Finding Corner Portfolios with Markowitz’s Critical Line Algorithm
Summary
The document asks how to find corner portfolios for a mean-variance optimization problem with weight constraints. These portfolios mark transitions along the minimum-variance frontier: within a segment, the same assets are held and their weights change at constant rates. Adjacent corner portfolios can be combined to trace portfolios on the efficient frontier, with the global minimum-variance portfolio serving as one such point.
The responses identify Markowitz’s Critical Line Algorithm as the relevant approach and describe it as an active-set method for quadratic programming. They point to an R implementation and several research papers, including work that calls corner portfolios turning points. The material offers references rather than a derivation or comparison of algorithms, and the cited resources would need to be consulted for implementation details and assumptions.
Key ideas
- Corner portfolios divide the efficient frontier into segments with stable asset membership and linear weight changes.
- Adjacent corner portfolios can be used to trace portfolios along the frontier.
- The global minimum-variance portfolio is a corner portfolio.
- Markowitz’s Critical Line Algorithm is presented as an active-set method for constrained quadratic programming.
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Full text
# Are there any tools or useful algos for identifying corner portfolios? # Are there any tools or useful algos for identifying corner portfolios? Let's say I am performing mean-variance optimization subject to some weight constraints. I'd like to identify the set of corner portfolios so that I can interpolate the entire efficient frontier. A corner portfolio defines a segment on the minimum-variance frontier within which i) portfolios hold identical assets, and ii) the rate of change of asset weights in moving from one portfolio to another is constant. Incidentally, The Global Minimum Variance portfolio is a corner portfolio. Any convex combination of two adjacent corner portfolios is also a portfolio on the efficient frontier. So these corner portfolios can drastically improve the performance of tracing out the frontier. Are there tools in R to identify the corner portfolios, or a research paper on an efficient algorithm to identify the portfolios? Markowitz himself introduced the critical line algorithm, however, I recall Sharpe and others have some approaches as well. R or matrix calculus approaches are preferred but I'll take research citations as well. ## Answer by Marco Breitig (score 5) https://quant.stackexchange.com/a/16467 7 years ago I had to solve the problem of a efficiency frontier under linear constraints on the asset weights and also stumbled upon Markowitz Critial Line Algorithm. I still have a directory with some resources in it. Since Bryce already gave a practical implementation with R code by Eric Zivot, I will concentrate on some papers which might help. I think one of the best papers is Applying Markowitz's Critical Line Algorithm by Andras and Daniel Niedermayer. There you have some nice matrix algebra. The corner portfolios are called turning points. There is also the german paper Einige Bemerkungen zum Critical Line Algorithmus von Markowitz by Detlef Mertens that is comparable to the paper of Niedermayer but has very much on corner portfolios. Also helpful might be the paper Portfolio Optimization: Part 2 - Constrained Portfolios by John Norstad and maybe this dissertation on Active-Set Methods for Quadratic Programming by Elizabeth Wong, too. More generally, the Critical Line Algorithm is an instance of the active set method in quadratic programming. You can search in this direction to find much more. For example in this slides from page 16 onwards there is a nice explanation. ## Answer by Bryce (score 2) https://quant.stackexchange.com/a/4829 This piece of research provides everything you need: http://faculty.washington.edu/ezivot/econ424/portfoliofunctions.pdf
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