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An Algorithm for Portfolio Optimization with Variable Transaction Costs, Part 1: Theory

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  • M. J. Best

    (University of Waterloo)

  • J. Hlouskova

    (Institute for Advanced Studies)

Abstract

A portfolio optimization problem consists of maximizing an expected utility function of n assets. At the end of a typical time period, the portfolio will be modified by buying and selling assets in response to changing conditions. Associated with this buying and selling are variable transaction costs that depend on the size of the transaction. A straightforward way of incorporating these costs can be interpreted as the reduction of portfolios’ expected returns by transaction costs if the utility function is the mean-variance or the power utility function. This results in a substantially higher-dimensional problem than the original n-dimensional one, namely (2K+1)n-dimensional optimization problem with (4K+1)n additional constraints, where 2K is the number of different transaction costs functions. The higher-dimensional problem is computationally expensive to solve. This two-part paper presents a method for solving the (2K+1)n-dimensional problem by solving a sequence of n-dimensional optimization problems, which account for the transaction costs implicitly rather than explicitly. The key idea of the new method in Part 1 is to formulate the optimality conditions for the higher-dimensional problem and enforce them by solving a sequence of lower-dimensional problems under the nondegeneracy assumption. In Part 2, we propose a degeneracy resolving rule, address the efficiency of the new method and present the computational results comparing our method with the interior-point optimizer of Mosek.

Suggested Citation

  • M. J. Best & J. Hlouskova, 2007. "An Algorithm for Portfolio Optimization with Variable Transaction Costs, Part 1: Theory," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 563-581, December.
  • Handle: RePEc:spr:joptap:v:135:y:2007:i:3:d:10.1007_s10957-007-9252-7
    DOI: 10.1007/s10957-007-9252-7
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    References listed on IDEAS

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    1. Michael J. Best & Jaroslava Hlouskova, 2005. "An Algorithm for Portfolio Optimization with Transaction Costs," Management Science, INFORMS, vol. 51(11), pages 1676-1688, November.
    2. Grauer, Robert R & Hakansson, Nils H, 1986. "A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills, with and without Small Stocks," The Journal of Business, University of Chicago Press, vol. 59(2), pages 287-318, April.
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    Cited by:

    1. Zhang, Wei-Guo & Xiao, Wei-Lin & Xu, Wei-Jun, 2010. "A possibilistic portfolio adjusting model with new added assets," Economic Modelling, Elsevier, vol. 27(1), pages 208-213, January.
    2. Woodside-Oriakhi, M. & Lucas, C. & Beasley, J.E., 2013. "Portfolio rebalancing with an investment horizon and transaction costs," Omega, Elsevier, vol. 41(2), pages 406-420.
    3. Areski Cousin & Jérôme Lelong & Tom Picard, 2023. "Mean-variance dynamic portfolio allocation with transaction costs: a Wiener chaos expansion approach," Working Papers hal-04086378, HAL.
    4. Areski Cousin & J'er^ome Lelong & Tom Picard, 2023. "Mean-variance dynamic portfolio allocation with transaction costs: a Wiener chaos expansion approach," Papers 2305.16152, arXiv.org, revised Jun 2023.
    5. Tiago P. Filomena & Miguel A. Lejeune, 2014. "Warm-Start Heuristic for Stochastic Portfolio Optimization with Fixed and Proportional Transaction Costs," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 308-329, April.
    6. Jiuping Xu & Xiaoyang Zhou & Steven Li, 2011. "A Class of Chance Constrained Multi-objective Portfolio Selection Model Under Fuzzy Random Environment," Journal of Optimization Theory and Applications, Springer, vol. 150(3), pages 530-552, September.
    7. Michael J. Best & Xili Zhang, 2011. "Degeneracy Resolution for Bilinear Utility Functions," Journal of Optimization Theory and Applications, Springer, vol. 150(3), pages 615-634, September.

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