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?