Short Selling and Covariance in Portfolio Return and Risk Trade-Offs
Summary
The document asks whether combining two portfolios can produce an expected return above that of the higher-return constituent while also producing volatility below that of the lower-volatility constituent. Its example gives one portfolio a higher expected return and the other a lower standard deviation, then asks how covariance affects the possibility of achieving both targets.
The replies distinguish long-only allocations from portfolios that allow short selling. With nonnegative weights, combining assets whose expected returns do not exceed the higher-return asset cannot exceed that asset’s expected return. Short selling can increase exposure to the higher-return portfolio, but whether the resulting portfolio also has sufficiently low variance depends on the covariance matrix. The response frames this as a constrained quadratic optimization problem, while noting that it provides no general closed-form condition. The discussion is conceptual and does not give a complete solution or account for estimation error, leverage limits, or other practical constraints.
Key ideas
- A long-only combination cannot have expected return above the highest-return constituent when returns are weighted averages.
- Short selling can raise exposure to the higher-return portfolio and increase expected return.
- Whether that leveraged combination lowers volatility depends on covariance between the portfolios.
- The question can be framed as minimizing variance subject to an expected-return constraint.
- The discussion does not derive a general condition or address implementation constraints.
Tags
Full text
# Is it possible to make a portfolio with higher expected return and lower standard deviation than constituent securities?
# 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?
## Answer by Kben59 (score 3)
https://quant.stackexchange.com/a/55200
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?
## Answer by Oscar (score 1)
https://quant.stackexchange.com/a/55205
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.