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I have heard some people say something like the following with regards to APT:

Let returns be given by the factor model

$r_t = B_tf_t + e$

with

$E(f_t) = \lambda_t$

Assume that factors are standardized so their variance is 1. Assume a linear sdf like in APT: $m_{t+1} = - \lambda_t f_{t+1}$

While I understand writing the sdf as a linear function of factors, I'm not sure why the coefficients of factors are price of risk to factor itself?

Is there a justification for this?

I found this note which seems to talk about it, but I'm still not clear on this. Specifically, page 7 top LHS states:

"When the entries of z are standardized to a have conditional variances equal to unity, the entries of λ become the conditional regression coefficients, the “betas” of the asset payoffs onto the alternative observable factors"

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