Formula, in Risk & Portfolio Construction
Annualized Volatility Formula
Annualized volatility is the standard deviation of per-period returns scaled by the square root of the number of periods in a year. Volatility scales with the square root of time, not linearly, because variance adds across independent periods while the mean adds linearly. This square-root-of-time rule is the standard convention for stating volatility on a comparable annual basis.
4 variables. Published May 26, 2026. By AI Fin Hub Research.
sigma_period = sqrt( (1/(n-1)) x sum_i (R_i - mean_R)^2 )
sigma_annual = sigma_period x sqrt(N)
where N = periods per year (252 daily, 52 weekly, 12 monthly) Variables
R_i
Period return
Return in period i. Log returns are often preferred for volatility because they add cleanly across periods.
mean_R
Mean return
Average of the per-period returns over the sample, the center around which dispersion is measured.
sigma_period
Per-period volatility
Sample standard deviation of returns at the native frequency (daily, weekly, monthly), using the n-1 divisor for an unbiased estimate.
N
Periods per year
Number of return periods per year. The square root appears because, for independent returns, variance grows linearly with the number of periods, so its square root (the standard deviation) grows with the square root of N.
Step by step
- 1
Compute the per-period return standard deviation from the sample.
Daily returns over a year have a standard deviation of 1.1%.
- 2
Identify the number of periods per year for the return frequency.
Daily equity returns use N = 252 trading days.
- 3
Multiply the per-period standard deviation by the square root of N.
0.011 x sqrt(252) = 0.011 x 15.875 = 0.1746.
- 4
Express as a percentage and treat it as the annualized volatility, valid under the assumption that returns are roughly independent across periods.
Annualized volatility of about 17.5%.
Worked example
Annualizing a stock's daily return volatility
Daily return standard deviation
1.1%
Trading days per year
252
sigma_annual = 0.011 x sqrt(252). sqrt(252) = 15.875. 0.011 x 15.875 = 0.17462 = 17.46%.
Annualized volatility of about 17.5%. The square-root-of-time scaling assumes returns are serially independent. If returns are positively autocorrelated (trending), this rule understates true annual volatility; if negatively autocorrelated (mean-reverting), it overstates it. Always check for autocorrelation before relying on the scaled figure.
Common variations
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Sources
- The Statistics of Sharpe Ratios , Andrew W. Lo, Financial Analysts Journal (2002)
- RiskMetrics Technical Document , J.P. Morgan / Reuters (1996)
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