Lesson 3 of 25 · 6 min
Annualising, and why the square root of time appears
Variances add across independent periods and volatilities do not, which is the entire reason a square root is involved.
The square root of time is usually presented as a convention to memorise. It is a consequence of one specific assumption, and the assumption is the interesting part of the lesson.
Start with what does add. If daily returns are independent of one another, the variance of a multi-day return is the sum of the daily variances. Variance is the squared quantity from lesson 2 — the average squared deviation, before the square root. So a year of independent sessions has 252 times the variance of one session.
Run it on the series from lesson 2. Its daily variance is 2.57991 in percent-squared terms. Multiply by 252 and the annual variance is 650.1379. Take the square root and the annual volatility is 25.4978 percent. That is the whole derivation, and the square root appears only because the additive quantity is the square.
annual volatility = daily volatility × square root of sessions in a year
- annual volatility
- The same measurement restated for a one-year interval. The convention most figures are quoted in.
- daily volatility
- The figure computed directly from daily returns, as in lesson 2.
- sessions in a year
- Trading sessions, conventionally 252. Its square root is 15.8745.
What happens if you skip the square root is worth seeing once. Multiplying 1.6062 percent by 252 gives 404.77 percent, which is not a number about anything. The same rule scales downward as well: 1.6062 times the square root of 5 is 3.5916 percent for a week, and times the square root of 21 is 7.3606 percent for a month.
Hold on to the word independent, because the rest of this track spends its time attacking it. Variances add only when sessions carry no information about each other. Lesson 5 measures the extent to which they do, and lesson 19 shows what that measurement does to a haircut built on this exact scaling rule.
Key terms
- Variance
- The average squared deviation, before the square root. The quantity that adds across independent periods, which is why it is the one doing the work.
- Annualising
- Restating a volatility for a one-year interval by multiplying by the square root of the number of periods in a year.
- Independence
- The condition that one session's return carries no information about the next. The assumption the square-root rule rests on.
Worked example
One series, four quoting periods
The ten-return series from lesson 2, whose daily volatility is 1.6062 percent.
- Daily variance
- 2.57991
- Annual variance
- 650.1379
- Annual volatility
- 25.4978%
- Weekly volatility
- 3.5916%
- Monthly volatility
- 7.3606%
- What scaling by 252 instead would give
- 404.77%
1.6062 squared.
2.57991 × 252.
The square root of 650.1379.
1.6062 × the square root of 5.
1.6062 × the square root of 21.
Four correct figures and one absurd one, all from the same series. The square root is not a refinement; it is the difference between a number and a nonsense.
Check yourself
A holding's daily volatility is measured at 0.94 percent. What is the annualised figure, and why is it not 236.88 percent?
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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.