3 sigma: One day in 741
Beyond 4 sigma: One day in 31,574
Beyond 5 sigma: One day in 3,488,556
Multiplier, 3 to 4 sigma: 42.6×
Multiplier, 4 to 5 sigma: 110.5×
Multiplier, 5 to 6 sigma: 290.5×

The cost of each additional sigma is not fixed. It rises by about a factor of 2.6 per step, which is why normal tail figures become absurd so quickly.

Check yourself

A haircut is widened from a 2.3263-sigma buffer to a 5-sigma buffer — roughly double in dollar terms. Under a normal model the implied safety improves by a factor of 34,886. What is the right reading of that figure?

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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.