Scaling Volatility Across Return Horizons and Annualization
Summary
The discussion distinguishes volatility measured over a return horizon from volatility annualized from that horizon. Under independent returns with additive variances, a 30-day return volatility can be converted to a 2-day return volatility by multiplying by the square root of the ratio of the horizons. When each horizon-specific volatility is then annualized consistently, the annualized values are equal; the proposed square-root factor does not convert one annualized volatility directly into another annualized volatility.
This relationship depends on assumptions about returns. Independence and variance additivity allow horizon scaling, while serial correlation can change the relationship between short- and long-horizon risk. Knowing only a 30-day volatility does not reveal the shorter-horizon correlation structure, so it cannot recover a precise 2-day volatility when those effects matter. The answers explain the algebra using a 360-day year as an illustration and note that the day-count convention should match the market and data. The result is a conditional scaling rule, not a way to infer missing return behavior from one volatility estimate.
Key ideas
- Under independent returns, variances scale with the length of the return horizon.
- A 30-day return volatility scales to a 2-day return volatility by the square root of the horizon ratio.
- Consistent annualization makes horizon-specific annualized volatility estimates equal under the stated assumptions.
- Serial correlation prevents reliable short-horizon volatility recovery from a single longer-horizon estimate.
- The annualization day count should reflect the convention relevant to the data.
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# Converting 30day annualized vol to 2day annualized vol
# Converting 30day annualized vol to 2day annualized vol
I would like to convert 30-day annualized volatility $\sigma_{30d}^a$ to 2-day annualized volatility $\sigma_{2d}^a$.
Am i right to say:
$$\sigma_{2d}^a = \sqrt{\frac{2}{30}} \cdot \sigma_{30d}^a$$
I don't have the returns but only 30-day annualized vol.
## Answer by Ami44 (score 4)
https://quant.stackexchange.com/a/46821
I assume with 30 day annualized volatility you mean, you calculated the volatility from the 30 day returns and then annualized it by multiplying with $\sqrt{252/30}$.
You can calculate the 2 day volatility only if you assume independence of the returns. In that case though all annualized volatilities are identical, especially $\sigma_{2d}^a = \sigma_{30d}^a$.
If you want to capture the effects of correlation in your data, there is no way to calculate the 2 day correlations from the 30 day correlations and hence there is no way to scale $\sigma_{30d}^a$ to get to $\sigma_{2d}^a$ apart from the first approximation that they are the same.
## Answer by Magic is in the chain (score 1)
https://quant.stackexchange.com/a/46832
Just to expand, let's say the assumptions needed for the variances to be identical and additive are satisfied, so the annual variance will be the sum of monthly variances. I am going to assume 360 days in a year but you will have to change it to reflect the local holidays/weekends etc. So the annual variance will be the sum of 12 monthly variances:
$\sigma^2_{360d}=12 \times \sigma^2_{30d}$
$\sigma^2_{360d}=\sigma^2_{30d}\frac{360}{30}$
You can similarly write for the 2-days horizons:
$\sigma^2_{360d}=\sigma^2_{2d}\frac{360}{2}$
Hence,
$\sigma^2_{360d}=\sigma^2_{30d}\frac{360}{30}=\sigma^2_{2d}\frac{360}{2}$
And if you rearrange you get:
$\sigma^2_{2d}=\sigma^2_{30d}\frac{2}{30}$
The square root of which is the relationship you have, but this is the relationship between the volatility of return over 30 days and the volatility of returns over 2 days as @Ami44 explained above. If the assumptions for variances to be additive are satisfied then both shall give the same annual volatility when annualised.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.