Annualizing Volatility from Daily Return Observations
Summary
The document clarifies how to annualize volatility estimated from daily stock returns over a rolling window. Under the stated assumption that returns are independent and identically distributed, with variance proportional to elapsed time, the standard deviation of daily returns estimates daily volatility. With 252 trading days per year, convert that daily estimate to annual volatility by multiplying by the square root of 252. The number of observations in the estimation window affects how much data goes into the estimate, not the annualization factor.
The apparent alternative, multiplying a six-day volatility by the square root of 252 divided by six, applies when the input is volatility for a six-day aggregate return. The key is to identify the time unit represented by the measured return before scaling. The explanation relies on the stated return-distribution and independence assumptions; dependence, changing volatility, or a different observation frequency may make simple square-root-of-time scaling inaccurate. The answer addresses daily returns, rather than detailing estimation choices such as sample versus population standard deviation.
Key ideas
- Annualization depends on the time interval represented by each return, not the length of the rolling estimation window.
- For daily return volatility, multiply by the square root of the number of trading days in a year.
- Use a square-root-of-time adjustment based on the interval when the input is volatility of an aggregate return.
- The scaling argument assumes independent returns with variance proportional to elapsed time.
- Confirm whether the input measures daily returns or a multi-day return before annualizing.
Tags
Full text
# Annualized rolling volatility?
# Annualized rolling volatility?
I have 600 days of closing prices of a stock. I want to calculate the annualized volatility for 6 day window. How do i do that?
If I calculate the std dev of the first 6 days, i get, say 1%. This is the daily volatility of the first 6 days. To annualize it, should i just multiply 1% with sqrt(252)? Or, is this 1% the daily volatility of 6 days? In that case i should multiply with sqrt(252/6)? Is this number 1% the daily volatility for 1 day, or is it the dayily volatility for 6 days?
I dont get it, I am missing a parameter. The period should also be involved in the calculation, right?
So to annualize 6 day volatility, i multiply with sqrt(252/6)? And when do i multiply with sqrt(252)?
UPDATE1: Ami44 writes that the correct procedure to annualize a 6 day window, is to multiply with sqrt (252/6). See Converting 30day annualized vol to 2day annualized vol
UPDATE2: in the answer below, ForeignVolatility says that I should multiply with sqrt (252). This is contradictory to "UPDATE1" above. So I am confused. Should I multiply with sqrt (252) or sqrt(252/6)? And, if I have a 30 day window, should I still multiply with sqrt (252), or should I use sqrt(252/30)? Great confusion. Some say Ba, and other say Bu.
## Answer by foreignvol (score 4, accepted)
https://quant.stackexchange.com/a/70867
We work in annual units because $T=1$ means one year. This means that the time units must be converted to portions of a year. For example, in the case of daily observations, $\Delta t = 1 / 252$. Hence, in your example, we multiply by $\sqrt{252}$ because it's assumed that the variance is measured daily.
More generally, say you have $n$ returns observed with frequency $\Delta t$ arbitrary. Assume they are i.i.d. and follow a distribution $N(0,\sigma^2\Delta t)$. You then have $$ \mathbb E\left[\frac 1n \sum_{t=1}^n r_t^2\right] = \frac 1n \sum_{t=1}^n\mathbb E\left[r_t^2\right] = \frac 1n \sum_{t=1}^n\sigma^2\Delta t = \sigma^2\Delta t. $$ What you described, the sdt dev of the first 6 days, corresponds to the square-root to an estimation of the LHS with $n=6$. The value you're looking for is $\sigma^2$. Hence, you need to multiply by $1/\sqrt{\Delta t} = 1/(1/\sqrt{252}) = \sqrt{252}$.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.