Overview of Portfolio Optimization Models

Document Type : Review Paper


Department of Financial Management, Faculty of Management, Islamic Azad University, Arak, Iran



Finding the best way to optimize the portfolio after Markowitz's 1952 article has always been and will continue to be one of the concerns of activists in the investment management industry. Researchers have come up with different solutions to overcome this problem. The introduction of mathematical models and meta-heuristic models is one of the activities that has influenced portfolio optimization in recent decades. Along with the growing use of portfolios and despite its rich literature, there are still many unanswered issues and questions in this area. Also, Iranian capital markets, as emerging markets, require native research to answer these questions and issues. The purpose of this study is to provide a useful and effective tool to assist professionals and researchers in portfolio selection theory. This study, while comprehensively reviewing the literature on the subject and the developments and expansions made in the area of portfolio selection and optimization, reviews the types of problems and optimization methods.


[1] Acerbi, C., Tasche, D., Expected shortfall: a natural coherent alternative value of risk, Journal of Economic Notes, 2002, 3(2), P. 379-388. Doi: 10.1111/1468-0300.00091.
[2] Bavi, O., Salehi, M., Genetic Algorithms and Optimization of composite structures, Tehran, Abed Publications, 2008, (In Persian).
[3] Bayat, A., Asadi, L., Stock Portfolio Optimization: The Benefits of Particle swarm optimization and Markowitz Model, Financial Engineering and Securities Management, 2017, 32, (In Persian).
[4] Campana, E.F., Diez, M., Fasano. G., Peri, D., Improving the initial particles position and parameters selection for PSO in bound constrained optimization problems, Springer Lecture Notes in Computer Science, 2013, 7928, P. 112-119. Doi: 10.1007/978-3-642-38703-6_13
[5] Chang, T.J., Meade, N., Beasley, J.E., Sharaiha, Y.M., Heuristics for cardinality constrained portfolio optimization, Computers & Operations Research, 2000, 27(13), P. 1271-1302. Doi:10.1016/S0305-0548(99)00074-X.
[6] Cong, F., Oosterlee, C.W., Multi-period mean–variance portfolio optimization based on Monte-Carlo simulation, Journal of Economic Dynamics and Control, 2016, 64, P. 23-38. Doi: 10.1016/j.jedc.2016.01.001
[7] DeMiguel, V., Mei, X., Nogales, F.J., Multiperiod portfolio optimization with multiple risky assets and general transaction costs, Journal of Banking & Finance, 2016, 69, P. 108-120. Doi: 10.1016/j.jbankfin.2016.04.002
[8] Deng, G.F., Lin, W.T., Lo, C.C., Markowitz-based portfolio selection with cardinality constraints using improved particle swarm optimization, Expert Systems with Applications, 2012, 39(4), P. 4558-4566. Doi:10.1016/j.eswa.2011.09.129.
[9] Doerner, K.F., Gutjahr W.J., Hartl R.F., Strauss, C., Stummer, C., Ant Colony Optimization in Multi Objective Portfolio Selection, Computer Science, MIC‟2001-4th Metaheuristics International Conference, 2001.
[10] Feshari, M., Mazaheri Far, P., Comparison of Forecasting and Portfolio Optimization Algorithms in Tehran Stock Exchange, Al-Zahra University Journal of Economic Development Policy, 2016, 4(11), (In Persian).
[11] Fobuzi, F., Modigliani, F., Ferry, M., Translated by Abdullah Tabrizi, Fundamentals of Markets and Financial Institutions, Pishboard Publishing, 1997, (In Persian).
[12] Gandomi, A.H., Alavi, A.H., Krill herd: A new bio-inspired optimization algorithm, Communications in Nonlinear Science and Numerical Simulation, 2012, 17(12), P. 4831-4845. Doi: 10.1016/j.cnsns.2012.05.010.
[13] Garkaz, M., Abbasi, A., Moghaddasi, M., Portfolio Selection and Optimization Using Genetic Algorithm Based on Different Definitions of Risk, Journal of Industrial Management, Faculty of Humanities, Azad University of Sanandaj, 2010, 5(11), (In Persian).
[14] Ghodousi, S., Tehrani, R., Bashiri, M., Portfolio Optimization Using Simulated Annealing Method, Financial Research, 2012, 17(1), P. 142-158. Doi: 10.22059/jfr.2015.52036, (In Persian).
[15] Goldenberg, D.E., Genetic Algorithms in Search, Optimization and Machine Learning, New York, Addison-Wesley Professional, 1989,  Doi:10.5860/choice.27-0936.
[16] Gutjahr, W.J., Katzensteiner, S., Reiter, P., Stummer, C., Denk, M., Competence_Driven Project Portfolio Selection, Scheduling and Staff Assignment, Central European Journal of Operations Research, 2008, 16(3), P. 281-306. Doi:10.1007/s10100-008-0057-z.
[17] Guo, S., Yu, L., Li, X., Kar, S., Fuzzy multi-period portfolio selection with different investment horizons, European Journal of Operational Research, 2016, 254(3), P. 1026-1035. Doi: 10.1016/j.ejor.2016.04.055
[18] Gupta, P., Mehlawat, M. K., Inuiguchi, M., Chandra, S., Fuzzy Portfolio Optimization, Springer-Verlag, Berlin, 2014. Doi: 10.1007/978-3-540-77926-1
[19] Izadikhah, M., Saen, R.F., Roostaee, R., How to assess sustainability of suppliers in the presence of volume discount and negative data in data envelopment analysis?, Annals of Operations Research, 2018, 269, P. 241–267, Doi: 10.1007/s10479-018-2790-6
[20] Izadikhah, M. Improving the Banks Shareholder Long Term Values by Using Data Envelopment Analysis Model. Advances in Mathematical Finance and Applications, 2018, 3(2), P. 27-41, Doi: 10.22034/amfa.2018.540829
[21] Jorion, P., Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition, New York: McGraw-Hill Education, 2006.
[22] Kandahari, M., Azar, A., Yazdanian, A., Gol Arazi, Gh., A Hybrid Model of Stochastic Dynamic Programming and Genetic Algorithm for Multistage Portfolio Optimization with GlueVaR Risk Measurement, Industrial Management, 2019, 11(3), P. 517-542. Doi: 10.22059/imj.2019.278912.1007579.
[23] Karaboga, D., Akay, B., A comparative study of Artificial Bee Colony algorithm, Applied Mathematics and Computation, 2009, 214(1), P. 108-132. Doi: 10.1016/j.amc.2009.03.090.
[24] Karimi, M., Ghalibaf Asl, H., Mohammadi, Sh., Portfolio Optimization Using Value at Risk Model in Tehran Stock Exchange, M.A. thesis, Al-Zahra University, Tehran, Iran, 2007, (In Persian).
[25] Kellerer, H., Mansini, R., Speranza, M.G., Selecting portfolios with fixed costs and minimum transaction lots, Annals of Operations Research, 2000, 99(1), P. 287-304. Doi: 10.1023/A:1019279918596.
[26] Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., Optimization by simulated annealing, Science, 1983, 220(4598), P. 671–680. Doi:10.1126/science.220.4598.671.
[27] Larsen, N., Mausser, H., Uryasev, S., Algorithms for optimization of value-at-risk, Vol. 70, Financial Engineering, E-commerce and Supply Chain. Kluwer Academic Publishers, 2002.
[28] Li, J., Xu, J., A novel portfolio selection model in a hybrid uncertain environment, Omega the International Journal of Management Science, 2009, 37(2). P. 439-449. Doi: 10.1016/j.omega.2007.06.002.
[29] Liebrman, J., Hielier, F., Operation Research, Translation: Yazdi, Vaziri. Tehran: Javan Publications, 2007, (In Persian).
[30] Liu, Y.-J., Zhang, W.-G., Zhang, P., A multi-period portfolio selection optimization model by using interval analysis, Economic Modelling, 2013, 33, P. 113-119. Doi: 10.1016/j.econmod.2013.03.006
[31] Liu, Y. J., Zhang, W. G., Xu, W. J., Fuzzy multi-period portfolio selection optimization models using multiple criteria, Automatica, 2012, 48(12), P. 3042-3053. Doi: 10.1016/j.automatica.2012.08.036
[32] Liu, Y.-J., Zhang, W.-G., Zhang, Q., Credibilistic multi-period portfolio optimization model with bankruptcy control and affine recourse, Applied Soft Computing, 2016, 38, P. 890-906. Doi: 10.1016/j.asoc.2015.09.023
[33] Liu, Y. J., Zhang, W. G., A multi-period fuzzy portfolio optimization model with minimum transaction lots, European Journal of Operational Research, 2015, 242(3), P. 933-941. Doi: 10.1016/j.ejor.2014.10.061
[34] Markowitz, H.M., Portfolio selection", Journal of Finance, 1952, 7(1), P. 77-91. Doi: 10.2307/2975974
[35] Mehdi Zadeh, S., Sabet, P., Optimal Selection of Petroleum Company Pension Fund Stock Portfolio Using Markowitz and VaR Models, Third Conference on Financial Mathematics and Applications, Semnan University, 2012, (In Persian).
[36] Mehlawat, M. K., Credibilistic mean-entropy models for multiperiod portfolio selection with multi-choice aspiration levels, Information Sciences, 2016, 345, P. 9-26. Doi: 10.1016/j.ins.2016.01.042
[37] Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E., Equation of State Calculations by Fast Computing Machines, Journal of Chemical Physics, 1953, 21(6), P.1087–1092. Doi: 10.1063/1.1699114.
[38] Mohammadi, E., Mohammadi, E., Ramtin Nia, Sh., Portfolio Optimization by Using the Symbiotic Organisms Search, Financial Research, 2016, 18(2), P. 369-390. Doi:10.22059/JFR.2016.61597, (In Persian).
[39] Najafi, A., Nopour, K., Ghahtarani, A., Portfolio interval optimization with conditional value at risk measure, Financial Research, 2017, 19(1), P. 157-172. Doi: 10.22059/JFR.2017.132312.1006053, (In Persian).
[40] Osborne, M.J., An introduction to game theory. New York: Oxford ‌ University Press. 2003.
[41] Osman, I.H., Kelly, J.P., Meta-Heuristics: An Overview, Boston: Kluwer Academic Publishers, 1996, P. 1-21. Doi: 10.1007/978-1-4613-1361-8_1.
[42] Paryabi, A., Salemi, Z., Stock Price Forecasting with Combined Data by Bee Colony Algorithm, First National Conference on Economics, Management and Accounting, 2016, (In Persian).
[43] Qasemi, J., Sarveh, F., A Review on the Application of Meta-Heuristic Algorithms in Financial Matters, Business Reviews, 2018, 96, (In Persian).
[44] Qu, B.Y., Suganthan. P.N., Constrained multi-objective optimization algorithm with an ensemble of constraint handling methods, Engineering Optimization, 2011, 43(4), P. 403-416.
[45] Radpour, M., Value at Risk and its Test in Tehran Stock Exchange, M.A. thesis, Shahid Beheshti University, Tehran, Iran, 2008, (In Persian).
[46] Raei, R., Talangi, A., Advanced Investment Management, The Organization for Researching and Composing University textbooks in the Humanities (SAMT), 2004, P. 105-142, (In Persian).
[47] Rockafeller, T., Uryasev, S., Conditional value-at-risk for general loss distributions, Journal of Banking and Finance, 2002, 26(7), P. 1443–1471. Doi: 10.1016/S0378-4266(02)00271-6.
[48] Rockafeller, T., Uryasev, S., Optimization of conditional value-at-risk, Journal of Risk, 2000, 2(3), P. 21-41.
[49] Schaerf, A., Local search techniques for constrained portfolio selection problems, Computational Economics, 2002, 20(3), P. 177-190. Doi: 10.1023/A:1020920706534.
[50] Shiri Ghohi, A., Didehkhani, H., Khalili Damghani, K., Saeedi, P., A Comparative Study of Multi-Objective Multi-Period Portfolio Optimization Models in a Fuzzy Credibility Environment Using Different Risk Measures, Financial Management Strategy, 2017, 5(18), P. 1-26, (In Persian).
[51] Talebi, A., Selection and optimization of portfolio using meta-heuristic methods and Comparison with Experts and Newcomers' Organizational Baskets in Tehran Stock Exchange, M.A thesis, Faculty of Management, Shahroud University of Technology, Shahroud, Iran, 2010, (In Persian).
[52] Tehrani, R., Falah Tafti, S., Asefi, S., Portfolio Optimization Using Krill Herd Metaheuristic Algorithm Considering Different Measures of Risk in Tehran Stock Exchange, Financial Research, 2018, 20(4), P. 409-426. DOI: 10.22059/frj.2019.244004.1006538, (In Persian).
[53] Tunchan, C., Particle swarm optimization approach to portfolio optimization, Nonlinear Analysis Real World Applications, 2009, 10(4), P. 2396-2406. Doi: 10.1016/j.nonrwa.2008.04.023.
[54] Vantsel, Y., Game theory and its application to strategic decision-making. Translation: Roshandel, Tayeb, Tehran: Ghoomes Publications, 1995, (In Persian).
[55] Yamai, Y., Yoshiba, T., On the Validity of Value-at-Risk: Comparative Analyses with Expected shortfall, Monetary and Economic Studies, 2002, 20(1), P. 57-85.
[56] Vercher, E., Bermúdez, J. D., Portfolio optimization using a credibility mean-absolute semi-deviation model, Expert Systems with Applications, 2015, 42(20), P. 7121-7131. Doi: 10.1016/j.eswa.2015.05.020
[57] Yao, H., Li, Z., Li, D., Multi-period mean-variance portfolio selection with stochastic interest rate and uncontrollable liability, European Journal of Operational Research, 2016, 252(3), P. 837-851. Doi: 10.1016/j.ejor.2016.01.049
[58] Yeganegi Dasjerdi, V., Game theory (1), Tehran: Encyclopedia of urban Economies, 2010, (In Persian).
[59] Zhang, W. G., Liu, Y. J., Xu, W. J., A possibilistic mean-semivarianceentropy model for multi-period portfolio selection with transaction costs, European Journal of Operational Research, 2012, 222(2), P. 341-349. Doi: 10.1016/j.ejor.2012.04.023