Compute monthly realized variance for Fama-French factor

Question

I need to compute monthly realized variance from daily data for Fama-French factors. Knowing that Fama-French factors is the difference of return between different type of stocks, for example SMB factor is the difference between returns on portfolios of small stocks and portfolios of big stocks, should I again take the sum of squared return from those factors to compute realized variance? I mean take the return from the data that is already return to compute monthly realized variance? Or is it better to follow the formula in Moreira, A., & Muir, T. (2017) (pictured below) to compute the monthly realized variance?

Answer 1

Fama-French factors are already returns so you should not calculate the returns on the Fama-French factors but simply calculate the variance in the normal way (by squaring the difference of each datapoint from the mean and summing).

Recall that the core of the Fama-French model is the equation $$ r=r_f+\beta_1(r_m-r_f)+\beta_2(SMB)+\beta_3(HML)+\epsilon $$ See the CFI website for definitions of terms and more details.

Since each $\beta$ is a dimensionless number, the quantities $r_m-r_f$, $SMB$ and $HML$ are in the same units as $r$. This supports the idea that, as you correctly point out, Fama-French factors are returns.

Therefore, if you wish to compute some statistic (e.g., variance) of $SMB$ you can proceed as if you were computing that statistic for returns.