How Square-Root-of-Time Scaling Expands Probability Cones

Four times the time horizon does not give four times the uncertainty. It gives about twice. That single fact shapes every probability cone you will ever read.

The rule

Volatility scales with the square root of time. To convert an annual volatility to a horizon of t years, multiply by √t. To annualise a daily volatility, multiply by √252, the approximate number of trading days in a year.

So a 16% annual volatility corresponds to roughly 16% ÷ √252 ≈ 1% per day, and a one-month (21 trading day) volatility of 16% × √(21÷252) ≈ 4.6%.

Why the square root and not simple multiplication

The underlying assumption is that successive returns are independent. Under independence, variances add — but volatility is the square root of variance.

So two days of independent returns have twice the variance of one day, and therefore √2 ≈ 1.41 times the volatility. Extending that logic across any horizon produces the square root rule. The intuition is that random moves partially cancel: some days up, some days down, so total displacement grows more slowly than total activity.

What this means visually

It is why a probability cone is a cone rather than a triangle. Near-dated bands are proportionally much wider than a linear intuition suggests, while distant bands widen more slowly.

It also explains why long-dated ranges become uninformatively wide fairly quickly. If the range at one month is roughly 5%, at a year it is not 60% — it is closer to 16% — but that is still wide enough that most specific price levels fall inside it, which is precisely why long-horizon probabilities cluster toward the middle.

Where the rule breaks

Returns are not independent. Volatility clusters: turbulent days follow turbulent days. This positive autocorrelation in volatility means realised multi-day ranges can exceed the square root estimate during stressed regimes.

Mean reversion cuts the other way. Assets that revert toward a level display less multi-period dispersion than the rule predicts, so the estimate runs wide.

Discrete catalysts break the smoothness. A scheduled earnings report or policy decision concentrates risk into a single session. Scaling smoothly across a window containing a known catalyst understates the risk on that day and overstates it on the others.