A Multi-objective Approach to Multi-period: Portfolio Optimization with Transaction Costs

  • Marius RadulescuEmail author
  • Constanta Zoie Radulescu
Part of the Multiple Criteria Decision Making book series (MCDM)


Portfolio optimization with transaction costs is a problem that involves non-smooth functions.

Transaction costs on each asset are usually assumed to be convex functions of the amount sold or bought. These functions can be non-differentiable in a finite number of points. In this chapter we intend to extend Markowitz’s portfolio selection model to multi-period models which include proportional transaction costs in the presence of initial holdings for the investor. Our approach is a novel one. We do research on an investor issue applicable to a business person who has some initial holdings and knows within the envisaged time frame the outbound and inbound cash flows as well as the exact points in time when these financial flows will occur.

A multi-period portfolio selection problem as a multi-objective programming problem with complementarity constraints is formulated. We prove that the above model is equivalent with a mixed-binary model. A goal programming approach to the multi-period multi-objective problem for portfolio selection is studied. In order to include the investor’s preferences, satisfaction functions are considered.


Portfolio optimization Multi-objective Multi-period Transaction costs Complementarity constraints 



This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS—UEFISCDI, project number PN-II-ID-PCE-2011-3-0908.


  1. Aouni, B., Colapinto, C., & La Torre, D. (2014). Portfolio management through goal programming: State-of-the-art. European Journal of Operations Research, 234, 536–545.CrossRefGoogle Scholar
  2. Aouni, B., Colapinto, C., & La Torre, D., Liuzzi, D., & Marsiglio, S. (2015). On dynamic multiple criteria decision making models: A goal programming approach. In M. Al-Shammari & H. Masri (Eds.), Multiple criteria decision making in finance, insurance and investment, multiple criteria decision making (pp. 31–48). Cham: Springer International Publishing.
  3. Aouni, B., & Kettani, O. (2001). Goal programming model: A glorious history and a promising future. European Journal of Operational Research, 133(2), 1–7.CrossRefGoogle Scholar
  4. Atta Mills, E. F. E., Yan, D., Yu, B., & Wei, X. (2016). Research on regularized mean–variance portfolio selection strategy with modified Roy safety-first principle. SpringerPlus, 5(919), 1–18.Google Scholar
  5. Bertsimas, D., & Pachamanova, D. (2008). Robust multiperiod portfolio management in the presence of transaction costs. Computers and Operations Research, 35(1), 3–17.CrossRefGoogle Scholar
  6. Bradley, S. P., & Crane, D. B. (1972). A dynamic model for bond portfolio management. Management Science, 19, 139–151.CrossRefGoogle Scholar
  7. Brodt, A. (1978). A dynamic balance sheet management model for a Canadian chartered bank. Journal of Banking and Finance, 2(3), 221–241.CrossRefGoogle Scholar
  8. Brodt, A. I. (1979). A multi-period portfolio theory model for commercial bank management. In E. Elton & M. J. Gruber (Eds.), Portfolio theory, TIMS studies in the management sciences (pp. 191–213). Amsterdam: North Holland.Google Scholar
  9. Brodt, A. I. (1984). Intertemporal bank asset and liability management. Journal of Bank Research, 15, 82–94.Google Scholar
  10. Chao, Z., Rui, H., & Lirong, W. (2017). Uncertain portfolio selection model considering transaction costs and minimum transaction lots requirement. Journal of Intelligent and Fuzzy Systems, 32(6), 4543–4554.CrossRefGoogle Scholar
  11. Chapados, N. (2011). Multiperiod problems, Chapter 3. In Portfolio choice problems: An introductory survey of single and multiperiod models. SpringerBriefs in electrical and computer engineering (Vol. 3, pp. 37–59). New York: Springer.Google Scholar
  12. Charnes, A., & Cooper, W. W. (1952). Chance constraints and normal deviates. Journal of the American Statistical Association, 57, 134–148.CrossRefGoogle Scholar
  13. Charnes, A., & Cooper, W. W. (1959). Chance-constrained programming. Management Science, 6, 73–80.CrossRefGoogle Scholar
  14. Charnes, A., Cooper, W. W., & Ferguson, R. (1955). Optimal estimation of executive compensation by linear programming. Management Science, 1, 138–151.CrossRefGoogle Scholar
  15. Colapinto, C., & La Torre, D. (2015). Multiple criteria decision making and goal programming for optimal venture capital investments and portfolio management. In M. Al-Shammari & H. Masri (Eds.), Multiple criteria decision making in finance, insurance and investment, multiple criteria decision making (pp. 9–30). Cham: Springer International Publishing. doi
  16. Cvitanic, J., & Karatzas, I. (1996). Hedging and portfolio optimization under transaction costs: A martingale approach. Mathematical Finance, 6, 133–165.CrossRefGoogle Scholar
  17. Daellenbach, H. G., & Archer, S. A. (1969). The optimal bank liquidity: A multi-period stochastic model. Journal of Financial and Quantitative Analysis, 4, 329–343.CrossRefGoogle Scholar
  18. Davis, M. H. A., & Norman, A. R. (1990). Portfolio selection with transaction costs. Mathematics of Operations Research, 15(4), 676–713.CrossRefGoogle Scholar
  19. Dempster, M. A. H., & Ireland, A. M. (1988). A financial expert decision support system. In G. Mitra (Ed.), Mathematical models for decision support. NATO ASI Series (Vol. F48, pp. 415–440). Berlin: Springer.Google Scholar
  20. Dumas, B., & Luciano, E. (1991). An exact solution to a dynamic portfolio choice problem under transaction costs. Journal of Finance, 46(2), 577–595.CrossRefGoogle Scholar
  21. Dupacova, J. (1999). Portfolio optimization via stochastic programming: Methods of output analysis. Mathematical Methods of Operations Research, 50, 245–270.CrossRefGoogle Scholar
  22. Elton, E., & Gruber, M. (1974a). The multi-period consumption investment problem and single period analysis. Oxford Economics Papers, 26(2), 289–301.CrossRefGoogle Scholar
  23. Elton, E., & Gruber, M. (1974b). On the optimality of some multi-period portfolio selection criteria. Journal of Business, 47(2), 231–243.CrossRefGoogle Scholar
  24. Fama, E. (1970). Multi-period consumption-investment decision. American Economic Review, 60, 163–179.Google Scholar
  25. Frauendorfer, K., & Siede, H. (1999). Portfolio selection using multistage stochastic programming. Central European Journal of Operations Research, 7(4), 277–289.Google Scholar
  26. Fu, Y. H., Ng, K. M., Huang, B., & Huang, H. C. (2015). Portfolio optimization with transaction costs: A two-period mean-variance model. Annals of Operations Research, 233–135.Google Scholar
  27. Guastaroba, G., Mansini, R., & Speranza, M. G. (2009). Models and simulations for portfolio rebalancing. Computational Economics, 33, 237–262.CrossRefGoogle Scholar
  28. Gülpinar, N., & Rustem, B. (2007). Worst-case decisions for multi-period mean-variance portfolio optimization. European Journal of Operational Research, 183(3), 981–1000.CrossRefGoogle Scholar
  29. Hakansson, N. H. (1970). Optimal investment and consumption strategies under risk for a class of utility functions. Econometrica, 38, 587–607.CrossRefGoogle Scholar
  30. Hakansson, N. H. (1974). Convergence to isoelastic utility and policy in multi-period portfolio choice. Journal of Financial Economics, 1, 201–224.CrossRefGoogle Scholar
  31. Hakansson, N. H. (1979). Optimal multi-period portfolio policies. In E. Elton & M. J. Gruber (Eds.), Portfolio theory, TIMS studies in the management sciences (p. 11). Amsterdam: North Holland.Google Scholar
  32. Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99, 287–304.CrossRefGoogle Scholar
  33. Konno, H., & Wijayanayake, A. (2001). Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming, 89, 233–250.CrossRefGoogle Scholar
  34. Korhonen, A. (1987). A dynamic bank portfolio planning model with multiple scenarios, multiple goals and changing priorities. European Journal of Operational Research, 30(1), 13–23.CrossRefGoogle Scholar
  35. Kusy, M. I., & Ziemba, W. T. (1986). A bank asset and liability management model. Operations Research, 34, 356–376.CrossRefGoogle Scholar
  36. Lane, M., & Hutchinson, P. (1980). A model for managing a certificate of deposit portfolio under uncertainty. In M. A. H. Dempster (Eds.), Stochastic programming (pp. 473–495). London: Academic Press.Google Scholar
  37. Li, Z. F., Li, Z. X., Wang, S. Y., & Deng, X. T. (2001). Optimal portfolio selection of assets with transaction costs and no short sales. International Journal of Systems Science, 32(5), 599–607.CrossRefGoogle Scholar
  38. Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234, 518–535.CrossRefGoogle Scholar
  39. Mansini, R., & Speranza, M. G. (2005). An exact approach for portfolio selection with transaction costs and rounds. IIE Transactions, 37, 919–929.CrossRefGoogle Scholar
  40. Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7, 77–91.Google Scholar
  41. Markowitz, H. M. (1959). Portfolio selection: Efficient diversification of investments. New York: Wiley.Google Scholar
  42. Markowitz, H. M. (1987). Mean-variance analysis in portfolio choice and capital markets. Cambridge, MA: Basil Blackwell Ltd.Google Scholar
  43. Martel, J. M., & Aouni, B. (1990). Incorporating the decision-maker’s preferences in the goal programming model. Journal of Operational Research Society, 41, 1121–1132.CrossRefGoogle Scholar
  44. Mei, Z., DeMiguel, V., & Nogales, F. J. (2016). Multiperiod portfolio optimization with multiple risky assets and general transaction costs. Journal of Banking and Finance, 69, 108–120.CrossRefGoogle Scholar
  45. Mossin, J. (1968). Optimal multi-period portfolio policies. Journal of Bussiness, 41, 215–229.Google Scholar
  46. Mulvey, J., Rosenbaum, D., & Shetty, B. (1997). Strategic financial management and operations research. European Journal of Operational Research, 97(1), 1–16.CrossRefGoogle Scholar
  47. Mulvey, J. M., & Vladimirou, H. (1989). Stochastic network optimization models for investment planning. Annals of Operations Research, 20, 187–217.CrossRefGoogle Scholar
  48. Najafi, A. A., & Mushakhian, S. (2015). Multi-stage stochastic mean–semivariance–CVaR portfolio optimization under transaction costs. Applied Mathematics and Computation, 256(1), 445–458.CrossRefGoogle Scholar
  49. Najafi, A. A., & Pourahmadi, Z. (2016). An efficient heuristic method for dynamic portfolio selection problem under transaction costs and uncertain conditions. Physica A: Statistical Mechanics and Its Applications, 448, 154–162.CrossRefGoogle Scholar
  50. Östermark, R. (1991). Vector forecasting and dynamic portfolio selection: Empirical efficiency of recursive multi-period strategies. European Journal of Operational Research, 55(1), 46–56.CrossRefGoogle Scholar
  51. Östermark, R. (2005). Dynamic portfolio management under competing representations. Kybernetes, 34(9/10), 1517–1550.CrossRefGoogle Scholar
  52. Peng, Z., & Jin, P. (2012). The optimization on the multiperiod mean average absolute deviation fuzzy portfolio selection. In: D. Jin & S. Lin (Eds.), Advances in electronic engineering, communication and management vol.2. Lecture notes in electrical engineering (Vol. 140). Berlin: Springer.Google Scholar
  53. Pogue, G. A. (1970). An extension of the Markowitz portfolio selection model to include transaction costs, short sales, leverage policies and taxes. Journal of Finance, 25, 1005–1027.Google Scholar
  54. Puopolo, G. W. (2017). Portfolio selection with transaction costs and default risk. Managerial Finance, 43(2), 231–241.CrossRefGoogle Scholar
  55. Radulescu, M., Radulescu, S., & Radulescu, C. Z. (2001). Computer simulation for portfolio selection models with transaction costs and initial holdings. In Proceedings of the EUROSIM 2001 Congress, “Shaping future with simulation”, Delft, pp. 1–6.Google Scholar
  56. Radulescu, C-Z, Radulescu, M., & Radulescu, S. (2002). Optimal decisions for portfolio selection in the presence of initial holdings and transaction costs. In Proceeding of the 4th International Workshop on Computer Science and Information Technologies CSIT’2002, Patras, pp. 1–6.Google Scholar
  57. Ruiz-Torrubiano, R., & Suárez, A. (2015). A memetic algorithm for cardinality-constrained portfolio optimization with transaction costs. Applied Soft Computing, 36, 125–142.CrossRefGoogle Scholar
  58. Shapiro, J. F. (1988). Stochastic programming models for dedicated portfolio selection. In: B. Mitra (Ed.), Mathematical models for decision support. NATO ASI Series (Vol. F48, pp. 587–611). Berlin: Springer.Google Scholar
  59. Shreve, S. E., & Soner, H. M. (1994). Optimal investment and consumption with transaction costs. The Annals of Applied Probability, 4, 609–692.CrossRefGoogle Scholar
  60. Steinbach, M. C. (2001). Markowitz, revisited: Mean-variance models in financial portfolio analysis. SIAM Review, 43(1), 31–85.CrossRefGoogle Scholar
  61. Taksar, M., Klass, M., & Assaf, D. (1988). A diffusion model for optimal portfolio selection in the presence of brokerage fees. Mathematics of Operations Research, 13, 277–294.CrossRefGoogle Scholar
  62. Wang, S., & Xia, Y. (2002). Portfolio selection and asset pricing. New York: Springer.Google Scholar
  63. Woodside-Oriakhi, M., Lucas, C., & Beasley, J. E. (2013). Portfolio rebalancing with an investment horizon and transaction costs. Omega, 41(2), 406–420.CrossRefGoogle Scholar
  64. Zhang, W. G., Liu, Y. J., & Xu, W. J. (2012). A possibilistic mean-semivariance-entropy model for multi-period portfolio selection with transaction costs. European Journal of Operational Research, 222(2), 341–349.Google Scholar
  65. Zhang, P., & Yu, L. (2011). The optimization on the multiperiod mean-VaR portfolio selection in friction market. In: D. Jin & S. Lin (Eds.), Advances in multimedia, software engineering and computing vol. 2. Advances in Intelligent and Soft Computing (Vol. 129). Berlin: Springer.Google Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute of Mathematical Statistics and Applied MathematicsOrganization Romanian AcademyBucharest 5Romania
  2. 2.National Institute for Research and Development in InformaticsBucharest 1Romania

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