# Why we can't lower a volatility of a portfolio (without changing expected return) by substituting a zero beta stock with a risk free asset?

part of the answer is that a zero-beta stock must be negatively correlated with other stocks in the portfolio. So having a zero beta stock can decrease the volatility. Does that mean that the volatility a zero beta stock is lower than a risk free asset?

• Hi: you can substitute the stock for a risk-free asset but the stock's expected return could be larger than the risk-fee asset in which case you'll be decreasing the variance of the portfolio but also decreasing the expected return. So, you'll be at a different point on the efficient frontier.. – mark leeds Oct 15 '18 at 16:34

If we assume that the starting portfolio is well diversified, then removing a zero beta stock from it will not reduce risk, since the only risk it carries is unsystematic risk but the only risk that matters for the portfolio is market risk. Of course this is only true in the limit of perfect diversification. For a more realistic case (good but not perfect diversification) the risk reduction would be negligible.

Not necessarily. Recall the form of covariance matrix when we calculate variance of whole portfolio (to make it simple say we have two assets $$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2 w_1w_2 cov(1,2)$$)

if your zero-beta stock is negatively correlated with the rest of the portfolio then of course variance is decreased since $$\rho$$ is negative.

However if risk free asset is substituted in, it is usually assumed to have 0 correlation with the rest of the asset, hence the overall return doesn't change (because expectation is calculated by the weight*individual stock return) but variance will increase.

• not a big deal but last term is $2 (1-w_{1}) w_{1} \times \rho \sigma_{1} \sigma_{2}$. and yes, if he doesn't include the new risk free return in his portolio's expected return, then it is possible to lower the variance and not effect the return ( or even improve the return ). But, my thinking is that, if it's going to be substiituted, then it should be included in the expected return also. – mark leeds Oct 15 '18 at 20:16
• @markleeds thanks for the note I've edited the original answer! However If we count risk free return into total return it would still be the same. if we consider a risky asset return to be $r_f + \beta(r_m-r_f)$, you see zero beta return is the same as risk free return – numerairX Oct 15 '18 at 20:38
• Thanks Jojo. I didn't know he was using CAPM to estimate his expected returns but, if he is doing that, then I totally agree with you. I would guess that that's a dangerous business to be in ( predicting returns using CAPM ) but nevertheless, if that's the method being used, then my apologies for confusion and noise. – mark leeds Oct 16 '18 at 2:57
• @markleeds not at all it's always good to have multiple views on these things. I just answered in a way that I think this is the model that can explain the phenomenon he described (constant mean but higher variance) – numerairX Oct 16 '18 at 12:35
• Gotcha and good to meet you. – mark leeds Oct 16 '18 at 16:17