Markowitz-Based Cardinality Constrained Portfolio Selection Using Asexual Reproduction Optimization (ARO)

Document Type : Research Paper

Authors

1 Department of Production and Operation Management, Faculty of Management, University of Tehran, Tehran, Iran

2 Department of Computing, Science and Engineering, University of Salford, Greater Manchester, UK

3 M.Sc. in Industrial Management, Department of Industrial Management, Faculty of Management, University of Tehran, Tehran, Iran

Abstract

The Markowitz-based portfolio selection turns to an NP-hard problem when considering cardinality constraints. In this case, existing exact solutions like quadratic programming may not be efficient to solve the problem. Many researchers, therefore, used heuristic and metaheuristic approaches in order to deal with the problem. This work presents Asexual Reproduction Optimization (ARO), a model-free metaheuristic algorithm inspired by the asexual reproduction, in order to solve the portfolio optimization problem including cardinality constraint to ensure the investment in a given number of different assets and bounding constraint to limit the proportions of fund invested in each asset. This is the first time that this relatively new metaheuristic is applied in the field of portfolio optimization, and we show that ARO results in better quality solutions in comparison with some of the well-known metaheuristics stated in the literature. To validate our proposed algorithm, we measured the deviation of the obtained results from the standard efficient frontier. We report our computational results on a set of publicly available benchmark test problems relating to five main market indices containing 31, 85, 89, 98, and 225 assets. These results are used in order to test the efficiency of our proposed method in comparison to other existing metaheuristic solutions. The experimental results indicate that ARO outperforms Genetic Algorithm (GA), Tabu Search (TS), Simulated Annealing (SA), and Particle Swarm Optimization (PSO) in most of test problems. In terms of the obtained error, by using ARO, the average error of the aforementioned test problems is reduced by approximately 20 percent of the minimum average error calculated for the above-mentioned algorithms.

Keywords

Main Subjects


Article Title [فارسی]

انتخاب پورتفولیوی محدود کاردینالیتی مبتنی بر مارکویتز با استفاده از بهینه‌سازی تولید مثل غیرجنسی

Authors [فارسی]

  • محمدرضا صادقی مقدم 1
  • طاها منصوری 2
  • مرتضی شیخی زاده 3
1 گروه مدیریت تولید و عملیات، دانشکده مدیریت، دانشگاه تهران، تهران، ایران
2 گروه محاسبات، علوم و مهندسی، دانشگاه سالفورد، منچستر بزرگ، بریتانیا
3 کارشناسی ارشد مدیریت صنعتی، گروه مدیریت صنعتی، دانشکده مدیریت، دانشگاه تهران، تهران، ایران
Abstract [فارسی]

انتخاب پورتفولیوی مبتنی بر مارکویتز هنگام در نظر گرفتن محدودیت‌های اصلی به یک مسئله با پیچیدیگی محاسباتی سخت تبدیل می‌شود. در این حالت، راه حل های دقیق موجود مانند مدل های برنامه ریزی درجه دوم ممکن است برای حل مشکل کارآمد نباشد. بنابراین بسیاری از محققین از رویکردهای ابتکاری و فراابتکاری برای مقابله با این مشکل استفاده کرده اند. دراین پژوهش، الگوریتم فراابتکاری بهینه سازی بازتولید غیرجنسی (ARO) که از بازتولید غیرجنسی الهام گرفته شده است، به منظور حل مشکل بهینه سازی پورتفولیو از جمله محدودیت کاردینالی برای اطمینان از سرمایه گذاری در تعداد معینی از دارایی های مختلف و محدود ساختن محدودیت ها به منظور منحصر ساختن نسبت های سرمایه گذاری شده هر سهم  مورد استفاده قرار گرفته است. این پژوهش اولین مطالعه مربوط به الگوریتم های فراابتکاری می باشد که در زمینه بهینه سازی پورتفولیو پیاده سازی شده است. براساس نتایج پژوهش، ARO در مقایسه با برخی از فراابتکاری های معروف بیان شده در ادبیات، راه حل های با کیفیت بهتری را به ارائه میدهد. برای صحت سنجی الگوریتم پیشنهادی، انحراف نتایج به‌دست‌آمده از مرز کارآمد استاندارد را اندازه‌گیری شده است. بدین منظور با استفاده از داده های مربوط به پنج بازار متشکل از 31، 85، 89، 98، و 225 دارایی که در دسترس عموم قرار دارند دقت مدل مورد ارزیابی قرار گرفت. نتایج پژوهش نشان می‌دهد که ARO از الگوریتم ژنتیک (GA)، جستجوی تابو (TS) ، بازپخت شبیه‌سازی شده (SA) و بهینه‌سازی ازدحام ذرات (PSO) در بیشتر مسائل آزمایشی بهتر عمل می‌کند. از نظر خطای به‌دست‌آمده، با استفاده از ARO، میانگین خطای آزمون مذکور تقریباً کاهش 20 درصدی از حداقل میانگین خطای محاسبه‌شده برای الگوریتم‌های فوق را نشان می دهد.

Keywords [فارسی]

  • بهینه‌سازی پورتفولیو
  • محدودیت‌های کاردینالیتی
  • مدل میانگین واریانس مارکوویتز
  • بهینه‌سازی بازتولید غیرجنسی
  • مرز کارآمد
Ahmadian, S., & Khanteymoori, A. R. (2015, May). Training back propagation neural networks using asexual reproduction optimization. In 2015 7th Conference on Information and Knowledge Technology (IKT) (pp. 1-6). IEEE.
Ahmadi-Javid, A., & Fallah-Tafti, M. (2019). Portfolio optimization with entropic value-at-risk. European Journal of Operational Research, 279, 225-241. http://doi.org/10.1016/j.ejor.2019.02.007
Anagnostopoulos, K. P., & Mamanis, G. (2010). A portfolio optimization model with three objectives and discrete variables. Computers and Operations Research, 37(7), 1285–1297. https://doi.org/10.1016/j.cor.2009.09.009
Anagnostopoulos, K. P., & Mamanis, G. (2011). The mean-variance cardinality constrained portfolio optimization problem: An experimental evaluation of five multiobjective evolutionary algorithms. Expert Systems with Applications, 38(11), 14208–14217. https://doi.org/10.1016/j.eswa.2011.04.233
Asl, A. N., Menhaj, M. B., & Sajedin, A. (2014). Control of leader–follower formation and path planning of mobile robots using Asexual Reproduction Optimization (ARO). Applied soft computing, 14, 563-576.
Babazadeh, H., & Esfahanipour, A. (2019). A novel multi period mean-VaR portfolio optimization model considering practical constraints and transaction cost. Journal of Computational and Applied Mathematics, 361, 313-342.
Baykasoğlu, A., Yunusoglu, M. G., & Burcin Özsoydan, F. (2015). A GRASP based solution approach to solve cardinality constrained portfolio optimization problems. Computers & Industrial Engineering, 90, 339–351. https://doi.org/http://dx.doi.org/10.1016/j.cie.2015.10.009
Beasley, J. E. (1990). OR-library : Distributing test problems by electronic mail author(s). J. E. Beasley Source: The Journal of the Operational Research Society, 41(11), 1069–1072.
Bertsimas, D., & Shioda, R. (2009). Algorithm for cardinality-constrained quadratic optimization. Computational Optimization and Applications, 43(1), 1–22. https://doi.org/10.1007/s10589-007-9126-9
Bienstock, D. (1996). Computational study of a family of mixed-integer quadratic programming problems. Mathematical Programming, 74(2), 121–140. https://doi.org/10.1007/BF02592208
Branke, J., Scheckenbach, B., Stein, M., Deb, K., & Schmeck, H. (2009). Portfolio optimization with an envelope-based multi-objective evolutionary algorithm. European Journal of Operational Research, 199(3), 684–693. https://doi.org/10.1016/j.ejor.2008.01.054
Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for Cardinality Constrained Portfolio Optimization, 27 
Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302.
Chang, T.-J., Yang, S.-C., & Chang, K.-J. (2009). Portfolio optimization problems in different risk measures using genetic algorithm. Expert Systems with Applications, 36(7), 10529–10537. https://doi.org/10.1016/j.eswa.2009.02.062
Chen, B., Zhong, J., & Chen, Y. (2020). A hybrid approach for portfolio selection with higher-order moments: Empirical evidence from Shanghai Stock Exchange. Expert Systems with Applications, 145, 113104. https://doi.org/https://doi.org/10.1016/j.eswa.2019.113104
Chiam, S. C., Tan, K. C., & Al Mamum, A. (2008). Evolutionary multi-objective portfolio optimization in practical context. International Journal of Automation and Computing, 5(1), 67–80. https://doi.org/10.1007/s11633-008-0067-2
Corazza, M., di Tollo, G., Fasano, G., & Pesenti, R. (2021). A novel hybrid PSO-based metaheuristic for costly portfolio selection problems. Annals of Operations Research,  1–29.
Corazza, M., di Tollo, G., Fasano, G., & Pesenti, R. (2021). A novel hybrid PSO-based metaheuristic for costly portfolio selection problems. Annals of Operations Research, 304(1), 109-137.
Crama, Y., & Schyns, M. (2003). Simulated annealing for complex portfolio selection problems. European Journal of Operational Research, 150(3), 546–571. https://doi.org/10.1016/S0377-2217(02)00784-1
Cura, T. (2009). Particle swarm optimization approach to portfolio optimization. Nonlinear Analysis: Real World Applications, 10(4), 2396–2406. https://doi.org/10.1016/j.nonrwa.2008.04.023
Deng, G.-F., Lin, W.-T., & Lo, C.-C. (2012). Markowitz-based portfolio selection with cardinality constraints using improved particle swarm optimization. Expert Systems with Applications, 39(4), 4558–4566. https://doi.org/10.1016/j.eswa.2011.09.129
Esfahani, H. N., hossein Sobhiyah, M., & Yousefi, V. R. (2016). Project portfolio selection via harmony search algorithm and modern portfolio theory. Procedia-Social and Behavioral Sciences, 226, 51-58.
Farasat, A., Menhaj, M. B., Mansouri, T., & Moghadam, M. R. S. (2010). ARO: A new model-free optimization algorithm inspired from asexual reproduction. Applied Soft Computing Journal, 10(4), 1284–1292. https://doi.org/10.1016/j.asoc.2010.05.011
Fernández, A., & Gómez, S. (2007). Portfolio selection using neural networks. Computers & Operations Research, 34(4), 1177–1191. https://doi.org/http://dx.doi.org/10.1016/j.cor.2005.06.017
Gulpinar, N., An, L. T. H., & Moeini, M. (2010). Robust investment strategies with discrete asset choice constraints using DC programming. Optimization, 59(1), 45–62. https://doi.org/10.1080/02331930903500274
Gupta, P., Mehlawat, M. K., Yadav, S., & Kumar, A. (2019). A polynomial goal
programming approach for intuitionistic fuzzy portfolio optimization using entropy and higher moments. Applied Soft Computing, 85, 105781. https://doi.org/10.1016/j.asoc.2019.105781
Kalayci, C. B., Ertenlice, O., Akyer, H., & Aygoren, H. (2017). An artificial bee colony algorithm with feasibility enforcement and infeasibility toleration procedures for cardinality constrained portfolio optimization. Expert Systems with Applications, 85, 61-75.
Kazemi, M., Najafi, J., & Bagher, M. (2012). Fuzzy PD cascade controller design for ball and beam system based on an improved ARO technique.  , 5(1), 1–6.
Kazemi, M., Najafi, J., & MENHAJ, M. B. (2012). Fuzzy PD cascade controller design for ball and beam system based on an improved ARO technique.
Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research,  , 287–304. https://doi.org/10.1023/a:1019279918596
Kellerer, H., Mansini, R., & Speranza, M. G. (2000). Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research, 99(1), 287-304.
Khanteymoori, A. R., Menhaj, M. B., & Homayounpour, M. M. (2011). Structure learning in Bayesian networks using asexual reproduction optimization. ETRI Journal, 33(1), 39-49.
 
Lee, E. K., & Mitchell, J. E. (1997). Computational experience of an interior-point SQP algorithm in a parallel branch-and-bound framework 1 introduction 2 interior-point based nonlinear solver.  1–13.
Lee, E. K., & Mitchell, J. E. (1997). Computational experience of an interior-point SQP algorithm in a parallel branch-and-bound framework. Proceedings of High Performance Optimization Techniques 1997, 97-08.
Liagkouras, K., & Metaxiotis, K. (2018). Multi-period mean–variance fuzzy portfolio optimization model with transaction costs. Engineering applications of artificial intelligence, 67, 260-269.
Li B, Zhu Y, Sun Y, Aw G, Teo KL (2018) Multi-period portfolio selection problem under uncertain environment with bankruptcy constraint. Appl Math Model 56:539–550
Ling, J. Sun and M. Wang, Robust multi-period portfolio selection based on downside risk with asymmetrically distributed uncertainty set, European J. Oper. Res., 285 (2020), 81-95. 
Li, D., Sun, X., & Wang, J. (2006). Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection. Mathematical Finance, 16(1), 83–101. https://doi.org/10.1111/j.1467-9965.2006.00262.x
Lwin, K., & Qu, R. (2013). A hybrid algorithm for constrained portfolio selection problems. Applied Intelligence, 39(2), 251–266. https://doi.org/10.1007/s10489-012-0411-7
Lwin, K., Qu, R., & Kendall, G. (2014). A learning-guided multi-objective evolutionary algorithm for constrained portfolio optimization. Applied Soft Computing, 24, 757–772. https://doi.org/10.1016/j.asoc.2014.08.026
Mansini, R., & Speranza, M. G. (1999). Heuristic algorithms for the portfolio selection problem with minimum transaction lots. European Journal of Operational Research, 114(2), 219–233. https://doi.org/10.1016/S0377-2217(98)00252-5
Mansouri, T., Farasat, A., Menhaj, M. B., & Sadeghi Moghadam, M. R. (2011). ARO: A new model free optimization algorithm for real time applications inspired by the asexual reproduction. Expert Systems with Applications, 38(5), 4866–4874. https://doi.org/10.1016/j.eswa.2010.09.084
Maringer, D. (2006). Portfolio management with heuristic optimization. In  , 53(9), …  https://doi.org/10.1017/CBO9781107415324.004
Maringer, D. G. (2005). Portfolio management with heuristic optimization (Vol. 8). Springer Science & Business Media.
Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91.
Metaxiotis, K., & Liagkouras, K. (2012). Multiobjective evolutionary algorithms for portfolio management: A comprehensive literature review. Expert Systems with Applications, 39(14), 11685–11698. https://doi.org/10.1016/j.eswa.2012.04.053
Meghwani, S. S., & Thakur, M. (2017). Multi-criteria algorithms for portfolio optimization under practical constraints. Swarm and evolutionary computation, 37, 104-125.
Mohammadi, S., & Nazemi, A. (2020). On portfolio management with value at risk and uncertain returns via an artificial neural network scheme. Cognitive Systems Research, 59, 247-263.
Moral-Escudero, R., Ruiz-Torrubiano, R., & Suarez, A. (2006). Selection of optimal investment portfolios with cardinality constraints. 2006 IEEE International Conference on Evolutionary Computation, 2382–2388. https://doi.org/10.1109/CEC.2006.1688603
Ni, Q., Yin, X., Tian, K., & Zhai, Y. (2017). Particle swarm optimization with dynamic random population topology strategies for a generalized portfolio selection problem. Natural Computing, 16(1), 31-44.
Ponsich, A., Jaimes, A. L., & Coello, C. A. C. (2012). A survey on multiobjective evolutionary algorithms for the solution of the portfolio optimization problem and other finance and economics applications. IEEE Transactions on Evolutionary Computation, 17(3), 321-344.
Ruiz-torrubiano, R., & Suárez, A. (2010).
Ruiz-Torrubiano, R., & Suarez, A. (2010). Hybrid approaches and dimensionality reduction for portfolio selection with cardinality constraints. IEEE Computational Intelligence Magazine, 5(2), 92-107.
 Rujeerapaiboon, N., Kuhn, D., & Wiesemann, W. (2016). Robust growth-optimal portfolios. Management Science, 62(7), 2090-2109.
Schaerf, A. (2002). Local search techniques for constrained portfolio selection problems. Computational Economics, 20(3), 22 . https://doi.org/10.1023/a:1020920706534
Schaerf, A. (2002). Local search techniques for constrained portfolio selection problems. Computational Economics, 20(3), 177-190. https://doi.org/10.1023/a:1020920706534
Shaw, D., Liu, S., & Kopman, L. (2008). Lagrangian relaxation procedure for cardinality-constrained portfolio optimization. Optimisation Methods & Software, 23(3), 411–420. https://doi.org/10.1080/10556780701722542
Streichert, F., & Tanaka-yamawaki, M. (2006). The effect of local search on the constrained portfolio selection problem. Evolutionary Computation,  , 2368–2374. https://doi.org/10.1109/CEC.2006.1688601
Streichert, F., & Tanaka-Yamawaki, M. (2006, July). The effect of local search on the constrained portfolio selection problem. In 2006 IEEE International Conference on Evolutionary Computation (pp. 2368-2374). IEEE.
Tollo, G., & Roli, A. (2008). Metaheuristics for the portfolio selection problem. International Journal of Operations Research, 5(1), 13–35.
Vielma, J. P., Ahmed, S., & Nemhauser, G. L. (2008). A lifted linear programming branch-and-bound algorithm for mixed integer conic quadratic programs. INFORMS Journal on Computing, 20, 438–450. https://doi.org/10.1287/ijoc.1070.0256
Woodside-Oriakhi, M., Lucas, C., & Beasley, J. E. (2011). Heuristic algorithms for the cardinality constrained efficient frontier. European Journal of Operational Research, 213(3), 538–550. https://doi.org/10.1016/j.ejor.2011.03.030
Yazdanparast, N., Shahbazian, M., Aghajani, M., & Abed, S. P. (2015). Design of nonlinear CSTR control system using active disturbance rejection control optimized by asexual reproduction optimization.  , 3(2), 36–42.
Yazdanparast, N., Shahbazian, M., Aghajani, M., & Abed, S. P. (2015). Design of nonlinear CSTR control system using active disturbance rejection control optimized by asexual reproduction optimization. Journal of Automation and Control, 3(2), 36-42. https://doi.org/10.12691/automation-3-2-1
Young, M. R. (1998). A minimax portfolio selection rule with linear programming solution. Management Science, 44(5), 673–683. https://doi.org/10.1287/mnsc.44.5.673
Zhang, J., & Li, Q. (2019). Credibilistic mean-semi-entropy model for multi-period portfolio selection with background risk. Entropy, 21(10), 944.