The square root of time
Independent days add in variance, so volatility scales with the square root of time. Daily to annual, annual to a ten-day horizon, a standard deviation to a value at risk: one rule, run in every direction.
- Annualize a daily volatility with √252, and go back
- Scale volatility to any horizon and turn it into a dollar move
- Compute a one-day 95% value at risk
- Solve for the days a target move needs, and say when √t fails
The intuition
If each day's return is independent of the last, the variances of the days add up. A year of 252 trading days has 252 times the variance of one day, and √252 ≈ 15.9 times the volatility. A stock with a 1.26% daily standard deviation is a 20% annual stock. Traders use 16: divide annual vol by 16 for a typical daily move.
The same root runs everywhere. Twenty percent annual over ten trading days is 20% × √(10 ÷ 252) = 4%, so a $50M position has a one-sigma ten-day move of about $2M. A one-day 95% value at risk is 1.645 daily standard deviations: about $1M on that position. And the rule is a default, not a law: trending returns make long-horizon risk larger than √t says, mean-reverting ones make it smaller, and smoothed prices on illiquid assets make daily vol look far too low.
σ_annual = σ_daily × √252. σ over h days = σ_annual × √(h ÷ 252). One-sigma dollar move = position × σ_h. One-day 95% VaR = 1.645 × σ_daily × position. Days for a one-sigma move to reach x: h = 252 × (x ÷ σ_annual)².
Why it works
- The conventions here: independent daily returns with constant volatility, 252 trading days a year. VaR is one-sided at 95%, normal returns, zero mean.
- Variances add, volatilities do not. Var(sum of h independent days) = h × daily variance, so σ_h = σ_daily √h. Multiplying a daily vol by 252 instead of √252 is the intern's mistake, and it is off by a factor of 16.
- Every direction is the same formula. Daily from annual divides by √252; a horizon multiplies by √(h ÷ 252); a target move squares the ratio to get days back.
- VaR is a quantile, not a worst case. 1.645 is the one-sided 95% point of a normal. Losses exceed VaR about one day in twenty; VaR says nothing about how large those losses are, and real returns have fatter tails. Expected shortfall answers that.
- When √t understates risk. Positively autocorrelated returns, momentum or trending markets, cluster the bad days. Illiquid assets with smoothed marks are the classic case.
- When it overstates. Mean-reverting returns partly cancel over long horizons. Neither correction is large for liquid equities over days to weeks, which is why the rule is the default.
| Annual vol: 1.26% × √252 = 1.26% × 15.87 | 20.0% |
| Rough check: 1.26% × 16 | 20.2% |
| 10-day vol: 20% × √(10 ÷ 252) | 3.98% |
| One-sigma 10-day move: $50M × 3.98% | $1.99M |
| One-day 95% VaR: 1.645 × 1.26% × $50M | $1.04M |
| Days for a one-sigma move to reach 10%: 252 × (10% ÷ 20%)² | 63 trading days |
Quadruple the horizon to 40 days and the vol doubles to 7.97%: four times the time, twice the volatility.
The formulas
Independent days add in variance.
Any horizon in trading days.
The one-sided 95% point of a normal.
Days for a one-sigma move to reach x: square the ratio.
Worked example
Multiply the daily figure by the square root of 252. The follow-up is the mental shortcut: times 16.
See it move
Same asset. Change its annual volatility, the horizon, the size of the position and the target move.
- Raise the annual volatility. Every volatility and every dollar figure rises, and the days a target move needs fall as the square.
- Lengthen the horizon. The horizon volatility and the dollar move rise with the root of the days; the one-day VaR does not move.
- Increase the position. Both dollar figures scale with it; no volatility changes.
- Raise the target move. The days needed rise as its square.
Run it backwards
The annual volatility is quoted. What is a single day's standard deviation?
Divide by √252, about 15.87. Or, in your head, divide by 16.
The harder inverse is in the check: the trading days over which a one-sigma move reaches a stated size, which squares the ratio of the two volatilities.
Traps
Say it in the interview
“Daily vol is 1.26%. What is the annual vol, and the ten-day risk on $50M?”
Check yourself
5 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- σ scales with √time: × √252 to annualize, × √(h ÷ 252) for a horizon.
- Divide annual vol by 16 for a typical daily move.
- One-day 95% VaR = 1.645 × daily σ × position, a quantile with a fat tail beyond it.
- √t is a default: trending or smoothed returns break it.