Applying Markowitz’s Critical Line Algorithm

  • Andras Niedermayer
  • Daniel Niedermayer


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.


Covariance Matrix Turning Point Portfolio Optimization Portfolio Selection Expected Return 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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
  2. Jorion, P. (1992): “Portfolio Optimization in Practice,” Financial Analysts Journal, 48(1), 68–74.CrossRefGoogle Scholar
  3. Markowitz, H. M. (1952): “Portfolio Selection,” Journal of Finance, 7(1), 77–91.CrossRefGoogle Scholar
  4. Markowitz, H. M. (1956): “The Optimization of a Quadratic Function Subject to Linear Constraints,” Naval Research Logistics Quarterly, III, 111–133.Google Scholar
  5. Markowitz, H. M. (1959): Portfolio Selection: Efficient Diversification of Investments. Wiley, New York, and 1991 2nd ed., Basil Blackwell, Cambridge, MA.Google Scholar
  6. 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
  7. Markowitz, H., and N. Usmen (2003): “Resampled Frontiers vs Diffuse Bayes: An Experiment,” Journal Of Investment Management, 1(4), 9–25.Google Scholar
  8. Michaud, R. (1998): Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization. Oxford University Press, Oxford.Google Scholar
  9. 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
  10. Scherer, B. (2002): “Portfolio Resampling: Review and Critique,” Financial Analysts Journal, 58(6), 98–109.CrossRefGoogle Scholar
  11. 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
  12. 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
  13. Wolf, M. (2006): Resampling vs. Shrinkage for Benchmarked Managers, IEW – Working Papers iewwp263, Institute for Empirical Research in Economics – IEW, Available at
  14. Wolfe, P. (1959): “The Simplex Method for Quadratic Programming,” Econometrica, 27(3), 382–398.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Kellogg School of Management, Managerial Economics and Decision SciencesNorthwestern UniversityEvanstonUSA
  2. 2.Credit Suisse, Asset Management and WWZUniversity of BaselBaselSwitzerland

Personalised recommendations