Volatility of monthly returns annualized vs volatility of annual returns

Question

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?

Answer 1

Let the monthly log return be $m_{i,j}$ where $i$ is the year and $j$ is the month. The monthly volatility is $\sqrt{\mathrm{Var}(m_{i,j})}$ which we assume is constant in time. The annual log return is $a_i = \sum_{j=1}^{12}m_{i,j}.$ The annual volatility is $\sqrt{\mathrm{Var}(a_i)}.$ In order to compute it we must make another assumption: that the monthly returns are uncorrelated. (We don't need to make this precise assumption, but we do need to make an assumption about the correlation and our answer depends on it.) Then we get $$\mathrm{Var}(a_i) = 12\mathrm{Var}(m_{i,j})$$ so taking square roots, the annual volatility is the monthly times $\sqrt{12}.$

Note that this is based on both a time-homogeneity assumption and an uncorrelated assumption. Thus converting volatility across time periods is not a model-independent thing.

Where you go wrong is in assuming $a=m*12$ and $m$ constant. This takes away all the randomness of the monthly returns. If $a$ and $m$ were just linearly related by $a=12m$, then of course the volatility of $a$ is $12$ times the volatility of $m$.