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Usually for CAPM we have:

$$E(R_i)-\gamma=\frac{cov(R_i,R_m)}{var(R_m)}\times E(R_m-\gamma)$$

We know that

$$\beta=\frac{cov(R_i,R_m)}{var(R_m)}$$

When can we replace $\beta$ with $\left(\frac{E(R_iR_m)}{E(R_m)}\right)^2$ ?

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To be honest I don't quite understand the question, but it's here for a week so I'll give a short explanation of CAPM. The CAPM is an attempt to explain the risk and return on an asset i.e. it is a risk/return analysis. I don't understand why do we need to transform beta in first place.

The CAPM Formula is the following:

$E(R_i)=r_f+β_i[E(R_m)-r_f]$ $(1)$

or

$(R_{it}-R_{ft})=α_i+β_i(R_{mt}-R_{ft})+ε_i$ $(2)$

Where,

  1. $E(R_i)$ is an asset $i$ expected return,
  2. $r_f$ is the rate on a risk-free asset,
  3. $E(R_m)$ is the expected return on the market index (benchmark),
  4. $β_i$ is the estimate of risk for asset $i$.

As you may already know, we can either regress the $(2)$ if we are interested on both $α_i$ and $β_i$ or alternatively we just use the formula $β_i=Cov(R_i,R_m)/Var(R_m)$, because $α$ should be close to zero (transaction costs).

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