Annualizing Daily Volatility with Trading-Day Assumptions
Summary
The document explains why daily volatility is commonly scaled by the square root of the number of trading days in a year. Under constant daily variance and uncorrelated returns, variances add across trading days, so annual volatility equals daily volatility multiplied by the square root of the trading-day count. The discussion distinguishes this volatility conversion from annualizing a return, for which a simple approximation may use the full day count.
The answers mention conventions such as 252 or 253 trading days and note that using 250, as in the question’s example, may reflect a different assumption. The resulting difference between nearby day-count conventions is usually small. The square-root rule depends on the assumptions of stable variance and negligible return correlation; covariance terms would matter when daily returns are correlated.
Key ideas
- Annualize daily volatility by multiplying it by the square root of the assumed trading days per year.
- The scaling follows from adding daily variances under constant variance and uncorrelated returns.
- Annualizing returns uses a different scaling idea from annualizing volatility.
- The chosen trading-day count is a convention, and nearby counts usually produce small differences.
- Return correlation or changing variance can make the simple square-root rule inaccurate.
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Full text
# Why multiply stock returns with $\sqrt{252}$?
# Why multiply stock returns with $\sqrt{252}$?
When converting daily volatilities to annual volatilities one need to multiply with $\sqrt{252}$.
But I found this piece of code this piece of code who calculate log-returns in the following way: In MATLAB:
```
y=price2ret(CrixData(:,2))*sqrt(250);
```
The documentation for `price2ret` function is given here, and what this function does is calculating returns of historical data. Why does one also need to multiply with $\sqrt{252}$ in this case?
## Answer by Stéphane (score 3)
https://quant.stackexchange.com/a/53340
On average, there should be about 253 trading days per year, though some people will use 252. If we assume $\sigma^2 = N_{\text{trading days}} \sigma_{\text{daily}}^2$, then obviously we will have $\sigma = \sqrt{N_{\text{trading days}}} \sigma_{\text{daily}}$.
Now, why 253? You could look up the data from NYSE or NASDAQ and compute the average number of trading days per year. Or, you can look up this page from Wikipedia that explains it.
To be frank, the difference in most computations of using 252 or 253 is going to be almost nothing. For volatility, you're looking at something that should be in the 20% range or so over a year, so you get 7.94e-4 or 7.91e-4... I mean, you're down to a difference at the 6th decimal. Everything else in your computation will swamp that easily. A for 250 days, it's the first time I see this.
## Answer by actuarialboi9 (score 3)
https://quant.stackexchange.com/a/53342
if daily returns are $r_1,r_2,...,r_{252}$ then the annual return,$R$, is given by $1+R=(1+r_1)(1+r_2)...(1+r_{252})$. By "logging" and using the approximation $ln(1+x)\approx x$ we get $R=\sum_{i=1}^{252}r_i$ Then the annual variance is $$\sigma^2=\sum_{i=1}^{252}\sigma_i^2+\sum_{i\neq j}\sigma_{ij}$$ where $\sigma_i$ are daily variances and $\sigma_{ij}$ are covariances.
Assuming lack of correlation and constant variance $\sigma_i=\sigma_d,\forall i$ we get $$\sigma^2=252*\sigma_d^2 \iff \sigma=\sqrt{252}\sigma_d$$
Assuming constant daily returns you should multiply by 252 to get the annual return, not the square root of that.
It's 252 because of trading days. Basically a year without weekends.
## Answer by Sławomir Jarek (score 0)
https://quant.stackexchange.com/a/51680
Perhaps someone assumed that there are 250 trading days per year for this time series instead of 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.