Scaling daily volatility to a month

Question

An asset's daily return volatility is 2\%. Assume daily returns are independent and a month has 21 trading days. What is the monthly volatility?

Accepted Answer

A monthly return is the sum of 21 daily returns. Because independent variances add, the monthly variance is 21 times the daily variance, and so the volatility grows by \sqrt{21}: \sigma{	ext{month}} = \sigma{	ext{day}} 	imes \sqrt{21} = 2\% 	imes \sqrt{21}. Since \sqrt{21} \approx 4.58, \sigma{	ext{month}} \approx 2\% 	imes 4.58 \approx 9.2\%. The swing grows by only about 4.6 times over a month, not 21 times, because it scales with the square root of the number of days. Takeaway. Monthly volatility = \sigma{	ext{day}}\sqrt{21}: the \sqrt{n} scaling means a full month of trading multip…

Scaling daily volatility to a month

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