Why Two Efficient Portfolios Span the Mean–Variance Frontier
Summary
The document asks whether linear combinations of two efficient portfolios in a universe of risky assets cover the efficient frontier. One response reports a numerical example in which combinations of two frontier portfolios, including combinations with short positions, appear to remain on the frontier. Another response describes a plotted combination of the global minimum-variance portfolio and a second efficient portfolio as an approximation, so the examples do not provide a unified proof.
In the standard unconstrained mean–variance framework, the minimum-variance portfolio for a target return varies affinely with that target. Consequently, affine combinations of two distinct portfolios on the efficient branch can trace the branch, provided the portfolios are valid points on the same frontier and weights sum to one; combinations outside their interval may require short positions. Portfolio constraints or other departures from the standard assumptions can change this result. The document’s plotted evidence is illustrative, and it does not lay out the mathematical assumptions needed for a general conclusion.
Key ideas
- In the standard unconstrained mean–variance setting, frontier portfolios vary affinely with target return.
- Affine combinations of two distinct portfolios on the efficient branch can trace that branch when weights sum to one.
- Combinations extending beyond the two selected portfolios may require short positions.
- Constraints on portfolio weights can alter the spanning result.
- The document offers conflicting plots and does not provide a proof or fully specify assumptions.
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# Do linear combinations of two efficient portfolios cover the entire efficient frontier? # Do linear combinations of two efficient portfolios cover the entire efficient frontier? Note : We are considering the case of N risky assets. I think the answer is 'Yes', although I am not sure as I am unable to prove it. The reasons for me thinking that the answer is 'Yes' are - 1) The two portfolios being considered are efficient, so they obviously lie on the efficient frontier. 2) We know that the linear combinations of any two portfolios form a parabola in the E-V space. So as a special case, the linear combinations of the two efficient portfolios being considered by us will also form a parabola in the E-V space. 3) The Efficient frontier for N risky assets is also a parabola in the E-V space. 4) So the only way the answer to my original question is 'No' is when the parabolas in (2) and (3) are not the same, which I think won't be possible geometrically. (I think so because if the parabola in (2) is different than the one in (3), it will have to be below the one in (3), so that it stays in the efficient frontier, but at the same time pass through the two efficient portfolios being considered.) ## Answer by Dan (score 1) https://quant.stackexchange.com/a/78161 Here is an example for five stocks: The black concave is the efficient frontier as estimated in R. I have by the power of my brain conjured two portfolios that always seem to lie on the efficient frontier. These are the green circles on the efficient frontier. I have then computed linear combinations of the two efficient portfolios marked by purple circles. The interior purple circle (between the two green circles) is a positive linear combination of the two portfolios (both long 50%). The exterior purple circles are long one of the green circle portfolios and short the other (the sum of their shares adding up to 1). All linear combinations lie on the frontier. Regards, Dan ## Answer by crl_qjx (score -1) https://quant.stackexchange.com/a/63303 I am not a math guy but I tried to plot the efficient frontier using the linear combination between the global min var portfolio and another efficient portfolio and I have this result: Points represent efficient portfolios from variance minimization given a target return The linear combination looks more like an approximation
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