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I am a little confused about the units of the variance of returns. One way to compute that would be to look at the units of returns-

$$r=\frac{1}{\Delta t}\ln\frac{P(t+\Delta t)}{P(t)}=\text{Dimension }(\text{time})^{-1}$$

$$\text{Cov}(r_i,r_j)=E[(r_i-\bar{r}_i)(r_j-\bar{r}_j)]=\text{Dimension }(\text{time})^{-2}$$

But the above seems incorrect. For starters variance scales with time, i.e. annual variance is 12 times the monthly variance (assuming iid returns). Also stochastic calculus tells us that $\sigma$ or standard deviation scales with $(\text{time})^{-1/2}$. How do I reconcile that with the above?

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There is nothing incorrect with your formulas, so let's look at the units when you annualize the volatility.

As an example, assume you have 252 daily return data. Their dimension is $(time)^{-1}$ and their variance is given in $(time)^{-2}$ (as you already stated). You may look at Chris Taylor's answer here on the underlying assumptions, why one can annualize volatility as $$\sigma_{\rm annual} = \sigma_{\rm daily} \times \sqrt{252}.$$

In fact, the scaling factor $\sqrt{252}$ is expressed with no unit. This arrises from the central limit theorem:

When using log-returns, the annualized return is calculated as $R_{year} = r_1 + r_2 + \cdots + r_n$ with $n=252$ daily log-return data. This sum of daily return data converges towards a normal distribution with parameters $N\left( n \mu ,n\sigma_{daily} \right)$, where $\mu$ is the mean of daily returns. This scaling with the factor $n$ is done without any unit. So on both sides of the above formula you have the unit $(time)^{-1}$ for the standard deviation of returns and the daily one scaled dimensionless by $\sqrt{252}$.

In a statistical meaning, the factor $n$ is just the number of data points you observe, although they represent single points in time in the context of financial calculus.

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