What are the units of the variance of returns?

Question

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?

Answer 1

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.

Answer 2

I agree with @skoestlmeier's answer that the $\sqrt{252}$ should be considered dimensionless. However, using the result from your dimensional analysis (i.e. returns and volatility having dimension $time^{-1}$), you would conclude that the black scholes (BS) formula is dimensionally inconsistent (see this question).

I find it more helpful to think of returns as dimensionless (i.e. ratios) and therefore volatility, given a measurement frequency. A naive analogy would be, if you have a car the travels at constant velocity $10\,m/s$, then it will have travelled $10\,m$ after $1\,s$. The $10\,m$ distance is the way of thinking about returns, given you always use some frequency (1s, 1m, 1y etc..).

Now if you must give them dimensions, then I suggest thinking the notions of returns and variance having a "unit" $time^{-1}$ (much like the velocity in my naive example) since they are usually quoted in annual terms (which is a choice of frequency). As a consequence volatility must have dimension $time^{-1/2}$ which is in agreement with your stochastic calculus source and you have a dimensionally correct BS formula.

(P.S. I do recognise the inconsistency there would be when plugging in the variance formula, so I welcome any better intuitions.)