# What is the canonical reference for Minimum Variance Portfolio's uniqueness?

I am writing a white paper in which I am trying to compare a strategy to different well-known - and classic - asset allocation optimization approaches.

One of the methods I chose is the minimum variance portfolio $w_\text{MV}$ defined as follows:

$$w_\text{MV} = \underset{w}{\arg \min} ~ w' \Sigma w$$

where $\Sigma$ is the covariance matrix of the assets and under the linear constraints $Aw \leq b$ and $E w = d$.

I have always heard that the MV portfolio was unique, and I know that this problem is linked to quadratic programming which I believe guarantees a unique solution as long as $\Sigma$ is positive-definite.

I wanted to add a reference to another paper where this uniqueness was discussed (proved), and I found several ones written quite recently. However, I was wondering if there was one paper thas was more famously known for discussing that particular property?

• You should look at papdog's comment. When I read you question, my first thought was "Merton, has to be Merton..." At least on the econ side, he's the big name attached to early portfolio choice research. Jan 17, 2013 at 1:54

For academic references, you will likely have to look in the very early optimization literature.

Uniqueness of the MV portfolio follows immediately from the lemma that a strictly convex function on a convex set has no local minima.

The standard textbook reference is Convex Optimization by Boyd and Vandenberghe. See section 4.2.2 in particular. A free online copy is available at stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf.

• Actually I don't think that's true. Uniqueness of a minimal variance is guaranteed by convexity but nothing says that there might not be several portfolios giving the same minimal value.
– SRKX
Jan 13, 2013 at 13:32
• That's not correct; if the covariance matrix is strictly positive definite then the variance is a strictly convex function of the portfolio weights, so it can only have one minimum on the (convex) constraint set. If you had two distinct min-var portfolio weights, then all linear combinations of those two would also have the same minimum variance, contracting strict convexity. Jan 13, 2013 at 13:41
• Yes, but the covariance matrix is not strictly positive definite, it is positive semi-definite isn't it?
– SRKX
Jan 13, 2013 at 14:35
• If you could add references to what you just sayed it would be great by the way (not to say it's wrong but it was the base of my question).
– SRKX
Jan 13, 2013 at 14:37
• If your covariance matrix is only positive semi-definite, that would mean there are non-zero portfolio weights so that the variance of that portfolio is zero. I don't know what kind of universe of securities you're considering, but in the equity space that doesn't usually happen. Jan 13, 2013 at 15:33

This article by Eric Falkenstein is exactly what you are looking for:

Early Low Vol Literature Now Everywhere

EDIT
Falkenstein has a new post out on the academic origins of the approach: Here

• I had a look at the blog posts, but none of them really mention the discussion of the uniqueness of the MV portfolio do they? This really the main point of my question in fact...
– SRKX
Jan 11, 2013 at 10:03
• No, not in the posts themselves but I thought in the papers that are referenced there. I will check on that... Jan 15, 2013 at 9:44

If short sell is allowed, I remember there's a unique analytical solution, otherwise it has to be solved numerically. Is your approache different? IMHO the issue of min variance approach is really not how to solve this constrained optimization problem, but how to estimate asset return and var/covar matrix accurately.

• Well, I was looking for a reference...
– SRKX
Nov 20, 2012 at 13:24
• JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS September 1972 AN ANALYTIC DERIVATION OF THE EFFICIENT PORTFOLIO FRONTIER Robert C. Merton* In this paper, the efficient portfolio frontiers are derived explicitly, and the characteristics claimed for these frontiers are verified. Nov 26, 2012 at 1:37