Lesson 5 of 20 · 6 min

Standard deviation as a unit of surprise

Standard deviation converts a move in dollars into a statement about how unusual it is.

A three percent fall means nothing on its own. In a quiet market it is remarkable; in a turbulent one it is a Tuesday. Standard deviation is the unit that makes the comparison possible.

The computation is a division. Take the move, subtract the typical move, and divide by the standard deviation. The result is expressed in sigmas, and it says how many typical-sized steps this move represents. That is why the same dollar move can be a one-sigma event in one asset and a five-sigma event in another.

z = (observed move − expected move) ÷ standard deviation

z: The move expressed in standard deviations. A unit of surprise rather than a unit of money.
observed move: What actually happened, as a return rather than a dollar amount.
expected move: The centre of the distribution. Often near zero for a single day.
standard deviation: The typical size of a move over the same period. Must match the period of the observed move.

The essential practical detail is period matching. Volatility is normally quoted annually, and converting it to a shorter period requires dividing by the square root of the number of periods — the same square-root-of-time rule from Track 3. A 25 percent annual volatility is about 1.6 percent daily, using 252 trading days. Compare a daily move against an annual figure and the answer is wrong by a factor of sixteen.

One caution that carries over from the previous lessons. Converting a move into sigmas is descriptive and reliable. Converting sigmas into a probability requires a distributional assumption, and in the tails that assumption is where the errors live. Saying a move was five sigma is a fact about scale; saying it should occur once in three million days is a claim about the model.

Key terms

Standard deviation: The typical size of deviation from the average. The natural unit for describing how large a move was.
Z-score (sigma): A move expressed in standard deviations. Comparable across assets and time periods.
Period matching: Ensuring the move and the standard deviation cover the same interval. Annual volatility divided by √252 gives a daily figure.

Worked example

The same 3% day in three different markets

A 3% single-day decline, measured against three different annual volatility levels.

Calm underlying, 12% annual volatility: Daily sigma 0.76%
Typical underlying, 25% annual volatility: Daily sigma 1.57%
Volatile underlying, 60% annual volatility: Daily sigma 3.78%
Same move, range of surprise: 0.8 to 3.9 sigma
Error from using the annual figure directly: 16×

One identical move is unremarkable in one market and a once-in-years event in another. The dollar figure carried none of that.

Check yourself

An underlying with 25% annual volatility falls 5% in a day. How many standard deviations is that, and what does the answer not tell you?

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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.