# 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? • A very interesting question. I will conjecture that if there are N negatives in the solution you have to enumerate all $2^N$ ways of fixing weights to zero. But maybe there is a shortcut along the lines you suggest. ... I'll try to think about it... May 9 '20 at 16:28
• Thank you, I think it's worth pointing out that after fixing some weights to zero you may very well end up with M new negative weights in your new portfolio, at which point you'd have to repeat the process and end up with 2^M additional possible portfolios (some of which may also have negative weights). So at the end you may end up with a very large amount of possible portfolios of which you'd chose the one with lowest variance. Possible, sure, but it definitely seems like a task. And I'm still not convinced that will give you the correct portfolio based on the concern in my first paragraph. May 9 '20 at 16:38
• "how can it that be that the portfolio with only 3 stocks has a lower variance than all possible portfolios including a 4th stock". It is not true: the portfolio with only 3 stocks [0,0.27,0.17,0.56] has standard deviation 0.125445008 which is higher than the 4 stock portfolio [-0.311,0.448,0.372,0.491] std dev of 0.11017802. But the second solution is not acceptable because it has a short position. We are forced to accept a higher variance if we don't allow short positions (makes sense). (Still working on a fuller explanation). May 9 '20 at 20:39
• Hey, yes that I understand, I meant how can it be lower than any long only 4 equity portfolio. I suppose I could check that myself but since you seem to already have it plugged in, what is the variance of the 4 stock portfolio with weights ($\epsilon$, $w_2$ * (1-$\epsilon$), $w_3$ * (1-$\epsilon$), $w_4$ * (1-$\epsilon$)) where $w_i$ are as in the 3 equity example and $\epsilon$ is a positive very small number. I would've imagined a portfolio like that to always have smaller variance than the 3 stock portfolio because of the diversification effect. May 9 '20 at 20:40
• I was about to look into that ... ;) May 9 '20 at 20:41

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: