# 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.)

• Although I don't have a reference, I believe it has been proved in Portfolio Theory that the efficient frontier can be generated by linear combinations of two portfolios on the efficient frontier. – Alex C Jan 29 '19 at 16:11
• @Alex C, even I remember reading somewhere that it has been proved. But I am not sure. Please upvote the question so that it gets some traction. If we aren't sure about the answer, I think it deserves it. – Dhruv Gupta Jan 29 '19 at 16:41
• This is known as the Two Fund Theorem and there is a proof here mycourses.aalto.fi/pluginfile.php/554618/mod_resource/content/5/… – Alex C Jan 29 '19 at 16:45
• To split hairs, when we say "cover the efficient frontier", we mean "can achieve the mean and volatility of any point on the frontier." Obviously, if the covariance matrix is rank deficient, the portfolios on the efficient frontier could be described as the linear combination of $k>2$ different portfolios. – steveo'america Apr 9 at 17:20 