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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?

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    $\begingroup$ Is shorting allowed? $\endgroup$
    – Bob Jansen
    Jun 24 '20 at 20:19
  • $\begingroup$ Yes, shorting is allowed. $\endgroup$
    – Darby Bond
    Jun 25 '20 at 12:42
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This is not possible because if you want a portfolio with an expected return higher than 10% you need an asset with an expected return higher than 10%. However both asset A and B are lower or equal to 10%. So you can’t create the portfolio, or is leveraging allowed?

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  • $\begingroup$ Let's say leveraging were allowed, would you still be able to create a portfolio with a lower standard deviation than 5% and a greater expected return than 10%? $\endgroup$
    – Darby Bond
    Jun 25 '20 at 7:13
  • $\begingroup$ With leveraging, you can create a portfolio with an greater expected return but you will also increase your risk so you won’t have a standard deviation lower than 5%. So it’s true that even with leveraging you won’t be able to create the portfolio you want. $\endgroup$
    – Kben59
    Jun 25 '20 at 16:48
  • $\begingroup$ That's not true @Kben59. If the correlation is negative your risk will increase but if the correlation is positive then you can achieve a lower risk. $\endgroup$
    – Oscar
    Jun 25 '20 at 16:59
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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":

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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.

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