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

With uncorrelated returns and constant variance, uncertainty grows with √time. Those assumptions need to accompany the number.

Why variance adds

For a sum of returns, variance equals the sum of individual variances plus covariance terms. If those covariance terms vanish and each period has the same variance, n periods have n times one-period variance. Standard deviation is its square root, giving √n scaling.

Independence is a sufficient assumption, but zero return autocovariance is enough for this variance calculation. It does not by itself make the resulting distribution normal.

Worked example with trading sessions

For hypothetical 16% annualised volatility using 252 trading sessions, daily volatility is 16% / √252 ≈ 1.008%. For five sessions it is 16% × √(5/252) ≈ 2.254%. Four times the horizon doubles the scale under the same assumptions.

For a calendar-time input using 365 days, use that matching convention instead. Do not put five calendar days into a 252-session formula without explaining the conversion.

Where the approximation can fail

Correlated returns add covariance terms. Changing volatility means period variances are not equal. Jumps, scheduled events and heavy tails can make a smooth constant-volatility cone miss important risks. Volatility clustering alone is not the same as positive return autocorrelation, so keep those concepts separate.

A scale is not a calibrated probability

Aiovel uses session counts for exchange-traded instruments and calendar days for Bitcoin. The interface’s month and year labels represent configured counts, rather than literal month-end or year-end deadlines. Read the horizon labels on the S&P 500 sample.

Applying √time does not independently validate touch probabilities or establish the model’s accuracy. Those require a defined forward test.