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Lets say that I have a stock with annual returns, $a_i $ for year $i\in \left\{1,...n\right\}$ and monthly returns $m_{i,j}$ for month $j\in \left\{1,...12\right\}$. Lets define monthly returns to be equal each month within a year so $m_{i,j} = \frac{a_i}{12}$.

Let $\bar{a}$ and $\bar{m}$ denote the means of yearly and monthly returns respectively so $$\bar{a}=\frac{1}{n}\sum_{i=1}^na_i\quad\text{and}\quad{}\bar{m}=\frac{1}{12n}\sum_{i=1}^n\sum_{j=1}^{12} m_{i,j}=\frac{1}{12n}\sum_{i=1}^n\sum_{j=1}^{12}\frac{a_i}{12}=\frac{1}{12n}\sum_{i=1}^na_i=\frac{\bar{a}}{12}$$

Then the monthly volatlity of returns is: $$\begin{align*} \sigma_{m} &= \sqrt{\frac{1}{12n}\sum_{i=1}^n\sum_{j=1}^{12} \left(m_{i,j}-\bar{m} \right)^2}\\ &= \sqrt{\frac{1}{12n}\sum_{i=1}^n\sum_{j=1}^{12} \left(\frac{a_i}{12}-\frac{\bar{a}}{12}\right)^2 }\\ &= \sqrt{\frac{1}{12n}\left(\frac{1}{144}\right)\sum_{i=1}^n\sum_{j=1}^{12} \left(a_i-\bar{a}\right)^2 }\\ &= \sqrt{\frac{1}{12n}\left(\frac{1}{144}\right) \sum_{i=1}^n12\left(a_i-\bar{a}\right)^2 }\\ &= \sqrt{\frac{1}{n}\left(\frac{1}{144}\right) \sum_{i=1}^n\left(a_i-\bar{a}\right)^2 }\\ &= \frac{1}{12}\sqrt{\frac{1}{n} \sum_{i=1}^n\left(a_i-\bar{a}\right)^2 }\\ &= \frac{\sigma_a}{12} \end{align*}$$

So in other words $$\sigma_a = 12\sigma_m$$. This seems to contradict https://en.wikipedia.org/wiki/Volatility_(finance) where they say generalized volatility in $T$ time periods is $\sigma_T = \sigma \sqrt{T}$. What am I doing wrong?

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closed as off-topic by Quantuple, LocalVolatility, vanguard2k, Bob Jansen Feb 19 '17 at 13:58

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    $\begingroup$ " Lets define monthly returns to be equal each month within a year " it seems like you are regarding (log-)returns as deterministic (I am assuming log because otherwise you could not sum monthly returns to get an annual one). They are not. Thus I don't understand what $m_{i,j} = \frac{a_i}{12}$ means. The assertion you refer to is just that the variance of a sum of iid variables is the sum of their variances. $\endgroup$ – Quantuple Feb 17 '17 at 8:11
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i think you are assuming 100% correlation between returns, whereas for a stochastic process, increments are independent

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