# 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,

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.