Document Type : Article

**Authors**

Faculty of Industrial and Systems Engineering, Tarbiat Modares University, Tehran, P.O. Box 14115-111, Iran.

**Abstract**

The multi-period portfolio optimization models were introduced to overcome the weaknesses of the single-period models via considering a dynamic optimization system. However, due to the nonlinear nature of the problem and rapid growth of the size complexity with increasing the number of periods and scenarios, this study is devoted to developing a novel league championship algorithm (LCA) to maximize the portfolio’s mean-variance function subject to different constraints. A Vector Auto Regression model is also developed to estimate the return on risky assets in different time periods and to simulate different scenarios of the rate of return accordingly. Besides, we proved a valid upper bound of the objective function based on the idea of using surrogate relaxation of constraints. Our computational results based on sample data collected from S&P 500 and 10-year T. Bond indices indicate that the quality of portfolios, in terms of the mean-variance measure, obtained by LCA is 10 to 20 percent better than those of the commercial software. This sounds promising that our method can be a suitable tool for solving a variety of portfolio optimization problems.

**Keywords**

**Main Subjects**

References:

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3. Best, M.J. and Hlouskova, J. "Portfolio selection and transactions costs", Computational Optimization and Applications, 24(1), pp. 95-116 (2003).

4. Liu, S., Wang, S.Y., and Qiu, W. "Mean-varianceskewness model for portfolio selection with transaction costs", International Journal of Systems Science, 34(4), pp. 255-262 (2010).

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6. Mulvey, J.M. and Shetty, B. "Financial planning via multi-stage stochastic optimization", Computers & Operations Research, 31(1), pp. 1-20 (2004).

7. Carino, D.R., Myers, D.H., and Ziemba, W.T. "Concepts, technical issues, and uses of the Russell-Yasuda Kasai financial planning model", Operations Research, 46(4), pp. 450-462 (1998).

8. Ertenlice, O. and Kalayci, C.B. "A survey of swarm intelligence for portfolio optimization: Algorithms and applications", Swarm and Evolutionary Computation, 39, pp. 36-52 (2018).

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10. Woodside-Oriakhi, M., Lucas, C., and Beasley, J.E. "Heuristic algorithms for the cardinality constrained efficient frontier", European Journal of Operational Research, 213, pp. 538-550 (2011).

11. Baykasoglu, A., Yunusoglu, M.G., and Ozsoydan, F.B. "A GRASP based solution approach to solve cardinality constrained portfolio optimization problems", Computers & Industrial Engineering, 90, pp. 339-351 (2015).

12. Kalayci, C.B., Ertenlice, O., Akyer, H., and Aygoren, H. "An artificial bee colony algorithm with feasibility enforcement and infeasibility toleration procedures for cardinality constrained portfolio optimization", Expert Systems with Applications, 85, pp. 61-75 (2017).

13. Bradley, S.P. and Crane, D.B. "A dynamic model for bond portfolio management", Management Science, 19(2), pp. 139-151 (1972).

14. Kallberg, J.G. and Ziemba, W.T. "Comparison of alternative utility functions in portfolio selection problems", Management Science, 29(11), pp. 1257-1276 (1983).

15. Mulvey, J.M. and Vladimirou, H. "Stochastic network optimization models for investment planning", Annals of Operations Research, 20(1), pp. 187-217 (1989).

16. Wei, S.Z. and Ye, Z.X. "Multi-period optimization portfolio with bankruptcy control in stochastic market", Applied Mathematics and Computation, 186(1), pp. 414-425 (2007).

17. Bertsimas, D. and Pachamanova, D. "Robust multiperiod portfolio management in the presence of transaction costs", Computers & Operations Research, 35(1), pp. 3-17 (2008).

18. C akmak, U. and Ozekici, S. "Portfolio optimization in stochastic markets", Mathematical Methods of Operations Research, 63(1), pp. 151-168 (2006).

19. Li, D. and Ng, W.L. "Optimal dynamic portfolio selection: Multiperiod mean-variance formulation", Mathematical Finance, 10(3), pp. 387-406 (2000).

20. Zhu, S.S., Li, D., and Wang, S.Y. "Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation", Automatic Control, IEEE Transactions on, 49(3), pp. 447-457 (2004).

21. Pinar, M.C . "Robust scenario optimization based on downside-risk measure for multi-period portfolio selection", OR Spectrum, 29(2), pp. 295-309 (2007).

22. Zhang, W.G., Liu, Y.J., and Xu, W.J. "A possibilistic mean-semivariance-entropy model for multi-period portfolio selection with transaction costs", European Journal of Operational Research, 222(2), pp. 341-349 (2012).

23. Fang, Y., Lai, K.K., and Wang, S.Y. "Portfolio rebalancing model with transaction costs based on fuzzy decision theory", European Journal of Operational Research, 175(2), pp. 879-893 (2006).

24. Sadjadi, S.J., Seyedhosseini, S.M., and Hassanlou, K. "Fuzzy multi period portfolio selection with different rates for borrowing and lending", Applied Soft Computing, 11(4), pp. 3821-3826 (2011).

25. Zhang, P., and Zhang, W.G. "Multiperiod mean absolute deviation fuzzy portfolio selection model with risk control and cardinality constraints", Fuzzy Sets and Systems, 255, pp. 74-91 (2014).

26. Yao, H., Li, Z., and Li, D. "Multi-period meanvariance portfolio selection with stochastic interest rate and uncontrollable liability", European Journal of Operational Research, 252(3), pp. 837-851 (2016).

27. Berger, A.J., Glover, F., and Mulvey, J.M. "Solving global optimization problems in long-term financial planning", Statistics and Operation Research Technical Report, Princeton University (1995).

28. Berger, A.J. and Mulvey, J.M. "Integrative risk management for individual investors", Worldwide Asset and Liability Modeling, Cambridge University Press (1996).

29. Chan, M.C., Wong, C.C., Cheung, B.K.S., and Tang, G.N. "Genetic algorithms in multi-stage asset allocation system", In Systems, Man and Cybernetics, 2002 IEEE International Conference on, 3, p. 6, IEEE (October, 2002).

30. Sun, J., Fang, W., Wu, X., Lai, C.H., and Xu, W. "Solving the multi-stage portfolio optimization problem with a novel particle swarm optimization", Expert Systems with Applications, 38(6), pp. 6727- 6735 (2011).

31. Yan, W., Miao, R., and Li, S. "Multi-period semivariance portfolio selection: Model and numerical solution", Applied Mathematics and Computation, 194(1), pp. 128-134 (2007).

32. Zhang, X., Zhang, W., and Xiao, W. "Multi-period portfolio optimization under possibility measures", Economic Modelling, 35, pp. 401-408 (2013).

33. Liu, Y.J., Zhang, W.G., and Zhang, Q. "Credibilistic multi-period portfolio optimization model with bankruptcy control and affine recourse", Applied Soft Computing, 38, pp. 890-906 (2016).

34. Liu, J., Jin, X., Wang, T., and Yuan, Y. "Robust multi-period portfolio model based on prospect theory and ALMV-PSO algorithm", Expert Systems with Applications, 42(20), pp. 7252-7262 (2015).

35. Wang, B., Li, Y., and Watada, J. "Multi-period portfolio selection with dynamic risk/expected-return level under fuzzy random uncertainty", Information Sciences, 385-386, pp. 1-18 (2017).

36. Li, B., Zhu, Y., Sun, Y., Aw, G., and Teo, K.L. "Multiperiod portfolio selection problem under uncertain environment with bankruptcy constraint", Applied Mathematical Modelling, 56, pp. 539-550 (2018).

37. Zhao, Y., and Ziemba, W.T.A. "Stochastic programming model using an endogenously determined worst case risk measure for dynamic asset allocation", Mathematical Programming, 89(2), pp. 293-309 (2001).

38. Jacobson, H.I. "The maximum variance of restricted unimodal distributions", The Annals of Mathematical Statistics, 40(5), pp. 1746-1752 (1969).

39. Husseinzadeh Kashan, A. "An efficient algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA)", Computer-Aided Design, 43(12), pp. 1769-1792 (2011).

40. Husseinzadeh Kashan, A. "League championship algorithm: a new algorithm for numerical function optimization", In 2009 International Conference of Soft Computing and Pattern Recognition, pp. 43-48, IEEE (December, 2009).

41. Husseinzadeh Kashan, A. "League Championship Algorithm (LCA): An algorithm for global optimization inspired by sport championships", Applied Soft Computing, 16, pp. 171-200 (2014).

42. Alimoradi, M.R. and Husseinzadeh Kashan, A. "A league championship algorithm equipped with network structure and backward Q-learning for extracting stock trading rules", Applied Soft Computing, 68, pp. 478- 493 (2018).

43. Husseinzadeh Kashan, A., Abbasi-Pooya, A., and Karimiyan, S. "A rig-based formulation and a league championship algorithm for helicopter routing in off- shore transportation", Proceedings of the 2nd International Conference on Data Engineering and Communication Technology: ICDECT 2017, Volume 828 of Advances in Intelligent Systems and Computing (2017).

44. Husseinzadeh Kashan, A. "A new metaheuristic for optimization: optics inspired optimization (OIO)", Computers & Operations Research, 55, pp. 99-125 (2015).

45. Husseinzadeh Kashan, A. "An effective algorithm for constrained optimization based on optics inspired optimization (OIO)", Computer-Aided Design, 63, pp. 52-71 (2015).

2. Yoshimoto, A. "The mean-variance approach to portfolio optimization subject to transaction costs", Journal of the Operations Research Society of Japan, 39(1), pp. 99-117 (1996).

3. Best, M.J. and Hlouskova, J. "Portfolio selection and transactions costs", Computational Optimization and Applications, 24(1), pp. 95-116 (2003).

4. Liu, S., Wang, S.Y., and Qiu, W. "Mean-varianceskewness model for portfolio selection with transaction costs", International Journal of Systems Science, 34(4), pp. 255-262 (2010).

5. Favaretto, D. "On the existence of solutions to the quadratic mixed-integer mean-variance portfolio selection problem", European Journal of Operational Research, 176(3), pp. 1947-1960 (2007).

6. Mulvey, J.M. and Shetty, B. "Financial planning via multi-stage stochastic optimization", Computers & Operations Research, 31(1), pp. 1-20 (2004).

7. Carino, D.R., Myers, D.H., and Ziemba, W.T. "Concepts, technical issues, and uses of the Russell-Yasuda Kasai financial planning model", Operations Research, 46(4), pp. 450-462 (1998).

8. Ertenlice, O. and Kalayci, C.B. "A survey of swarm intelligence for portfolio optimization: Algorithms and applications", Swarm and Evolutionary Computation, 39, pp. 36-52 (2018).

9. Deng, G.-F., Lin, W.-T., and Lo, C.-C. "Markowitzbased portfolio selection with cardinality constraints using improved particle swarm optimization", Expert Systems with Applications, 39, pp. 4558-4566 (2012).

10. Woodside-Oriakhi, M., Lucas, C., and Beasley, J.E. "Heuristic algorithms for the cardinality constrained efficient frontier", European Journal of Operational Research, 213, pp. 538-550 (2011).

11. Baykasoglu, A., Yunusoglu, M.G., and Ozsoydan, F.B. "A GRASP based solution approach to solve cardinality constrained portfolio optimization problems", Computers & Industrial Engineering, 90, pp. 339-351 (2015).

12. Kalayci, C.B., Ertenlice, O., Akyer, H., and Aygoren, H. "An artificial bee colony algorithm with feasibility enforcement and infeasibility toleration procedures for cardinality constrained portfolio optimization", Expert Systems with Applications, 85, pp. 61-75 (2017).

13. Bradley, S.P. and Crane, D.B. "A dynamic model for bond portfolio management", Management Science, 19(2), pp. 139-151 (1972).

14. Kallberg, J.G. and Ziemba, W.T. "Comparison of alternative utility functions in portfolio selection problems", Management Science, 29(11), pp. 1257-1276 (1983).

15. Mulvey, J.M. and Vladimirou, H. "Stochastic network optimization models for investment planning", Annals of Operations Research, 20(1), pp. 187-217 (1989).

16. Wei, S.Z. and Ye, Z.X. "Multi-period optimization portfolio with bankruptcy control in stochastic market", Applied Mathematics and Computation, 186(1), pp. 414-425 (2007).

17. Bertsimas, D. and Pachamanova, D. "Robust multiperiod portfolio management in the presence of transaction costs", Computers & Operations Research, 35(1), pp. 3-17 (2008).

18. C akmak, U. and Ozekici, S. "Portfolio optimization in stochastic markets", Mathematical Methods of Operations Research, 63(1), pp. 151-168 (2006).

19. Li, D. and Ng, W.L. "Optimal dynamic portfolio selection: Multiperiod mean-variance formulation", Mathematical Finance, 10(3), pp. 387-406 (2000).

20. Zhu, S.S., Li, D., and Wang, S.Y. "Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation", Automatic Control, IEEE Transactions on, 49(3), pp. 447-457 (2004).

21. Pinar, M.C . "Robust scenario optimization based on downside-risk measure for multi-period portfolio selection", OR Spectrum, 29(2), pp. 295-309 (2007).

22. Zhang, W.G., Liu, Y.J., and Xu, W.J. "A possibilistic mean-semivariance-entropy model for multi-period portfolio selection with transaction costs", European Journal of Operational Research, 222(2), pp. 341-349 (2012).

23. Fang, Y., Lai, K.K., and Wang, S.Y. "Portfolio rebalancing model with transaction costs based on fuzzy decision theory", European Journal of Operational Research, 175(2), pp. 879-893 (2006).

24. Sadjadi, S.J., Seyedhosseini, S.M., and Hassanlou, K. "Fuzzy multi period portfolio selection with different rates for borrowing and lending", Applied Soft Computing, 11(4), pp. 3821-3826 (2011).

25. Zhang, P., and Zhang, W.G. "Multiperiod mean absolute deviation fuzzy portfolio selection model with risk control and cardinality constraints", Fuzzy Sets and Systems, 255, pp. 74-91 (2014).

26. Yao, H., Li, Z., and Li, D. "Multi-period meanvariance portfolio selection with stochastic interest rate and uncontrollable liability", European Journal of Operational Research, 252(3), pp. 837-851 (2016).

27. Berger, A.J., Glover, F., and Mulvey, J.M. "Solving global optimization problems in long-term financial planning", Statistics and Operation Research Technical Report, Princeton University (1995).

28. Berger, A.J. and Mulvey, J.M. "Integrative risk management for individual investors", Worldwide Asset and Liability Modeling, Cambridge University Press (1996).

29. Chan, M.C., Wong, C.C., Cheung, B.K.S., and Tang, G.N. "Genetic algorithms in multi-stage asset allocation system", In Systems, Man and Cybernetics, 2002 IEEE International Conference on, 3, p. 6, IEEE (October, 2002).

30. Sun, J., Fang, W., Wu, X., Lai, C.H., and Xu, W. "Solving the multi-stage portfolio optimization problem with a novel particle swarm optimization", Expert Systems with Applications, 38(6), pp. 6727- 6735 (2011).

31. Yan, W., Miao, R., and Li, S. "Multi-period semivariance portfolio selection: Model and numerical solution", Applied Mathematics and Computation, 194(1), pp. 128-134 (2007).

32. Zhang, X., Zhang, W., and Xiao, W. "Multi-period portfolio optimization under possibility measures", Economic Modelling, 35, pp. 401-408 (2013).

33. Liu, Y.J., Zhang, W.G., and Zhang, Q. "Credibilistic multi-period portfolio optimization model with bankruptcy control and affine recourse", Applied Soft Computing, 38, pp. 890-906 (2016).

34. Liu, J., Jin, X., Wang, T., and Yuan, Y. "Robust multi-period portfolio model based on prospect theory and ALMV-PSO algorithm", Expert Systems with Applications, 42(20), pp. 7252-7262 (2015).

35. Wang, B., Li, Y., and Watada, J. "Multi-period portfolio selection with dynamic risk/expected-return level under fuzzy random uncertainty", Information Sciences, 385-386, pp. 1-18 (2017).

36. Li, B., Zhu, Y., Sun, Y., Aw, G., and Teo, K.L. "Multiperiod portfolio selection problem under uncertain environment with bankruptcy constraint", Applied Mathematical Modelling, 56, pp. 539-550 (2018).

37. Zhao, Y., and Ziemba, W.T.A. "Stochastic programming model using an endogenously determined worst case risk measure for dynamic asset allocation", Mathematical Programming, 89(2), pp. 293-309 (2001).

38. Jacobson, H.I. "The maximum variance of restricted unimodal distributions", The Annals of Mathematical Statistics, 40(5), pp. 1746-1752 (1969).

39. Husseinzadeh Kashan, A. "An efficient algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA)", Computer-Aided Design, 43(12), pp. 1769-1792 (2011).

40. Husseinzadeh Kashan, A. "League championship algorithm: a new algorithm for numerical function optimization", In 2009 International Conference of Soft Computing and Pattern Recognition, pp. 43-48, IEEE (December, 2009).

41. Husseinzadeh Kashan, A. "League Championship Algorithm (LCA): An algorithm for global optimization inspired by sport championships", Applied Soft Computing, 16, pp. 171-200 (2014).

42. Alimoradi, M.R. and Husseinzadeh Kashan, A. "A league championship algorithm equipped with network structure and backward Q-learning for extracting stock trading rules", Applied Soft Computing, 68, pp. 478- 493 (2018).

43. Husseinzadeh Kashan, A., Abbasi-Pooya, A., and Karimiyan, S. "A rig-based formulation and a league championship algorithm for helicopter routing in off- shore transportation", Proceedings of the 2nd International Conference on Data Engineering and Communication Technology: ICDECT 2017, Volume 828 of Advances in Intelligent Systems and Computing (2017).

44. Husseinzadeh Kashan, A. "A new metaheuristic for optimization: optics inspired optimization (OIO)", Computers & Operations Research, 55, pp. 99-125 (2015).

45. Husseinzadeh Kashan, A. "An effective algorithm for constrained optimization based on optics inspired optimization (OIO)", Computer-Aided Design, 63, pp. 52-71 (2015).

Transactions on Industrial Engineering (E)

March and April 2020Pages 829-845