# Comparing annualised volatility from monthly and annual data

I fear there is a very simple answer to this question and its killing me that I can't see it. I am interested in calculating historical volatility:

I have monthly index values starting in Jan 2005 and ending in August 2013. I have calculated Ln(m+1/m) for each month. I have also calculated the year-on-year returns (8 data points in total for my data set).

If I calculate the standard deviation of the monthly return series I get 1.74%. On an annualised basis, I make it 1.74%*sqrt(12)=6.01%. The standard deviation of the annual series is 16.25% (i.e the standard deviation of my 8 data points).

can someone please give me an explanation as to why the annualised volatility values are so different? I know the sqrt(t) rule is an approximation and subject to constraints such as i.i.d and no autocorreclation of the series but should it not be the case that:

standard deviation of monthly returns*sqrt(12) approx = standard deviation of annual returns

Thanks in advance for any help!!

## 1 Answer

Returns are percentages, therefore the montly returns need to be multiplied, not added:

$$\%R_{\text{year}_i}=\prod\limits_{i=1}^{11}L_i-1$$ Hence, the variance of the annual returns is the variance of the product of 11 iid rvs.

The effect of compounding will increase the variability of your returns. What you are modeling are simple returns, which would happen if you cashed-out your returns each month, keeping the principal untouched (i.e, you could have a large IOU at the end of the year, or go into debt to keep the interest-bearing value the same each month)

• Does this take into account that data are logarithms of ratios of values? – Henry Aug 7 '17 at 17:30