Abstract
This paper proposes a novel multiobjective evolutionary Algorithm (MOEA) for the solution of the cardinality constrained portfolio optimization problem (CCPOP). The proposed algorithm introduces an efficient encoding scheme specially designed for dealing with the difficulties of the CCPOP. Also, the proposed algorithm incorporates a new mutation and recombination operator tailor-made to work well with the new encoding scheme. Datasets from seven different stock markets are utilized for testing the efficiency of the proposed approach. In particular, the performance of the proposed efficiently encoded multiobjective portfolio optimization solver (EEMPOS) is assessed in comparison with two well-known MOEAs, namely NSGAII and MOEA/D. The experimental results indicate that the proposed EEMPOS outperforms the two other MOEAs for all examined performance metrics when is applied to the solution of the CCPOP for a fraction of time required by the other techniques.
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Abbreviations
- \(\Omega \) :
-
The search space
- \(\hbox {f}_{\mathrm{i}}\) :
-
Objective function i
- \(\hbox {R}^{\mathrm{N}} \) :
-
Set of all N-tuples of real numbers (N-dimentional space of the decision variables)
- \(\rho _{ij} \) :
-
Correlation
- N :
-
Number of assets available
- \(w_i\) :
-
Decision variable denoting the proportion of asset i in the portfolio
- \(\sigma _i \) :
-
Standard deviation of stock returns
- \(K_{min}\) :
-
Minimum number of assets that a portfolio can hold
- \(K_{max} \) :
-
Maximum number of assets that a portfolio can hold
- \(l_i\) :
-
Floor constraint
- \(u_i \) :
-
Ceiling constraint
- \(P_{c}\) :
-
Crossover probability
- \(P_{m}\) :
-
Mutation probability
- \(\eta _{c}\) :
-
Distribution index for the crossover operator
- \(\eta _{m}\) :
-
Distribution index for the mutation operators
- F:
-
Real constant factor, \(\hbox {F}~\in ~[0, 2]\) which controls the amplification of the differential variation
- CR:
-
Crossover constant \(\hbox {CR}~\in ~[0, 1]\)
- \(\hbox {N}^{\mathrm{pop}}\) :
-
Population size
- HV:
-
Hypervolume indicator
- IGD:
-
Inverted Generational distance
- \(\hbox {I}_{\upvarepsilon }\) :
-
Epsilon indicator
References
Anagnostopoulos, K. P., & Mamanis, G. (2010). A portfolio optimization model with three objectives and discrete variables. Computers & Operations Research, 37(2010), 1285–1297.
Anagnostopoulos, K. P., & Mamanis, G. (2011a). The mean-variance cardinality constrained portfolio optimization problem: An experimental evaluation of five multiobjective evolutionary algorithms. Expert Systems with Applications, 38(2011), 14208–14217.
Anagnostopoulos, K. P., & Mamanis, G. (2011b). Multiobjective evolutionary algorithms for complex portfolio optimization problems. Computational Management Science, 8, 259–279.
Andriosopoulos, K., Doumpos, M., Papapostolou, N. C., & Pouliasis, P. K. (2013). Portfolio optimization and index tracking for the shipping stock and freight markets using evolutionary algorithms. Transportation Research Part E, 52(2013), 16–34.
Beasley, J. E. (1990). OR-Library: Distributing test problems by electronic mail. Journal of the Operational Research Society, 41(11), 1069–1072.
Beasley, J. E., Meade, N., & Chang, T. J. (2003). An evolutionary heuristic for the index tracking problem. European Journal of Operational Research, 148(2003), 621–643.
Bertsimas, D., & Shioda, R. (2009). Algorithm for cardinality-constrained quadratic opti-mization. Computational Optimization and Applications, 43(1), 1–22.
Bienstock, D. (1996). Computational study of a family of mixed-integer quadratic pro-gramming problems. Mathematical Programming, 74(2), 121–140.
Bird, R., & Tippett, M. (1986). Naive diversification and portfolio risk–A note. Management Science, 32(2), 244–251.
Bloomfield, T., Leftwich, R., & Long, J. (1977). Portfolio strategies and performance. Journal of Financial Economics, 5, 201–218.
Brands, S., & Gallagher, D. R. (2005). Portfolio selection, diversification and fund-of-funds: A note. Accounting & Finance, 45(2), 185–197.
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, 684–693.
Calvo, C., Ivorra, C., & Liern, V. (2016). Fuzzy portfolio selection with non-financial goals: exploring the efficient frontier. Annals of Operations Research, 245(1), 31–46.
Cesarone, F., Scozzari, A., & Tardella, F. (2008). Efficient algorithms for mean-variance portfolio optimization with hard real-world constraints. In The 18th AFIR colloquium: Financial risk in a changing world, Rome, September 30–October 3, 2008.
Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(2000), 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(2009), 10529–10537.
Chiam, S. C., Tan, K. C., & Al Mamum, A. (2008). Evolutionary multi-objective portfolio optimization in practical context. International Journal of Automation and Computing, 05(1), 67–80.
Corazza, M., Fasano, G., & Gusso, R. (2013). Particle Swarm Optimization with non-smooth penalty reformulation, for a complex portfolio selection problem. Applied Mathematics and Computation, 224(2013), 611–624.
Crama, Y., & Schyns, M. (2003). Simulated annealing for complex portfolio selection problems. European Journal of Operational Research, 150(2003), 546–571.
Deb, K., & Agrawal, R. B. (1995). Simulated binary crossover for continuous search space. Complex Systems, 9(2), 115–148.
Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA. II. IEEE Transactions on Evolutionary Computation, 6(2), 182–197.
Deb, K., & Tiwari, S. (2008). Omni-optimizer: A generic evolutionary algorithm for single and multi-objective optimization. European Journal of Operational Research., 185, 1.
Deng, G. F. & Lin, W. T. (2010). Ant colony optimization for markowitz mean-variance portfolio model, swarm, evolutionary, and memetic computing: First international conference on swarm, evolutionary, and memetic computing, SEMCCO 2010, Chennai, India, December 16–18, 2010, Volume 6466 of the series Lecture Notes in Computer Science (pp. 238–245). Berlin, Heidelberg: Springer.
Evans, J. L. (2004). Wealthy Investor Attitudes, Expectations, and Behaviors toward Risk and Return. The Journal of Wealth Management, 7(1), 12–18.
Evans, J. L., & Archer, S. H. (1968). Diversification and the reduction of dispersion: An empirical analysis. Journal of Finance, 23(5), 761–767.
Fabozzi, F. J., Huang, D., & Zhou, G. (2010). Robust portfolios: Contributions from operations research and finance. Annals of Operations Research, 176(1), 191–220.
Fielitz, B. D. (1974). Indirect versus direct diversification. Financial Management, 3(4), 54–62.
Fisher, L., & Lorie, J. H. (1970). Some studies of Variability of Returns on Investments in Common stocks. Journal of Business, 43(2), 99–134.
Hirschberger, M., Steuer, R. E., Utz, S., Wimmer, M., & Qi, Y. (2013). Computing the nondominated surface in tri-criterion portfolio. Operations Research, 61(1), 169–183.
Jennings, E. H. (1971). An empirical analysis of some aspects of common stock diversification. Journal of Financial and Quantitative Analysis, 6(02), 797–813.
Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative Finance, 1(2001), 1–13.
Johnson, K. H., & Shannon, D. S. (1974). A note on diversification and the reduction of dispersion. Journal of Financial Economics, 1(4), 365–372.
Krzemienowski, A. (2009). Risk preference modeling with conditional average: an application to portfolio optimization. Annals of Operations Research, 165(1), 67–95.
Krzemienowski, A., & Szymczyk, S. (2016). Portfolio optimization with a copula-based extension of conditional value-at-risk. Annals of Operations Research, 237(1), 219–236.
Lai, T.-Y. (1991). Portfolio selection with skewness: A multiple-objective approach. Review of Quantitative Finance and Accounting, 1(3), 293–305. doi:10.1007/BF02408382.
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.
Li, J., & Xu, J. (2013). Multi-objective portfolio selection model with fuzzy random returns and a compromise approach-based genetic algorithm. Information Sciences, 220(2013), 507–521.
Liagkouras, K., & Metaxiotis, K. (2013) An elitist polynomial mutation operator for improved performance of MOEAs in computer networks. In 2013 22nd international conference on computer communications and networks (ICCCN) (pp. 1–5). doi:10.1109/ICCCN.2013.6614105.
Liagkouras, K., & Metaxiotis, K. (2014). A new Probe Guided Mutation operator and its application for solving the cardinality constrained portfolio optimization problem. Expert Systems with Applications, 41(2014), 6274–6290.
Liagkouras, K., & Metaxiotis, K. (2015a). An experimental analysis of a new two-stage crossover operator for multiobjective optimization. Soft Computing,. doi:10.1007/s00500-015-1810-6.
Liagkouras, K., & Metaxiotis, K. (2015b). An experimental analysis of a new interval-based mutation operator. international journal of computational intelligence and applications, 14(3), 1550018. doi:10.1142/S1469026815500182.
Liagkouras, K., & Metaxiotis, K. (2015c). Efficient portfolio construction with the use of multiobjective evolutionary algorithms: Best practices and performance metrics. International Journal of Information Technology & Decision Making, 14(3), 535–564.
Lwin, K., Qu, R., & Kendall, G. (2014). A learning-guided multi-objective evolutionary algorithm forconstrained portfolio optimization. Applied Soft Computing, 24(2014), 757–772.
Maringer, D. (2005). Portfolio management with heuristic optimization (Vol. 8). Berlin: Springer. Advanced in Computational Management Science Series.
Maringer, D., & Parpas, P. (2009). Global optimization of higher order moments in portfolio selection. Journal of Global Optimization, 43(2–3), 219–230. doi:10.1007/s10898-007-9224-3.
Markowitz, H. (1952). Portfolio selection. 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(2012), 11685–11698.
Metaxiotis, K., & Liagkouras, K., (2013) A fitness guided mutation operator for improved performance of MOEAs. In 2013 IEEE 20th international conference on electronics, circuits, and systems (ICECS) (pp. 751–754). doi:10.1109/ICECS.2013.6815523.
Mishra, S. K., Panda, G., & Majhi, R. (2014). A comparative performance assessment of a set of multiobjective algorithms for constrained portfolio assets selection. Swarm and Evolutionary Computation, 16(2014), 38–51.
Qi, Y., Steuer, R. E., & Wimmer, M. (2015). An analytical derivation of the efficient surface in portfolio selection with three criteria. Annals of Operations Research, 1–17. doi:10.1007/s10479-015-1900-y.
Saborido, R., Ruiz, A. B., Bermúdez, J. D., Vercher, E., & Luque, M. (2016). Evolutionary multi-objective optimization algorithms for fuzzy portfolio selection. Applied Soft Computing, 39, 48–63.
Shaw, D. X., Liu, S., & Kopman, L. (2008). Lagrangian relaxation procedure for cardinality-constrained portfolio optimization. Optimisation Methods & Software, 23(3), 411–420.
Smimou, K. (2014). International portfolio choice and political instability risk: A multi-objective approach. European Journal of Operational Research, 234(2014), 546–560.
Solnik, B. H. (1974). Why not diversify internationally rather than domestically? Financial Analysts Journal, 30(4), 48–52+54.
Steuer, R. E., Qi, Y., & Hirschberger, M. (2007). Suitable-portfolio investors, nondominated frontier sensitivity, and the effect of multiple objectives on standard portfolio selection. Annals of Operations Research, 152(1), 297–317.
Storn, R., & Price, K. (1997). Differential evolution–A simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11, 341–359.
Streichert, F., Ulmer, H., & Zell, A. (2004). Evaluating a hybrid encoding and three crossover operators on the constrained portfolio selection problem. Proceedings of the congress on evolutionary computation (CEC 2004) (pp. 932–939). Oregon: Portland.
Suksonghong, K., Boonlong, K., & Goh, K. L. (2014). Multi-objective genetic algorithms for solving portfolio optimization problems in the electricity market. Electrical Power and Energy Systems, 58(2014), 150–159.
Tang, G. Y. N. (2004). How efficient is naive portfolio diversification? An educational note. Omega, 32(2), 155–160.
Wierzbicki, A. P. (1980). The use of reference objectives in multiobjective optimization. In G. Fandel & T. Gal (Eds.), Multiple criteria decision making, theory and applications (Vol. 177, pp. 468–486). Lecture Notes in Economics and Mathematical Systems. Berlin: Springer.
Wierzbicki, A. P. (1986). On the completeness and constructiveness of parametric characterizations to vector optimization problems. OR Spektrum, 8, 73–87.
Woodside-Oriakhi, M., Lucas, C., & Beasley, J. E. (2011). Heuristic algorithms for the cardinality constrained efficient frontier. European Journal of Operational Research, 213, 538–550.
Woodside-Oriakhi, M., Lucas, C., & Beasley, J. E. (2013). Portfolio rebalancing with an investment horizon and transaction costs. Omega, 41(2013), 406–420.
Zhang, Q., & Li, H. (2007). MOEA/D: A multiobjective evolutionary algorithm based on decomposition. IEEE Transactions on Evolutionary Computation, 11(6), 712–731.
Zitzler, E., Deb, K., & Thiele, L. (2000). Comparison of multiobjective evolutionary algorithms: Empirical results. Evolutionary Computation, 8(2), 173–195.
Zitzler, E., & Thiele, L. (1999). Multiobjective evolutionary algorithms: A comparative case study and the strength pareto approach. IEEE Transactions on Evolutionary Computation, 3(4), 257–271.
Zitzler, E., Thiele, L., Laumanns, M., Foneseca, C. M., & Fonseca, V. G. (2003). Performance assessment of multiobjective optimizers: An analysis and review. IEEE Transactions on Evolutionary Computation, 7(2), 117–132.
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Liagkouras, K., Metaxiotis, K. A new efficiently encoded multiobjective algorithm for the solution of the cardinality constrained portfolio optimization problem. Ann Oper Res 267, 281–319 (2018). https://doi.org/10.1007/s10479-016-2377-z
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DOI: https://doi.org/10.1007/s10479-016-2377-z