# Standard Deviation Annualized

Say I take the standard deviation of 730 data points, representing two years worth of data. How would I convert this standard deviation to an "annualized" one?

Thanks for the help.

It depends on the frequency of the data points. In general, if there are $T$ data points per year (and subject to some conditions*) then the conversion formula is

$$\sigma_{\rm annual} = \sigma_{\rm measured} \sqrt{T}$$

For example, in finance it is common to measure the return on a stock every day, but to quote volatility (aka standard deviation of returns) as an annual figure. There are about 260 trading days in a year, so you commonly see

$$\sigma_{\rm annual} = \sigma_{\rm daily} \times \sqrt{260}$$

[*] The conditions are as follows:

1. The annual quantity can be expressed as a sum of the quantities measured on a smaller timescale, that is, $$X_{\rm Year\,1} = x_1 + x_2 + \cdots + x_T$$

2. There is no autocorrelation among the quantities on the smaller timescale.

The the square root is most easily explained by noting that, subject to the conditions above, the variance increases in proportion to the elapsed time, and the variance is the square of the standard deviation. This is not too difficult to prove by starting from the definition of the annualized variance in terms of the micro-quantities $x_t$.

• Thanks! How about annualizing return from those data points? – icobes Jan 30 '12 at 6:44
• Add them up (or multiply the expected return by $T$). – Chris Taylor Jan 30 '12 at 6:46
• Last question: So for my data set, would I do the standard deviation of all 730 daily data points and multiply by root 730? or by root 260 (since I want an annual number)? Thanks! – icobes Jan 30 '12 at 6:50
• You'd multiply by $\sqrt{260}$ to get an annual number (assuming that there are 260 observations per year, of course...) – Chris Taylor Jan 30 '12 at 10:08
• @kookster Yes, still true (note that with only 30 data points the confidence intervals on your measured standard dev will be quite wide). – Chris Taylor Feb 18 '14 at 12:23

icobes - Your 730 data points representing 2 years worth of data sounds like you have daily data points (i.e., 365 data points per year). Using the formula provided by Chris Taylor, the annualized standard deviation is calculated as

  [standard deviation of the 730 data points] x [square root of 365]


If you had 520 data points representing 2 years worth of data (i.e., 260 data points per year), then the annualized standard deviation is calculated as

  [standard deviation of the 520 data points] x [square root of 260].


Hope this clarifies the 260 number provided in Chris Taylor's example.

In theory it should be sqrt(252) not 260 or 365. Remember there is a lot of "noise" in daily returns so it is good practice to analyze vol on a daily, monthly, and annual level.

If you use 365 then you are accounting for variability that happens on the days markets are closed, which is zero b/c markets are closed on weekends, holidays, exc..

Obviously, Rtsqrt(252) > Rtsqrt(365) so it is an unbiased estimate of annualized daily volatility.

Hope this helps