Applying Markowitz’s Critical Line Algorithm
This paper derives a numerically enhanced version of Markowitz’s Critical Line Algorithm for computing the entire mean variance frontier with arbitrary lower and upper bounds on weights. We show that this algorithm computationally outperforms standard software packages and a recently developed quadratic optimization algorithm. As an illustration: For a 2,000-asset universe, our method needs less than a second to compute the whole frontier whereas the quickest competitor needs several hours. This paper can be considered as a didactic alternative to the Critical Line Algorithm such as presented by Markowitz and treats all steps required by the algorithm explicitly. Finally, we present a benchmark of different optimization algorithms’ performance.
KeywordsCovariance Matrix Turning Point Portfolio Optimization Portfolio Selection Expected Return
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- Hirschberger, M., Y. Qi, and R. E. Steuer (2004): Quadratic Parametric Programming for Portfolio Selection with Random Problem Generation and Computational Experience, Working papers, Terry College of Business, University of Georgia.Google Scholar
- Markowitz, H. M. (1956): “The Optimization of a Quadratic Function Subject to Linear Constraints,” Naval Research Logistics Quarterly, III, 111–133.Google Scholar
- Markowitz, H. M. (1959): Portfolio Selection: Efficient Diversification of Investments. Wiley, New York, and 1991 2nd ed., Basil Blackwell, Cambridge, MA.Google Scholar
- Markowitz, H. M., and P. Todd (2000): Mean–Variance Analysis in Portfolio Choice and Capital Markets. Frank J. Fabozzi Associates, New Hope, PA.Google Scholar
- Markowitz, H., and N. Usmen (2003): “Resampled Frontiers vs Diffuse Bayes: An Experiment,” Journal Of Investment Management, 1(4), 9–25.Google Scholar
- Michaud, R. (1998): Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization. Oxford University Press, Oxford.Google Scholar
- Niedermayer, D. (2005): Portfolio Theory and the Cross-sectional Relation between Expected Returns and Betas, Master’s Thesis, Department of Economics, University of Bern.Google Scholar
- Scherer, B., and D. R. Martin (2006): Introduction to Modern Portfolio Optimization with NuOPT, S-PLUS and S+Bayes, 1st edn. Springer, New York.Google Scholar
- Steuer, R. E., Y. Qi, and M. Hirschberger (2006): “Portfolio Optimization: New Capabilities and Future Methods,” Zeitschrift für BWL, 76(2), 199–219.Google Scholar
- Wolf, M. (2006): Resampling vs. Shrinkage for Benchmarked Managers, IEW – Working Papers iewwp263, Institute for Empirical Research in Economics – IEW, Available at http://ideas.repec.org/p/zur/iewwpx/263.html