# CAPM derivation - active portfolio management

In the book Active Portfolio Management by Grinold and Kahn, the author presents the derivation of CAPM. I am rather confused by the notation of $$\beta$$ in this derivation and hope to seek some help. On page 27, the author defines $$\beta$$ to be the vector of asset betas. I came to notice that when talking about beta, one has to specify a portfolio that the beta is with respect to, but this is not specified here. On page 31, the author assumes $$\beta$$ to be the attribute defined by some benchmark portfolio $$B$$. So my understanding is that $$\beta$$ throughout the derivation is the beta with respect to portfolio $$B$$, is this correct? On page 34, in line (2A.36), the author mentions $$\beta_{w.r.t. Q}$$, and in line (2A.37), the author mentions $$\beta_Q$$. My understanding is that these are the same, and in general, the exposure to a portfolio is the same as the beta with respect to that portfolio, is this correct? Finally, on page 38, in line (2A.49), the author again mentions $$\beta$$ here. Is this still the asset betas with respect to portfolio $$B$$? If so, what is portfolio $$B$$? Or is this the asset betas with respect to the market portfolio $$M$$?