Annals of Operations Research

, Volume 152, Issue 1, pp 341–365 | Cite as

Portfolio optimization with linear and fixed transaction costs

  • Miguel Sousa LoboEmail author
  • Maryam Fazel
  • Stephen Boyd


We consider the problem of portfolio selection, with transaction costs and constraints on exposure to risk. Linear transaction costs, bounds on the variance of the return, and bounds on different shortfall probabilities are efficiently handled by convex optimization methods. For such problems, the globally optimal portfolio can be computed very rapidly. Portfolio optimization problems with transaction costs that include a fixed fee, or discount breakpoints, cannot be directly solved by convex optimization. We describe a relaxation method which yields an easily computable upper bound via convex optimization. We also describe a heuristic method for finding a suboptimal portfolio, which is based on solving a small number of convex optimization problems (and hence can be done efficiently). Thus, we produce a suboptimal solution, and also an upper bound on the optimal solution. Numerical experiments suggest that for practical problems the gap between the two is small, even for large problems involving hundreds of assets. The same approach can be used for related problems, such as that of tracking an index with a portfolio consisting of a small number of assets.


Portfolio optimization Transaction costs Convex programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alizadeh, F., J.P. Haeberly, M.V. Nayakkankuppam, M.L. Overton, and S. Schmieta. (1997, June). sdppack User’s Guide, Version 0.9 Beta. NYU.Google Scholar
  2. Andersen, E. (1999). MOSEK v1.0b User’s manual. Available from the url Scholar
  3. Ben-Tal, A. and A. Nemirovski. (2001). Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. SIAM.Google Scholar
  4. Bertsimas, D., C. Darnell, and R. Soucy. (1999, January–February). “Portfolio Construction Through Mixed-Integer Programming at Grantham, Mayo, Van Otterloo and Company.” Interfaces, 29(1), 49–66.Google Scholar
  5. Blog, B., G.V.D. Hoek, A.H.G.R. Kan, and G.T. Timmer. (1983, July). “The Optimal Selection of Small Portfolios.” Management Science, 29(7), 792–798.Google Scholar
  6. Boyd, S. and L. Vandenberghe. (2004). Convex Optimization. Cambridge University Press.Google Scholar
  7. Chen, S.S., D. Donoho, and M.A. Saunders. (2001). “Atomic Decomposition by Basis Pursuit.” SIAM Review, 43.Google Scholar
  8. Delaney, A.H. and Y. Bresler. (1998, February). “Globally Convergent Edge-Preserving Regularized Reconstruction: An Application to Limited-Angle Tomography.” IEEE Transactions on Signal Processing, 7(2), 204–221.Google Scholar
  9. Fazel, M. (2002). Matrix Rank Minimization with Applications. Ph. D. thesis, Dept. of Electrical Engineering, Stanford University. Available at Scholar
  10. Fiacco, A. and G. McCormick. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley. Reprinted 1990 in the SIAM Classics in Applied Mathematics series.Google Scholar
  11. Karmarkar, N. (1984). “A New Polynomial-Time Algorithm for Linear Programming.” Combinatorica, 4(4), 373–395.Google Scholar
  12. Kellerer, H., R. Mansini, and M.G. Speranza. (2000). “Selecting Portfolios with Fixed Costs and Minimum Transaction Lots.” Annals of Operations Research, 99, 287–304.CrossRefGoogle Scholar
  13. Lawler, E.L. and D.E. Wood. (1966). “Branch-and-Bound Methods: A Survey.” Operations Research, 14, 699–719.Google Scholar
  14. Leibowitz, M.L., L.N. Bader, and S. Kogelman. (1996). Return Targets and Shortfall Risks: Studies in Strategic Asset Allocation. Chicago: Salomon Brothers, Irwin publishing.Google Scholar
  15. Lintner, J. (1965, February). “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets.” The Review of Economics and Statistics, 47(1), 13–37.CrossRefGoogle Scholar
  16. Lobo, M.S., L. Vandenberghe, and S. Boyd. (1997). socp : Software for Second-Order Cone Programming. Information Systems Laboratory, Stanford University.Google Scholar
  17. Lobo, M.S., L. Vandenberghe, S. Boyd, and H. Lebret. (1998, November). “Applications of Second-Order Cone Programming.” Linear Algebra and Applications, 284(1–3), 193–228.CrossRefGoogle Scholar
  18. Lucas, A. and P. Klaasen. (1998, Fall). “Extreme Returns, Downside Risk, and Optimal Asset Allocation.” The Journal of Portfolio Management, 71–79.Google Scholar
  19. Luenberger, D.G. (1998). Investment Science. New York: Oxford University Press.Google Scholar
  20. Markowitz, H.M. (1952). “Portfolio Selection.” The Journal of Finance, 7(1), 77–91.CrossRefGoogle Scholar
  21. Markowitz, H.M. (1959). Portfolio Selection. New York: J. Wiley & Sons.Google Scholar
  22. Meyer, R.R. (1974). “Sufficient Conditions for the Convergence of Monotonic Mathematical Programming Algorithms.” Journal of Computer and System Sciences, 12, 108–121.Google Scholar
  23. Nesterov, Y. and A. Nemirovsky. (1994). Interior-Point Polynomial Methods in Convex Programming, Volume 13 of Studies in Applied Mathematics. Philadelphia, PA: SIAM.Google Scholar
  24. Patel, N.R. and M.G. Subrahmanyam. (1982). “A Simple Algorithm for Optimal Portfolio Selection with Fixed Transaction Costs.” Management Science, 28(3), 303–314.CrossRefGoogle Scholar
  25. Perold, A.F. (1984, October). “Large-Scale Portfolio Optimization.” Management Science, 30(10), 1143–1160.Google Scholar
  26. Roy, A.D. (1952). “Safety First and the Holding of Assets.” Econometrica, 20, 413–449.Google Scholar
  27. Rudolph, M. (1994). Algorithms for Portfolio Optimization and Portfolio Insurance. Switzerland: Haupt.Google Scholar
  28. Schattman, J.B. (2000, May). Portfolio Selection Under Nonconvex Transaction Costs and Capital gain Taxes. Ph. D. thesis, Rutgers University.Google Scholar
  29. Schrijver, A. (1986). Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons.Google Scholar
  30. Sharpe, W.F. (1964, September). “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk.” Journal of Finance, 19(3), 425–442.CrossRefGoogle Scholar
  31. Sturm, J. (1999). “Using Sedumi 1.02, A Matlab Toolbox for Optimization Over Symmetric Cones.” Optimization Methods and Software, (11–12), 625–653. Special Issue on Interior Point Methods (CD Supplement with Software).Google Scholar
  32. Telser, L.G. (1955). “Safety First and Hedging.” Review of Economics and Statistics, 23, 1–16.CrossRefGoogle Scholar
  33. Tibshirani, R. (1996). “Regression Shrinkage and Selection Via the LASSO.” Journal of the Royal Statistical Society Series B, 267–288.Google Scholar
  34. Vandenberghe, L. and S. Boyd. (1996, March). “Semidefinite Programming”. SIAM Review, 38(1), 49–95.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • Miguel Sousa Lobo
    • 1
    Email author
  • Maryam Fazel
    • 2
  • Stephen Boyd
    • 3
  1. 1.Fuqua School of BusinessDuke UniversityDurhamUSA
  2. 2.California Institute of TechnologyControl and Dynamical Systems DepartmentCaliforniaUSA
  3. 3.Information Systems LabStanford UniversityStanfordUSA

Personalised recommendations