Lesson 6 of 20 · 6 min
What a two-sigma or three-sigma move means in practice
Sigma counts translate into frequencies only under an assumption, and the translation gets worse as the count rises.
Sigma language is used constantly and rarely converted into anything concrete. The conversion is worth doing once so the vocabulary carries weight.
Under a normal distribution, single-tail probabilities are: beyond two sigma, about 2.3 percent; beyond three sigma, about 0.13 percent; beyond four sigma, about 0.003 percent; beyond five sigma, about 0.00003 percent. Translated into trading days at 252 per year: a three-sigma day about once in three years, a four-sigma day about once in 125 years, a five-sigma day about once in 13,000 years.
Now compare those frequencies against experience. Days beyond four sigma have occurred multiple times within the working lives of people currently employed in markets, which is not compatible with a once-in-125-years frequency. The model's frequencies are usable near the centre and unusable in the tail.
There is a second confusion worth separating: one tail against two. A two-sigma move in either direction covers about 4.6 percent of outcomes, but a two-sigma decline specifically is about 2.3 percent. Risk questions are almost always one-tailed, since a large gain does not trigger a maintenance threshold. Quoting the two-tailed figure doubles the apparent likelihood of the thing you were asking about.
Use sigma to describe how big a move was. Be sceptical the moment anyone converts a sigma count above three into a frequency.
Key terms
- One-tailed versus two-tailed
- Whether you are asking about a move in one direction or either. Risk questions are one-tailed; a large gain triggers nothing.
- Implied frequency
- How often an outcome should occur, derived from a probability. Reliable near the centre of a distribution and not in the tail.
- Tail probability
- The chance of exceeding a given sigma level. Falls extremely fast under a normal distribution, which is the source of the error.
Worked example
Sigma counts as frequencies, and where they break
Normal distribution, one tail, 252 trading days per year.
- Beyond 2 sigma
- 2.3%
- Beyond 3 sigma
- 0.13%
- Beyond 4 sigma
- 0.003%
- Beyond 5 sigma
- 0.00003%
- Observed 4+ sigma days in recent decades
- Several
- Conclusion
- The frequencies fail somewhere between 3 and 4 sigma
About 6 days a year.
About once in 3 years.
About once in 125 years.
About once in 13,000 years.
A four-sigma day should be a once-in-a-lifetime-of-lifetimes event under the model. Anyone who has watched markets for twenty years has seen several.
Check yourself
Someone says a decline was a two-sigma event, and quotes a 4.6% probability. What is wrong with the figure?
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.