Let's take a look at the standard CAPM: $$ r_{i} -r_F = \alpha+\beta(r_{MKT}-r_F) + \varepsilon $$ I would like to consider the alternative formulation: $$ r_{i} = \alpha+\beta(r_{MKT}-r_F) + \varepsilon $$ here the return of the asset $i$ is not corrected for the risk free rate. Is there any literature that refers to this formulation? What is the difference in the economic interpretation of the estimates?

  • $\begingroup$ If $r_F$ does not appear anywhere, then you have the single index model due to Sharpe. I have not seen a case where $r_F$ is subtracted on one side but not the other. $\endgroup$
    – nbbo2
    Jun 28, 2022 at 15:35
  • $\begingroup$ Yes the latter is my case $\endgroup$
    – Barbab
    Jun 28, 2022 at 15:47

1 Answer 1


I think that if you do a regression with these two equations, the beta will be the same, but what will change is of course the "alpha". In first equation the alpha is the well-know alpha in the industry, which means excess-return or abnormal return. In the second equation the alpha is not anymore only the excess-return, but the excess-return + the risk-free rate (rf). The proof is simple as you only have to do + rf to the first equation.

  • $\begingroup$ Of course, two questions: here the alpha is excess return + risk free rate which can be decomposed in market risk free + specific idiosyncratic risk free relative to the considered asset? So if the asset perceived risk freeness equals the market risk free rate, than alpha - EV(Rf) gives the well-known alpha (abnormal return), if not the alpha will be market risk free rate + perceived specific risk free rate on the individual stock? $\endgroup$
    – Barbab
    Jun 29, 2022 at 8:20
  • $\begingroup$ In both cases the residuals can be interpreted as idiosyncratic shocks, they are equal if the market risk free rate equals the perceived specific risk freeness of the asset, in the case where the perceived specific risk free rate on the asset does not match the market risk free rate they are idiosyncratic shocks net to the market risk free rate and the perceived specific risk freeness on the individual stock? $\endgroup$
    – Barbab
    Jun 29, 2022 at 8:24
  • $\begingroup$ Where the perceived risk freeness of the individual stock can depend on many factors such as age, size, profitability, earnings persistence, industry... $\endgroup$
    – Barbab
    Jun 29, 2022 at 8:26

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