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The standard deviation of an asset is less relevant if you hold a portfolio and not a single asset. In the CAPM, every investor holds the same portfolio of risky assets - the market portfolio. The CAPM implies further that the market portfolio has the highest expected portfolio return per unit of portfolio standard deviation of all possible portfolios. ...

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In the CAPM, $E(r_i) = r_f$, if $\beta_i=0$ holds ALWAYS. BUT: recall the calculation of $\beta$: $\beta_i=\cfrac{\sigma_{iM}}{\sigma_M^2}$ and ${\sigma_{iM}}=\Sigma_jw_j\sigma_{ij}$ if your asset is uncorrelated with all other assets in the market, the last expression simplifies to: ${\sigma_{iM}}=\Sigma_jw_j\sigma_{ij}=w_i\sigma_i^2$ so for ...

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No. The CAPM is an equilibrium model. It describes relationships between expectations. With OLS you typically estimate a distribution for realized returns in the future (or sometimes even only in the past...). $R^i_{t+1} - R^f = \delta (R^{MVE}_{t+1}-R^f) + \varepsilon_{t+1}$ is not an equation from the CAPM. The CAPM SML is: \$r+E(r_i - r) = r+\beta_i ...

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The CAPM is by no means a failure. While not a normative theory like MPT, but a positive one describing a capital market equilibrium under conditions of risk, the CAPM, if applied carefully, still provides a reasonable basis for methods to value uncertain cash flows, e.g. in capital budgeting. Further, it is at the core of models for derivatives pricing - ...

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