Annualizing a daily volatility

Question

A stock's daily return has a standard deviation (volatility) of 1.5\%. Assume daily returns are independent, and there are 252 trading days in a year. What is the annualized volatility?

Accepted Answer

An annual return is the sum of many daily returns. When you add independent random pieces, their variances add, not their standard deviations. So over 252 days the variance is 252 times as large, and the standard deviation grows by the square root of 252. This is the same \sqrt{n} scaling as a standard error, just applied to accumulating returns over time. \sigma{	ext{year}} = \sigma{	ext{day}} 	imes \sqrt{252} = 1.5\% 	imes \sqrt{252}. Since \sqrt{252} \approx 15.87, \sigma{	ext{year}} \approx 1.5\% 	imes 15.87 \approx 23.8\%. Note the swing does not grow 252-fold, only about 16-fold, …

Annualizing a daily volatility

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