# Is it possible to make a portfolio with higher expected return and lower standard deviation than constituent securities?

Assume we are working in the framework of modern portfolio theory. Now, let's say we have two securities (they could also be portfolios themselves) A and B. Portfolio A has expected return 10% and standard deviation 10% while portfolio B has expected return of 5% and a standard deviation of 5%.

Is there a way to combine A and B, to make a portfolio C that has higher expected return than 10% and lower standard deviation than 5%. Are conditions on the covariance matrix that would make this possible?

• Is shorting allowed? Commented Jun 24, 2020 at 20:19
• Yes, shorting is allowed. Commented Jun 25, 2020 at 12:42

If short selling is not allowed then no, clearly you can never get a higher expected return than by putting more weight in a security with a lower expected return. If short selling is allowed then you can immediately get a higher rate of return by shorting some of portfolio B to allow you to buy more of portfolio A. Whether or not this new portfolio will have a lower standard deviation than portfolio B depends on the correlation between the two portfolios, specifically can you find some weights $$\bar{w} = (w_A, w_B)$$ with $$w_A$$ negative such that $$\bar{w} \Sigma \bar{w}^T < \sigma_A$$ Where $$\Sigma$$ is the covariance matrix of portfolio A and B. This resembles the (what I've heard called) minimization-of-variance problem in quadratic optimization where you aim to minimize variance under the constraint $$\bar{w} R > \mu_0$$ for a given volatility $$\sigma_0$$. A formulation is given in "Risk and Portfolio Analysis Principles and Methods":
I'm not able to find any closed form solution online to this problem however so I'm not sure what the general rule would be for it to hold, certainly there is a solution for some values of $$\sigma_A, \sigma_B, \mu_A, \mu_b, \rho_{A,B}$$ and breaks down at a certain point.