Calculating Variance from Fama-French Factor Returns

Summary

The document clarifies how to calculate variance for daily Fama-French factor data aggregated over a month. Because factors such as SMB are already expressed as portfolio return differences, they are returns in their own right; there is no need to calculate another return series from them first.

To estimate variance, treat the factor observations as return observations and compute the usual variance around their mean, using squared deviations. The factor-model equation supports this interpretation: factor coefficients are dimensionless, so the factor terms share the units of the modeled return. The answer distinguishes this ordinary sample variance calculation from summing squared returns, which is not the stated procedure. The discussion is brief and does not specify choices such as sample versus population normalization, annualization, or other realized-variance conventions, so those details depend on the intended analysis.

Key ideas

• Fama-French factors are returns, including factors formed as differences between portfolio returns.
• Compute a factor’s variance from its observations by summing squared deviations from their mean.
• Do not calculate a second return series from factor data that are already returns.
• The factor-model equation is consistent with factors sharing the units of the modeled return.
• Normalization and annualization choices are not specified and must be set for the analysis.

Tags

# Compute monthly realized variance for Fama-French factor


# Compute monthly realized variance for Fama-French factor












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 by Jamie Ballingall (score 1)

https://quant.stackexchange.com/a/74345

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.