Annualizing Volatility from Monthly Returns
Summary
The document explains why annualized volatility is commonly calculated by multiplying monthly return volatility by the square root of twelve. Under independent, identically distributed monthly log returns with finite variance, annual log return is the sum of twelve monthly returns, so its variance is twelve times the monthly variance and its standard deviation scales by the square root of twelve.
It contrasts this with compounding each individual monthly return over twelve periods, which changes the quantity being measured and does not represent the volatility of a year of sequential returns. The discussion also notes that the scaling rule follows from independence and finite variance, rather than requiring normally distributed returns. The treatment is simplified: dependence between returns or changing volatility can invalidate the basic square-root-of-time relationship, and simple returns require care because they compound multiplicatively.
Key ideas
- Annual log returns are sums of monthly log returns.
- For independent monthly returns with finite variance, annual standard deviation is monthly standard deviation multiplied by the square root of twelve.
- Compounding each monthly return over a year is not the same as calculating the volatility of sequential annual returns.
- The standard scaling relationship depends on assumptions about return dependence and variance.
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Full text
# Calculating annualized volatility of stock returns
# Calculating annualized volatility of stock returns
Suppose I have a sequence of monthly returns of a stock, $r_1,r_2,\ldots$. Suppose further that this is an i.i.d. sequence with with finite second moments.
In every paper, report, lecture note etc. the annualized volatility of the return of this stock is given as $\sigma(r_1)\sqrt{12}$.
On the other hand, if I annualize the monthly returns first, that is if I consider $(1+r_1)^{12}-1,(1+r_2)^{12}-1,\ldots$, then I get $\sigma((1+r_1)^{12}-1) \approx 12\sigma(r_1)$ since $(1+x)^{12}-1 \approx 12x$.
My question is what is wrong with what I am doing? Is it only a matter of convention that people use the first formula to report annualized volatility?
## Answer by nbbo2 (score 2)
https://quant.stackexchange.com/a/30524
It makes no difference whether you work with annualized numbers or not.
If you work with monthly logarithmic returns $\{r_1,r_2,\cdots,r_{12}\}$ then the return for the year is $R=r_1+r_2+\cdots+r_{12}$. Assuming only that the returns are i.i.d and the standard deviation $\sigma$ exists, then the standard deviation of the annual return $R$ is $\sqrt {12} \sigma$.
If you prefer to work with annualized returns, then you are looking at $\{12 r_1,12 r_2,\cdots,12r_{12}\}$. The return for the full year is $\frac{12r_1+12r_2+\cdots+12r_{12}}{12}$ which is the identical expression as before and its volatility is again $\sqrt {12} \sigma$.
## Answer by Guga (score -1)
https://quant.stackexchange.com/a/30520
Actually what you are referring as a conventions comes from an assumption that the returns are driven by a normal distribution. If you consider a stochastic variable (time-series of a random variable) that is normally distributed you can demonstrate that the variance of the distribution grows linearly with time. It means that if you go from a 1 day return time-series to a two-days return time series, the variance of the later is going to be twice as the former.
$\sigma^2_{annual} = 12 \sigma^2_{month} $
However, the standard deviation (which is simplest estimation of the volatility) of the later is going to be:
$\sigma_{annual} = \sqrt{12} \sigma_{month}$
The definition of standard deviation, as being the square-root of the variance, is what makes financial industry, as a whole, to consider the square root of the time (compared to the daily volatility, for instance) as the correct way to convert to an annual volatility.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.