# What is the meaning of Beta of an individual asset in relation to a portfolio, not the market?

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.

• 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) Jan 7 at 12:47
• Generally the attractiveness of adding a security to a well diversified portfolio is measured by the Alpha (not the Beta). Jan 7 at 12:55