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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 $M$?

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