Why Independent Monthly Returns Scale Volatility by Square Root of Twelve

Summary

The document derives the familiar annualization rule for standard deviation from the variance of a sum. If twelve monthly observations are independent and identically distributed, the variance of their sum is twelve times the variance of one observation. Taking square roots makes the standard deviation of the annual sum equal to the monthly standard deviation multiplied by the square root of twelve.

The result depends on the independence assumption, as well as the stated identical distribution of the monthly observations. It describes the scaling of a sum across periods; it does not establish that real financial returns satisfy those assumptions. The brief explanation gives the mathematical basis for volatility annualization but does not discuss autocorrelation, changing volatility, or estimation error.

Key ideas

• The variance of a sum of independent observations is the sum of their variances.
• For twelve identically distributed monthly observations, summed variance is twelve times one-month variance.
• Taking the square root yields the square-root-of-twelve standard deviation scaling rule.
• The derivation assumes monthly observations are independent and identically distributed.

Tags

# Multiplying by the Square Root of Twelve to calculate annual standard deviation


# Multiplying by the Square Root of Twelve to calculate annual standard deviation












I failed to see the mathematics truism. Can someone care to elaborate.

## Answer by Phun (score 4)

https://quant.stackexchange.com/a/19239

If $$ X_1, X_2, \dots, X_{12} $$ are i.i.d. (stochastic independent identical distributed) it holds $$ var(\sum X_i) = \sum var(X_i) = \sum var(X_1) = 12var(X_1) $$. now take the square root to get the stated result.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.