# 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!

• Letting $X_i$ represent rate at year $i$ and remember rate from year 1 to year 2 would be $(1+X_1)(1+X_2)-1=X_1X_2+X_1+X_2$ (if compounded meaning you reinvested all of your earnings and original amount at from year 1). Thus want to find $Var(X_1X_2+X_1+X_2)$ Now if we assume that rates from year one to year to are independent and note that $X_1X_2$ is normally quite a small number we would have $Var(X_1X_2+X_1+X_2)\approx Var(X_1)+Var(X_2)$ – Kamster Jan 15 '15 at 18:31
• I would like to note I don't know if this is necessarily true but what I think is if anyone knows my analysis is wrong please correct me – Kamster Jan 15 '15 at 18:34
• If any of the answers were helpful please upvote and accept one of them. I have noticed that you haven't cast a single vote since you joined. Feedback is very valuable for the community - Thank you :-) – vonjd Feb 17 '15 at 17:55

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


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: $$\sigma^2 = sum(R_i^2)/nbDaysInTwoYears$$ $$= (\sigma1^2 *openDaysYear1 + \sigma2^2 * openDaysYear2)/(openDaysYear1 +openDaysYear2)$$

• Thanks, but what if I also have the return data? Does that change the calculation (then I don't have to assume the mean is zero). I've tested this on data from Apple on 2013 and 2014, but I get a different volatility when I calculate it at once based on the returns for the whole 2-year period and when I calculate it based upon the volatily in year and in year 2 (via the method in your answer). – AltTabsen Jan 15 '15 at 0:06
• =STDEV(returns from 2013)*SQRT(252 trading days) = volatility of 28.9% ||||| =STDEV(returns from 2014)*SQRT(252 trading days) = volatility of 21.7% ||||| =STDEV(returns from 2013 and 2014)*SQRT(504 trading days) = volatility of 36.1% ||||| (28.9% * 252 + 21.7% * 252)/(252+252) = 25.2% – AltTabsen Jan 15 '15 at 0:14
• If the mean return is not zero the formula of course has to take them into account - probably for the case of aapl. – hotsource Jan 15 '15 at 14:17