Factor investing papers typically report CAPM alphas for factor-mimicking portfolios (e.g. SMB, HML) or for quintile/decile portfolios formed by single or double sorts to show the outperformance of these strategies.
I'm interested in knowing what my absolute (not risk adjusted) excess (over market) returns for a dollar invested in such portfolios would be. I.e., returns on a dollar invested in these portfolios vs a dollar invested in total market.
How should I convert reported CAPM alphas to this absolute excess return over a market?
CAPM says that:
Ri - Rf = ai + bi(Rm - Rf)
where ai is the alpha of the portfolio and bi is the beta (or regression slope against market returns).
By subtracting Rm - Rf from both sides, it should be:
(Ri - Rf) - (Rm - Rf) = ai + (bi - 1) (Rm - Rf)
However bi is calculated ex post, and is typically not reported in papers. Is there another obvious or approximate relation between these two quantities that I'm missing?
Thanks!