When the Square-Root-of-Time Volatility Rule Applies

Summary

The document asks whether volatility scales with the square root of time when log returns are not normally distributed. Its answer derives the scaling rule in discrete time from the variance of a sum. When period returns are mutually uncorrelated and have the same variance, the variance of their total is the number of periods multiplied by the single-period variance. Taking the square root gives standard deviation proportional to the square root of elapsed periods.

The derivation does not require normality, so non-normal returns alone do not invalidate the rule under these assumptions. The explanation is conditional, however: equal variances and zero pairwise correlations are needed for the stated result. The document does not discuss what happens with serial dependence, changing variance, or other violations, and it offers no market data or empirical test. It concerns scaling standard deviation, not a guarantee that returns themselves follow a normal distribution.

Key ideas

• The square-root-of-time rule follows from variance addition for uncorrelated returns with equal variance.
• Normality is not required for that variance-scaling result.
• Serial correlation or unequal period variances can invalidate the assumptions behind the derivation.
• The rule describes standard deviation scaling and does not establish a normal return distribution.

Tags

# Square root of t rule for non-normal returns


# Square root of t rule for non-normal returns












It is well known that the square root of t rule holds for log returns that are normally distributed.

However, does it also hold for non-normally distributed log returns?

## Answer by Richard Hardy (score 5, accepted)

https://quant.stackexchange.com/a/80842

If you are interested in discrete time, variance of a sum of mutually uncorrelated random variables equals the sum of their variances: $$ \text{Var}(X_1 + \dots + X_T) = \text{Var}(X_1) + \dots + \text{Var}(X_T). $$ If all the variances are the same, $$ \text{Var}(X_1 + \dots + X_T) = T\sigma^2 $$ where $\sigma^2 := \text{Var}(X_1) = \dots = \text{Var}(X_T)$. Standard deviation is the square root of variance. When all variances are the same, that makes $$ \text{sd}(X_1 + \dots + X_T) = \sqrt{ \text{Var}(X_1 + \dots + X_T)} = \sqrt{T\sigma^2} = \sqrt{T}\sigma. $$ This does not assume normality.

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