I have the following well-known formula for Fama and French's three-factor model
My question is: Why do we not take the size (market capitalization) of the firm itself instead of constructing a risk factor based on the sizes of other firms and why are the correlation between the firm's excess return and the size factor (which concerns the other firms) an explanation for the size?
Most explanations concerning the size factor are about the size of the firm itself. For example, small firms are more riskier because their small size makes them more susceptible to outside influences. However, it is not the size of the firm that is featured in the excess return equation, but the return on a portfolio of other firms based on size. So what explanation could you give why the correlation with this portfolio yields a higher return.
In this question I have focused on the size factor but my question also holds for the value factor. Could someone help me out? Thanks in advance!