# 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!!

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