Beta less than 1 despite the market standard deviation is lower than the one of the stock

Question

I have a really dumb question.

As an exercise, I calculated the beta of a stock with respect to the market in which it is listed. However, I noticed that although the standard deviation (and coefficient of variation) of the stock's returns are greater than those of the market, the beta is less than 1.

$$\beta =\frac{cov(\text{returns}_{stock}\,\,, \text{returns}_{market}\,\,)}{var(\text{returns}_{stock}\,)}$$

Dividing the returns year by year, and re-estimating them in this way, despite the fact that SD is always greater than the market, only in some years does the beta appear to be greater than 1. Is this possible?

Answer 1

$$ \beta = \frac{\sigma_{stock,market}}{\sigma^2_{market}} = \frac{\rho\sigma_{stock}\sigma_{market}}{\sigma^2_{market}} = \rho\frac{\sigma_{stock}}{\sigma_{market}} $$ where $0\leq\rho\leq 1$ is the correlation between the stock and the market returns. (Note that your formula for $\beta$ got the denominator wrong.) As you can see, a small enough correlation $\rho$ will keep $\beta<1$ even if $\sigma_{stock}>\sigma_{market}$. More precisely, small enough is $|\rho| < \frac{\sigma_{stock}}{\sigma_{market}}$.