# What are the units of the variance of returns?

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?

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.