Combining Period Volatility

Article Quant Q&A · Author: AltTabsen

Summary

The document explains how to combine volatility estimates from consecutive periods. If returns across the periods are uncorrelated, their variances add, so the standard deviation over the full two-year span is the square root of the sum of the two annual variances. This differs from adding the volatilities directly.

To express the combined result as an annualized volatility, the total variance is averaged across the two years before taking the square root. A second answer gives the more general calculation when the years contain different numbers of observations: weight each period’s variance by its observation count, then divide by the total count. These formulas assume zero mean in the observation-based derivation and no autocorrelation for variance additivity. If returns are correlated across periods, covariance terms are needed; the short answers do not cover that case.

Key ideas

For uncorrelated period returns, total variance is the sum of the period variances.
The volatility across the full span is the square root of that summed variance.
Annualized volatility over equal-length years is the square root of the average squared volatility.
When periods have different observation counts, combine variances using observation-count weights.
Autocorrelation or cross-period dependence requires accounting for covariance.

Tags

Statistics Risk management Portfolio construction

Full text

# I have portfolio volatility for year 1 and for year 2. What is portfolio volatility for year 1 and 2 combined?


# I have portfolio volatility for year 1 and for year 2. What is portfolio volatility for year 1 and 2 combined?












Thanks for looking into this question.

Portfolio volatility in year 1 = 15%. Portfolio volatility in year 2 = 20%.

What is the portfolio volatility over the timespan year 1 and 2 combined?

Is it SQRT(0.5)*15% + SQRT(0.5)*20%?

Thanks!

## Answer by vonjd (score 2)

https://quant.stackexchange.com/a/16572

Assuming that we are talking about volatility as the standard deviation of uncorrelated random variables (in this case this would mean no autocorrelation) the variance is additive, which means that we get $\sqrt{.15^2+.2^2}=.25=25\%$.

You can illustrate this result by simulation in R:

```
> sd(rnorm(1e7,sd=.15)+rnorm(1e7,sd=.2))
[1] 0.2500001
```

If you want to annualize this number again you'd have to divide by $\sqrt{2}$ (because of the two one-year periods) which gives about $17.68\%$.

So putting it all together what you do is to calculate the square root of the average of the squared volatilities: $$\sqrt{\frac{.15^2+.2^2}{2}}\approx.1768=17.68\%$$.

This can again be illustrated by a simulation in R:

```
> sd(c(rnorm(1e7,sd=.15),rnorm(1e7,sd=.2)))
[1] 0.1767796
```

## Answer by hotsource (score 1)

https://quant.stackexchange.com/a/16213

So you have the vol of the first half and second half of the return series. Assume mean of returns are zero:

Vol of first year and second year: $$ \sigma1^2 = sum(R_1i^2)/openDaysYear1; $$ $$ \sigma2^2 = sum(R_2i^2)/openDaysYear2; $$

Vol of the entire series: $$ \sigma^2 = sum(R_i^2)/nbDaysInTwoYears $$ $$ = (\sigma1^2 *openDaysYear1 + \sigma2^2 * openDaysYear2)/(openDaysYear1 +openDaysYear2) $$

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.

Combining Period Volatility and Annualizing Across Years

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