The Square-Root-of-Time Rule

A regulator asks for 10-day 99% VaR; the risk system only computes a 1-day number efficiently. Rerunning the whole VaR calculation on 10-day return windows is possible but throws away most of the daily data (a 500-day history yields only 50 non-overlapping 10-day observations) and is slow to update. The square-root-of-time rule offers a shortcut: take the 1-day VaR and multiply by $\sqrt{10}$.

The analogy before any symbols

A drunkard's random walk away from a lamppost, one step left or right each second chosen independently at random, does not wander twice as far in twice the time. Distance from the start grows with the square root of the number of steps, because steps sometimes cancel each other out (a left step followed by a right step). Ten times as many independent random steps produces only $\sqrt{10} \approx 3.16$ times the typical distance, not ten times. Daily stock returns, if truly independent and identically distributed day to day, spread out exactly like the drunkard's steps, which is where the square-root-of-time rule for volatility comes from.

The mechanics

If daily returns are independent and identically distributed with standard deviation $\sigma_1$, the variance of an $h$-day return is the sum of $h$ independent daily variances (variances of independent random variables add):

$\sigma_h^2 = h\,\sigma_1^2 \quad\Rightarrow\quad \sigma_h = \sigma_1\sqrt{h}.$

In words: variance scales linearly with the number of days because independent shocks pile up additively, but standard deviation, the square root of variance, therefore scales with the square root of the number of days, not the number of days itself. Since VaR at a fixed confidence level is just a multiple of $\sigma$ (in the normal case, $\text{VaR} = z_\alpha \sigma$), the same scaling applies directly to VaR:

$\text{VaR}_h = \text{VaR}_1 \times \sqrt{h}.$

In words: to get 10-day VaR from 1-day VaR, multiply by $\sqrt{10} \approx 3.16$, not by 10. This is exactly the shortcut Basel's original market-risk framework wrote into regulation for scaling 1-day VaR up to the 10-day regulatory horizon.

Watch how the spread of these simulated paths widens over time: it widens roughly with the square root of elapsed time, not proportionally, which is this rule made visible. The paths fan out fast at first, then more slowly relative to how far they've already travelled.

Worked example: 1-day to 10-day

A book's 1-day 99% VaR is $500,000. Scaled by the square-root rule: $500{,}000 \times \sqrt{10} = 500{,}000 \times 3.162 = 1{,}581{,}139$, i.e. $1,581,139. A naive linear scaling (multiply by 10) would give $5,000,000, more than three times too large. The correct 10-day number reflects that daily moves partially offset each other over the ten days rather than piling up in the same direction every time.

Worked example: how much bigger is 1-year VaR?

The same $500,000 1-day VaR, scaled to roughly 256 trading days in a year: $\sqrt{256} = 16$, so annual VaR is approximately $500{,}000 \times 16 = 8{,}000{,}000$, i.e. $8,000,000. Compare a 4-day scaling: $\sqrt{4} = 2$, so 4-day VaR is $1,000,000, exactly double. This is the pattern: doubling the horizon only multiplies VaR by $\sqrt{2} \approx 1.41$, not by 2, at every step, which is easy to compute in your head for interview-style questions by remembering a few square roots: $\sqrt{4}=2$, $\sqrt{9}=3$, $\sqrt{16}=4$, $\sqrt{25}=5$.

What this means in practice

The square-root rule is used constantly for quick horizon conversions in risk reporting, capital calculations, and back-of-envelope sizing, precisely because it needs no extra data, just the 1-day number and a square root. It is also a fast interview mental-math trick: "if daily vol is 1.5%, what's monthly vol?" is $1.5\% \times \sqrt{21} \approx 1.5\% \times 4.58 \approx 6.9\%$, using roughly 21 trading days per month.