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I run the following regression:

$$r_i-r_f=a+b(r_m-r_f)+e $$ Now I see that the alpha is not zero and I construct two different zero cost portfolios:

1) 100% long in ($r_i-r_f$) and -b short in ($r_m-r_f$)

2) 100% long in $r_i$ -b short in r_m and borrow/short (1-b) on$ r_f$

I have problems determining the expected returns of the two portfolios and determining if they are the same.

I would appreciate your help a lot!

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I'm not sure I follow, but do the two portfolios mean the following?

For portfolio (1), $$ r_i - r_f - b(r_m - r_f) = r_i + (b-1) r_f - b r_m $$ and for portfolios (2) $$ r_i - b r_m - (1-b) r_f = r_i + (b-1) r_f - b r_m. $$ So, writing them out like this, they look the same.

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