# Monthly returns annualized vs annual returns [closed]

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?

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

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Basic financial questions are off-topic as they are assumed to be common knowledge for those studying or working in the field of quantitative finance." – Quantuple, LocalVolatility, vanguard2k, Bob Jansen
If this question can be reworded to fit the rules in the help center, please edit the question.

• " 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. – Quantuple Feb 17 '17 at 8:11