Skip to main content

Portfolio optimization with linear and fixed transaction costs

Abstract

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.

This is a preview of subscription content, access via your institution.

References

  • 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.

  • Andersen, E. (1999). MOSEK v1.0b User’s manual. Available from the url http://www.mosek.com.

  • Ben-Tal, A. and A. Nemirovski. (2001). Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. SIAM.

  • 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 

  • 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 

  • Boyd, S. and L. Vandenberghe. (2004). Convex Optimization. Cambridge University Press.

  • Chen, S.S., D. Donoho, and M.A. Saunders. (2001). “Atomic Decomposition by Basis Pursuit.” SIAM Review, 43.

  • 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 

  • Fazel, M. (2002). Matrix Rank Minimization with Applications. Ph. D. thesis, Dept. of Electrical Engineering, Stanford University. Available at http://www.cds.caltech.edu~maryam.

  • Fiacco, A. and G. McCormick. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley. Reprinted 1990 in the SIAM Classics in Applied Mathematics series.

  • Karmarkar, N. (1984). “A New Polynomial-Time Algorithm for Linear Programming.” Combinatorica, 4(4), 373–395.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Lawler, E.L. and D.E. Wood. (1966). “Branch-and-Bound Methods: A Survey.” Operations Research, 14, 699–719.

    Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • Lobo, M.S., L. Vandenberghe, and S. Boyd. (1997). socp : Software for Second-Order Cone Programming. Information Systems Laboratory, Stanford University.

  • 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.

    Article  Google Scholar 

  • Lucas, A. and P. Klaasen. (1998, Fall). “Extreme Returns, Downside Risk, and Optimal Asset Allocation.” The Journal of Portfolio Management, 71–79.

  • Luenberger, D.G. (1998). Investment Science. New York: Oxford University Press.

    Google Scholar 

  • Markowitz, H.M. (1952). “Portfolio Selection.” The Journal of Finance, 7(1), 77–91.

    Article  Google Scholar 

  • Markowitz, H.M. (1959). Portfolio Selection. New York: J. Wiley & Sons.

    Google Scholar 

  • 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 

  • Nesterov, Y. and A. Nemirovsky. (1994). Interior-Point Polynomial Methods in Convex Programming, Volume 13 of Studies in Applied Mathematics. Philadelphia, PA: SIAM.

  • 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.

    Article  Google Scholar 

  • Perold, A.F. (1984, October). “Large-Scale Portfolio Optimization.” Management Science, 30(10), 1143–1160.

    Google Scholar 

  • Roy, A.D. (1952). “Safety First and the Holding of Assets.” Econometrica, 20, 413–449.

    Google Scholar 

  • Rudolph, M. (1994). Algorithms for Portfolio Optimization and Portfolio Insurance. Switzerland: Haupt.

    Google Scholar 

  • Schattman, J.B. (2000, May). Portfolio Selection Under Nonconvex Transaction Costs and Capital gain Taxes. Ph. D. thesis, Rutgers University.

  • Schrijver, A. (1986). Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons.

  • Sharpe, W.F. (1964, September). “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk.” Journal of Finance, 19(3), 425–442.

    Article  Google Scholar 

  • 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).

  • Telser, L.G. (1955). “Safety First and Hedging.” Review of Economics and Statistics, 23, 1–16.

    Article  Google Scholar 

  • Tibshirani, R. (1996). “Regression Shrinkage and Selection Via the LASSO.” Journal of the Royal Statistical Society Series B, 267–288.

  • Vandenberghe, L. and S. Boyd. (1996, March). “Semidefinite Programming”. SIAM Review, 38(1), 49–95.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Miguel Sousa Lobo.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lobo, M.S., Fazel, M. & Boyd, S. Portfolio optimization with linear and fixed transaction costs. Ann Oper Res 152, 341–365 (2007). https://doi.org/10.1007/s10479-006-0145-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-006-0145-1

Keywords

  • Portfolio optimization
  • Transaction costs
  • Convex programming