Corner Portfolios and Constraints on the Efficient Frontier
Summary
The document explains how corner portfolios arise when portfolio optimization includes inequality constraints, such as prohibiting short sales. As expected return changes along the efficient frontier, asset weights shift; when a weight reaches zero, the constraint can force an asset out or allow another asset to enter. These points mark changes in portfolio composition and are called corner portfolios.
With only the budget constraint that weights sum to one, short positions are permitted and the document says any frontier portfolio can be formed as a linear combination of two frontier portfolios. With additional inequality constraints, the corner portfolios partition the frontier into regions with different active asset sets, so the unconstrained two-fund argument does not transfer unchanged across those regions. The Critical Line Algorithm is identified as a way to trace the frontier and locate corners. The discussion is conceptual and brief; it does not provide a derivation, numerical example, or a full treatment of distinctions between the minimum-variance and efficient portions of the frontier.
Key ideas
- Corner portfolios occur when inequality constraints change which assets can have nonzero weights.
- A weight reaching zero can mark an asset leaving or entering the constrained portfolio.
- Without inequality constraints, the document describes frontier portfolios as linear combinations of two frontier portfolios.
- The Critical Line Algorithm can scan the frontier to identify corner portfolios.
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Full text
# Corner portfolios # Corner portfolios This is more a theoretical problem rather than a technical one. I am looking for a clear and rigorous definition of corner portfolios and I like to understand more precisely their relation with the mean-variance/minimum variance frontier. I noticed divergent views while reading about this topic and the related two-funds theorem (i.e. this question). - Does the two-funds theorem apply to the entire minimum-variance frontier or just to the efficient frontier? In other words both frontiers can be derived with a linear combination of portfolios suitably chosen? - Is it correct asserting that deriving the efficient frontier by recurring to the two-funds theorem requires the use of two efficient portfolios that are also corner portfolios for the linear combination? - Is it correct that the two-funds separation theorem is valid only when there is no corner portfolio between the two used in the linear combination at least when deriving the efficient frontier? (Efficient Asset Management by Michaud) Unfortunately, in some books and papers "mean-variance optimal" and "mean-variance efficient" are used interchangeably. ## Answer by nbbo2 (score 2, accepted) https://quant.stackexchange.com/a/61835 A key consideration is whether the only constraint is "the weights sum to 1" or whether there are additional constraints (for example positive weights constraints, i.e. no shorting). The corner portfolios arise in the second case only (the extra constraints problem). As we move across the efficient frontier, the weights change. At some point a weight might decrease to zero and if we move further would become negative. That is a corner point or corner portfolio. The asset whose weight has decreased to zero must be kicked out of the portfolio to satisfy the positivity constraint (or other inequality constraint). Or at some point another asset can be brought in with $\ge 0$ weight. So the names of the assets in the portfolio change at each corner point, with a name dropping out or a new name coming in. On the other hand in the no inequality constraint problem every portfolio consists of all $n$ stocks (some with positive weights, some negative) and here it is true that an arbitrary portfolio on the frontier can be formed from linear combination of any two frontier portfolios. As mentioned in the post you linked the corner portfolios can be found using the Critical Line Algorithm invented by Markowitz, which scans the entire frontier looking for these portfolios.
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