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Assume I've got a portfolio "A" with an expected return of 14% and a volatility of 20% and my broker suggests to add a new share "H" to my portfolio which has an expected return of 20%, a volatility of 60% and a correlation of zero to my portfolio. Risk-free return is 3,8%.

I'm following my broker's advice since according to CAPM, we have a Beta of zero ($\beta_H = \frac{\sigma_H\sigma_A\rho_{H,A}}{\sigma_A^2}=0$) and are thus expecting a return of at least 3,8% ($E(r_H)=r_f+\beta_H(E(r_A) - r_f)=3,8\%$ and $20\% \gt 3,8\%=E(r_H)$).

Now, I've invested 40% of my portfolio into the new stock H. I've been told this is too much since

$$ \beta_H^A=\frac{\sigma_H\rho_{H,A}}{\sigma_A}=\frac{Cov(R_H,w_HR_H+w_AR_A)}{\sigma_A^2}=\frac{w_H\sigma_H^2+0}{\sigma_A^2}=2$$

and thus stock A adds a lot of systematic risk to my portfolio. I don't understand this explanation, what is the meaning of $\beta_H^A$ here? I only know how to interpret Beta in relation to a market portfolio but in this case, Beta is calculated as the change of systematic risk of stock A in relation to the new portfolio that includes stock A.

It seems like Beta is used as a measure of how much systematic risk is added to my portfolio by the new stock. I only know beta as a measure of sensitivity to the market. I don't understand the calculation and this interpretation.

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    $\begingroup$ It is not proper to compute Beta with respect to a given portfolio A and then apply the CAPM. That is not how the CAPM Beta is defined. (It is defined vis a vis the MArket Portfolio). And this is assuming CAPM is valid, which many doubt (see answer below) $\endgroup$
    – nbbo2
    Commented Jan 7 at 12:47
  • $\begingroup$ Generally the attractiveness of adding a security to a well diversified portfolio is measured by the Alpha (not the Beta). $\endgroup$
    – nbbo2
    Commented Jan 7 at 12:55

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Not directly answering your question, but is this theoretical or did you actually do this? If so, I hope you do know that beta is not stable over time, only measures a linear relationship and that the CAPM is generally highly questionable?

In the words of Fama & French, The Capital Asset Pricing Model: Theory and Evidence, Journal of Economic Perspectives—Volume 18, Number 3—Summer 2004—Pages 25–46,

The problems [of CAPM] are serious enough to invalidate most applications of the CAPM. The CAPM, like Markowitz’s (1952, 1959) portfolio model on which it is built, is nevertheless a theoretical tour de force. We continue to teach the CAPM as an introduction to the fundamental concepts of portfolio theory and asset pricing, to be built on by more complicated models like Merton’s (1973) ICAPM. But we also warn students that despite its seductive simplicity, the CAPM’s empirical problems probably invalidate its use in applications.

Wikipedia offers a good collection of problems of the CAPM model.

It is the second highest entry in the list of the most dangerous concepts in quantitative finance work on Quantitative Finance SE.

Irrespective, adding 40% of your portfolio into a single stock is never a good idea unless you are absolutely sure that this stock will be the best you can find and don't believe in diversification.

Ultimately, I think this answer is probably best asked on money stack exchange.

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