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**

1. Markowitz, H. Portfolio selection", The Journal of

Finance, 7(1), pp. 77{91 (1952).

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 nancial 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

ecient 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 articial 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. Pnar, 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 dierent

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).

844 A. Husseinzadeh Kashan et al./Scientia Iranica, Transactions E: Industrial Engineering 27 (2020) 829{845

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 nancial

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 ane 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 ecient 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 o-

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 eective algorithm for

constrained optimization based on optics inspired optimization

(OIO)", Computer-Aided Design, 63, pp.

52{71 (2015).

Volume 27, Issue 2

Transactions on Industrial Engineering (E)

March and April 2020Pages 829-845