Removing Short Positions in a No-Short Minimum-Variance Portfolio
Summary
The document explains why adding another stock does not always reduce the variance of an unconstrained minimum-variance portfolio. In its example, an added stock is highly correlated with and less attractive than an existing holding; the unconstrained solution shorts the new stock while increasing the position in the existing one, using the short position as a hedge. The example reports lower variance for the portfolio that includes the short position.
For a no-short constraint, the answer describes excluding a stock with a negative unconstrained weight and resolving the optimization, then gives an iterative procedure that adds eligible stocks to an active set and rejects those with negative weights. This offers intuition and a proposed workflow, but the document does not show a proof of the procedure or resolve the question about whether removing several negative-weight stocks in different orders always gives the same result. The method’s scope and assumptions are not fully discussed.
Key ideas
- Adding an asset does not necessarily lower minimum portfolio variance when it is correlated with existing holdings.
- An unconstrained optimizer may short one asset to hedge a larger position in another.
- The answer proposes removing negative-weight assets and re-solving for a no-short portfolio.
- The document gives an iterative active-set procedure but does not prove order independence for removing multiple assets.
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# Is this methodology for finding the minimum variance portfolio with no short-selling sound? # Is this methodology for finding the minimum variance portfolio with no short-selling sound? I have below here an excerpt from a book on (among other things) mean-variance analysis showing how to find the minimum variance portfolio (Risk and Portfolio Analysis: Principles and Methods, by Hult, Lindskog, Hammarlid, and Rehn). I am confused by the statement here saying that if short-selling isn't allowed, you can find the constrained minimum-variance portfolio simply by removing the offending equity and trying again. This goes against my intuition which says that unless your equities are perfectly correlated, you will always see a reduction in risk by diversification, so how can it that be that the portfolio with only 3 stocks has a lower variance than all possible portfolios including a 4th stocks? Further, using the methodology described here, if you have more than one equity that is given a weight less than zero would it not possibly affect your results if you remove them in different orders and try again? Or can you simply remove all of them together right away? Is it possible to see (or show) that the end result will be the same no matter if you remove them 1 by 1 or all together at once? ## Answer by nbbo2 (score 2, accepted) https://quant.stackexchange.com/a/54069 The intuition that "if I have an N stock portfolio and an (N+1)th stock becomes available, buying some of it will lower portfolio variance" is not correct. It is true if all stocks are uncorrelated, or if stock correlations are low. But it can fail in general, as the example given in your book demonstrates. Suppose you initially invest in stocks 2,3,4. The minimum var portfolio is [0.27,0.17,0.56] and the variance is 0.015736 ($\sigma=$ 0.125445). Now we add Stock 1, which has a higher std deviation that Stock 2 but has a high correlation with it (in this case 0.6). Essentially Stock 1 is a possible substitute for Stock 2 but it is an inferior substitute since it has a higher std deviation. The first thought might be "OK, so it is not a good idea to buy Stock 1 and decrease our holdings of Stock 2, that would worsen the variance". But it goes beyond this: it is actually advantageous to short Stock 1 and buy more of Stock 2. Essentially a short position in Stock 1 is being used as a hedge for the extra Stock 2 that you are buying. The optimal min variance portfolio turns out to be [-0.3115727,0.448340977,0.372268681,0.490963043] with a short position in Stock 1 and an increased long position in Stock 2. The variance of this portfolio is lower: 0.0121392 ($\sigma=$ 0.110178). If short positions are not allowed the procedure described in the book is correct: When there is 1 short position in the unconstrained portfolio, do not allow that Stock and re-solve the problem without that stock. This will give the optimal no short portfolio. It also suggests the following general procedure for finding the no-short min variance portfolio of N stocks: Step 1. Start with 2 stocks Step 2. Find the (unconstrained) min var portfolio of your stocks Step 3. If a negative holding appears, reject that stock, it should not be considered further. Step 4. Add another stock not previously examined to the active set of stocks and go back to Step 2. If there are no such stocks Stop.
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