Why Factor Regressions Use Portfolio Premia to Measure Exposure
Summary
The document explains why Fama–French style time-series regressions use returns on factor portfolios, such as SMB and HML, rather than a stock’s size or book-to-market characteristics directly. Factor portfolio returns represent the payoffs associated with those factors, while regression coefficients estimate how a security’s returns respond to those payoffs. This makes the coefficients analogous to market beta.
It distinguishes estimating factor exposures from testing whether firm characteristics are associated with returns. Cross-sectional regressions on characteristics can address the latter, while time-series regressions on factor returns estimate exposures. The answer also notes that size and value premia need not be compensation for systematic risk: behavioral explanations are possible. Double sorts are described as a way to reduce overlap between the size and value portfolios. The discussion is conceptual and does not provide empirical tests or establish that factor portfolios are uncorrelated in practice.
Key ideas
- Factor portfolio returns serve as regressors for estimating a security’s time-series exposure to factor payoffs.
- A stock’s size and book-to-market characteristics are distinct from its estimated factor betas.
- Cross-sectional characteristic regressions address a different question from time-series factor regressions.
- Size and value premia may have behavioral explanations rather than representing systematic risk compensation.
- Double sorts aim to reduce the influence of one characteristic when constructing the other factor portfolio.
Tags
Full text
# Why do we regress with respect to premiums in factor models like FF? # Why do we regress with respect to premiums in factor models like FF? Factor investing can be explained by factor models, via the factors exposures. For example Fama-French observed that Size and Book-to-Ratio were systematic risks of a portfolio and consequently they led the following regression $$R_i-R_f = \alpha_i + \beta_m(R_m-R_f)+\beta_s SMB + \beta_v HML + \varepsilon_i.$$ However, the thing that I don't get is how this equation allows them to prove that Size and Book-to-Ratio are systematic risks. Why did they regress with respect to SMB (size premium) and HML ? It would have been easier to just regress with respect to their size and book-to-ratio values directly, doesn't it ? Why is it important to know the correlation between a portfolio return $R$ and a SMB/HML portfolio ? Are not we rather more interested in the correlation between a portfolio return $R$ and the size of this stocks instead ? Thank you, ## Answer by Alex (score 1, accepted) https://quant.stackexchange.com/a/54164 Firstly note that Fama and French argue that size and book-to-market represent systematic risk. But they don't have to. There are many behavioural explanations that argue why small stocks outperfom large stocks and why value stocks outperform growth stocks. So, let's assume size and book-to-market carry a premium (regardless whether it's a risk premium or driven by something else than systematic risk). How big are these premia? Well, they are portfolio returns if the portfolios are (only) exposed to the size/book-to-market factor. That's, why Fama and French use their double sorts (to avoid that book-to-market influences $SMB$ and to avoid that size influences $HML$). Ideally, $SMB$ and $HML$ are uncorrelated with each other and the market excess returns. Regressing stock excess returns on $SMB$ and $HML$ gives you the exposure of that stock to the size factor and the value factor, just like market beta tells you how your stock returns relate to changes of the market portfolio. If you run Fama MacBeth (1973) regressions on (log-)market cap and (log-)book-to-market, you'll also find a positive estimate for size and a negative estimate for value. But if you want to find a firm's betas (factor exposures), you cannot regress on it's own size and book-to-market ratio. You, instead, run a time series regression on excess returns on market factor, size factor and value factor. Other popular factors include momentum, profitability, investment (asset growth), low market beta, idiosyncratic volatility, ...
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.