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Portfolio selection problems with Markowitz’s mean–variance framework: a review of literature

Fuzzy Optimization and Decision Making volume 17, pages 125–158 (2018)Cite this article

Abstract

Since the pioneering work of Harry Markowitz, mean–variance portfolio selection model has been widely used in both theoretical and empirical studies, which maximizes the investment return under certain risk level or minimizes the investment risk under certain return level. In this paper, we review several variations or generalizations that substantially improve the performance of Markowitz’s mean–variance model, including dynamic portfolio optimization, portfolio optimization with practical factors, robust portfolio optimization and fuzzy portfolio optimization. The review provides a useful reference to handle portfolio selection problems for both researchers and practitioners. Some summaries about the current studies and future research directions are presented at the end of this paper.

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References

  • Ammar, E. E. (2008). On solutions of fuzzy random multiobjective quadratic programming with applications in portfolio problem. Information Sciences, 178(2), 468–484.

    MathSciNet  MATH  Article  Google Scholar 

  • Anagnostopoulos, K. P., & Mamanis, G. (2010). A portfolio optimization model with three objectives and discrete variables. Computers and Operations Research, 37(7), 1285–1297.

    MathSciNet  MATH  Article  Google Scholar 

  • Bajeux-Besnainou, I., & Portait, R. (1998). Dynamic asset allocation in a mean-variance framework. Management Science, 44(11), 79–95.

    MATH  Article  Google Scholar 

  • Barry, C. B. (1974). Portfolio analysis under uncertain means, variances, and covariances. The Journal of Finance, 29(2), 515–522.

    Article  Google Scholar 

  • Basak, S., & Chabakauri, G. (2010). Dynamic mean-variance asset allocation. Review of Financial Studies, 23(8), 2970–3016.

    Article  Google Scholar 

  • Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management Science, 17(4), B-141.

    MathSciNet  MATH  Article  Google Scholar 

  • Ben-Tal, A., Margalit, T., & Nemirovski, A. (2000). Robust modeling of multi-stage portfolio problems. High performance optimization (pp. 303–328). New York: Springer.

    Chapter  Google Scholar 

  • Ben-Tal, A., & Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23(4), 769–805.

    MathSciNet  MATH  Article  Google Scholar 

  • Ben-Tal, A., & Nemirovski, A. (1999). Robust solutions of uncertain linear programs. Operations Research Letters, 25(1), 1–13.

    MathSciNet  MATH  Article  Google Scholar 

  • Bensoussan, A., Wong, K. C., Yam, S. C. P., & Yung, S. P. (2014). Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting. SIAM Journal on Financial Mathematics, 5(1), 153–190.

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  • Best, M. J., & Grauer, R. R. (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: Some analytical and computational results. Review of Financial Studies, 4(2), 315–342.

    Article  Google Scholar 

  • Bielecki, T. R., Jin, H., Pliska, S. R., & Zhou, X. Y. (2005). Continuous-time mean-variance portfolio selection with Bankruptcy prohibition. Mathematical Finance, 15(2), 213–244.

    MathSciNet  MATH  Article  Google Scholar 

  • Björk, T., & Murgoci, A. (2010). A general theory of Markovian time inconsistent stochastic control problems. Available at SSRN 1694759.

  • Björk, T., & Murgoci, A. (2014). A theory of Markovian time-inconsistent stochastic control in discrete time. Finance and Stochastics, 18(3), 545–592.

    MathSciNet  MATH  Article  Google Scholar 

  • Björk, T., Murgoci, A., & Zhou, X. Y. (2014). Mean-variance portfolio optimization with state-dependent risk aversion. Mathematical Finance, 24(1), 1–24.

    MathSciNet  MATH  Article  Google Scholar 

  • Bodnar, T., Mazur, S., & Okhrin, Y. (2015). Bayesian estimation of the global minimum variance portfolio. In Working papers in statistics.

  • Bodnar, T., & Schmid, W. (2008). A test for the weights of the global minimum variance portfolio in an elliptical model. Metrika, 67(2), 127–143.

    MathSciNet  MATH  Article  Google Scholar 

  • Brown, S. J. (1976). Optimal portfolio choice under uncertainty: A Bayesian approach. (Doctoral dissertation, University of Chicago, Graduate School of Business).

  • Candelon, B., Hurlin, C., & Tokpavi, S. (2012). Sampling error and double shrinkage estimation of minimum variance portfolios. Journal of Empirical Finance, 19(4), 511–527.

    Article  Google Scholar 

  • Carlsson, C., & Fullér, R. (2001). On possibilistic mean value and variance of fuzzy numbers. Fuzzy Sets and Systems, 122(2), 315–326.

    MathSciNet  MATH  Article  Google Scholar 

  • Carlsson, C., Fullér, R., & Majlender, P. (2002). A possibilistic approach to selecting portfolios with highest utility score. Fuzzy Sets and Systems, 131(1), 13–21.

    MathSciNet  MATH  Article  Google Scholar 

  • Ceria, S., & Stubbs, R. A. (2006). Incorporating estimation errors into portfolio selection: Robust portfolio construction. Journal of Asset Management, 7(2), 109–127.

    Article  Google Scholar 

  • Chan, L. K., Karceski, J., & Lakonishok, J. (1999). On portfolio optimization: Forecasting covariances and choosing the risk model. Review of Financial Studies, 12(5), 937–974.

    Article  Google Scholar 

  • Chang, H. (2015). Dynamic mean-variance portfolio selection with liability and stochastic interest rate. Economic Modelling, 51, 172–182.

    Article  Google Scholar 

  • Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers and Operations Research, 27(13), 1271–1302.

    MATH  Article  Google Scholar 

  • Chen, Y., Liu, Y. K., & Chen, J. (2006). Fuzzy portfolio selection problems based on credibility theory. Lecture Notes in Computer Science (pp. 377–386).

  • Chen, P., & Yang, H. (2011). Markowitz’s mean-variance asset-liability management with regime switching: A multi-period model. Applied Mathematical Finance, 18(1), 29–50.

    MathSciNet  MATH  Article  Google Scholar 

  • Chen, P., Yang, H., & Yin, G. (2008). Markowitz’s mean-variance asset-liability management with regime switching: A continuous-time model. Insurance: Mathematics and Economics, 43(3), 456–465.

    MathSciNet  MATH  Google Scholar 

  • Chen, W., & Zhang, W. G. (2010). The admissible portfolio selection problem with transaction costs and an improved PSO algorithm. Physica A: Statistical Mechanics and its Applications, 389(10), 2070–2076.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Chiu, M. C., & Li, D. (2006). Asset and liability management under a continuous-time mean-variance optimization framework. Insurance: Mathematics and Economics, 39(3), 330–355.

    MathSciNet  MATH  Google Scholar 

  • Chiu, M. C., & Wong, H. Y. (2011). Mean-variance portfolio selection of cointegrated assets. Journal of Economic Dynamics and Control, 35(8), 1369–1385.

    MathSciNet  MATH  Article  Google Scholar 

  • Chiu, M. C., & Wong, H. Y. (2013). Mean-variance principle of managing cointegrated risky assets and random liabilities. Operations Research Letters, 41(1), 98–106.

    MathSciNet  MATH  Article  Google Scholar 

  • Chiu, M. C., & Wong, H. Y. (2014). Mean-variance portfolio selection with correlation risk. Journal of Computational and Applied Mathematics, 263, 432–444.

    MathSciNet  MATH  Article  Google Scholar 

  • Chiu, M. C., & Wong, H. Y. (2015). Dynamic cointegrated pairs trading: Mean-variance time-consistent strategies. Journal of Computational and Applied Mathematics, 290, 516–534.

    MathSciNet  MATH  Article  Google Scholar 

  • Chopra, V. K. (1993). Improving optimization. The. Journal of Investing, 2(3), 51–59.

    Article  Google Scholar 

  • Chopra, V. K., & Ziemba, W. T. (1993). The effect of errors in means, variances, and covariances on optimal portfolio choice. Journal of Portfolio Management, 19(2), 6–11.

    Article  Google Scholar 

  • Costa, O. L., & Araujo, M. V. (2008). A generalized multi-period mean-variance portfolio optimization with Markov switching parameters. Automatica, 44(10), 2487–2497.

    MathSciNet  MATH  Article  Google Scholar 

  • Costa, O. L. V., & Nabholz, R. B. (2007). Multiperiod mean-variance optimization with intertemporal restrictions. Journal of Optimization Theory and Applications, 134(2), 257–274.

    MathSciNet  MATH  Article  Google Scholar 

  • Crama, Y., & Schyns, M. (2003). Simulated annealing for complex portfolio selection problems. European Journal of Operational Research, 150(3), 546–571.

    MATH  Article  Google Scholar 

  • Cui, X., Gao, J., Li, X., & Li, D. (2014a). Optimal multi-period mean-variance policy under no-shorting constraint. European Journal of Operational Research, 234(2), 459–468.

    MathSciNet  MATH  Article  Google Scholar 

  • Cui, X., Li, X., & Li, D. (2014b). Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection. IEEE Transactions on Automatic Control, 59(7), 1833–1844.

    MathSciNet  MATH  Article  Google Scholar 

  • Cui, X., Li, D., Wang, S., & Zhu, S. (2012). Better than dynamic mean-variance: Time inconsistency and free cash flow stream. Mathematical Finance, 22(2), 346–378.

    MathSciNet  MATH  Article  Google Scholar 

  • Cui, X., Li, X., Wu, X., & Yi, L. (2015). A mean-field formulation for optimal multi-period asset–liability mean–variance portfolio selection with an uncertain exit time. Available at SSRN 2680109.

  • Cura, T. (2009). Particle swarm optimization approach to portfolio optimization. Nonlinear Analysis: Real World Applications, 10(4), 2396–2406.

    MathSciNet  MATH  Article  Google Scholar 

  • Czichowsky, C. (2013). Time-consistent mean-variance portfolio selection in discrete and continuous time. Finance and Stochastics, 17(2), 227–271.

    MathSciNet  MATH  Article  Google Scholar 

  • Dai, M., & Zhong, Y. (2008). Penalty methods for continuous-time portfolio selection with proportional transaction costs. Available at SSRN 1210105.

  • Davis, M. H., & Norman, A. R. (1990). Portfolio selection with transaction costs. Mathematics of Operations Research, 15(4), 676–713.

    MathSciNet  MATH  Article  Google Scholar 

  • DeMiguel, V., & Nogales, F. J. (2009). Portfolio selection with robust estimation. Operations Research, 57(3), 560–577.

    MathSciNet  MATH  Article  Google Scholar 

  • Deng, X., & Li, R. (2010). A portfolio selection model based on possibility theory using fuzzy two-stage algorithm. Journal of Convergence Information Technology, 5(6), 138–145.

    MathSciNet  Article  Google Scholar 

  • Deng, X., & Li, R. (2012). A portfolio selection model with borrowing constraint based on possibility theory. Applied Soft Computing, 12(2), 754–758.

    MathSciNet  Article  Google Scholar 

  • Deng, X., & Li, R. (2014). Gradually tolerant constraint method for fuzzy portfolio based on possibility theory. Information Sciences, 259, 16–24.

    MathSciNet  MATH  Article  Google Scholar 

  • Deng, S., & Min, X. (2013). Applied optimization in global efficient portfolio construction using earning forecasts. The Journal of Investing, 22(4), 104–114.

    Article  Google Scholar 

  • Dubois, D., & Prade, H. (1988). Possibility theory. New York: Plenum Press.

    MATH  Book  Google Scholar 

  • Dumas, B., & Luciano, E. (1991). An exact solution to a dynamic portfolio choice problem under transactions costs. The Journal of Finance, 46(2), 577–595.

    Article  Google Scholar 

  • Ehrgott, M., Klamroth, K., & Schwehm, C. (2004). An MCDM approach to portfolio optimization. European Journal of Operational Research, 155(3), 752–770.

    MathSciNet  MATH  Article  Google Scholar 

  • Fabozzi, F. J., Huang, D., & Zhou, G. (2010). Robust portfolios: Contributions from operations research and finance. Annals of Operations Research, 176(1), 191–220.

    MathSciNet  MATH  Article  Google Scholar 

  • Fabozzi, F. J., Kolm, P. N., Pachamanova, D., & Focardi, S. M. (2007). Robust portfolio optimization and management. Hoboken: Wiley.

    Google Scholar 

  • Fernández, A., & Gómez, S. (2007). Portfolio selection using neural networks. Computers and Operations Research, 34(4), 1177–1191.

    MATH  Article  Google Scholar 

  • Frost, P. A., & Savarino, J. E. (1986). An empirical Bayes approach to efficient portfolio selection. Journal of Financial and Quantitative Analysis, 21(03), 293–305.

    Article  Google Scholar 

  • Fu, C., Lari-Lavassani, A., & Li, X. (2010). Dynamic mean-variance portfolio selection with borrowing constraint. European Journal of Operational Research, 200(1), 312–319.

    MathSciNet  MATH  Article  Google Scholar 

  • Fullér, R., & Majlender, P. (2003). On weighted possibilistic mean and variance of fuzzy numbers. Fuzzy Sets and Systems, 136(3), 363–374.

    MathSciNet  MATH  Article  Google Scholar 

  • Gao, J., Li, D., Cui, X., & Wang, S. (2015). Time cardinality constrained mean-variance dynamic portfolio selection and market timing: A stochastic control approach. Automatica, 54, 91–99.

    MathSciNet  MATH  Article  Google Scholar 

  • Garlappi, L., Uppal, R., & Wang, T. (2007). Portfolio selection with parameter and model uncertainty: A multi-prior approach. Review of Financial Studies, 20(1), 41–81.

    Article  Google Scholar 

  • Goldfarb, D., & Iyengar, G. (2003). Robust portfolio selection problems. Mathematics of Operations Research, 28(1), 1–38.

    MathSciNet  MATH  Article  Google Scholar 

  • Grauer, R. R., & Hakansson, N. H. (1993). On the use of mean-variance and quadratic approximations in implementing dynamic investment strategies: A comparison of returns and investment policies. Management Science, 39(7), 856–871.

    Article  Google Scholar 

  • Greyserman, A., Jones, D. H., & Strawderman, W. E. (2006). Portfolio selection using hierarchical Bayesian analysis and MCMC methods. Journal of Banking and Finance, 30(2), 669–678.

    Article  Google Scholar 

  • Guo, Z., & Duan, B. (2015). Dynamic mean-variance portfolio selection in market with jump-diffusion models. Optimization, 64(3), 663–674.

    MathSciNet  MATH  Google Scholar 

  • Hakansson, N. H. (1971). Capital growth and the mean-variance approach to portfolio selection. Journal of Financial and Quantitative Analysis, 6(01), 517–557.

    Article  Google Scholar 

  • Hao, F. F., & Liu, Y. K. (2009). Mean-variance models for portfolio selection with fuzzy random returns. Journal of Applied Mathematics and Computing, 30(1–2), 9–38.

    MathSciNet  MATH  Article  Google Scholar 

  • Hasuike, T., Katagiri, H., & Ishii, H. (2009). Portfolio selection problems with random fuzzy variable returns. Fuzzy Sets and Systems, 160(18), 2579–2596.

    MathSciNet  MATH  Article  Google Scholar 

  • Henri, W. Big data techniques can give institutional portfolio managers upper hand. http://www.wallstreetandtech.com/trading-technology/big-data-techniques-can-give-institutional-portfolio-managers-upper-hand/d/d-id/1268644?.

  • Huang, X. (2007a). Portfolio selection with fuzzy returns. Journal of Intelligent and Fuzzy Systems, 18(4), 383–390.

    MATH  Google Scholar 

  • Huang, X. (2007b). Two new models for portfolio selection with stochastic returns taking fuzzy information. European Journal of Operational Research, 180(1), 396–405.

    MathSciNet  MATH  Article  Google Scholar 

  • Huang, X. (2007c). A new perspective for optimal portfolio selection with random fuzzy returns. Information Sciences, 177(23), 5404–5414.

    MathSciNet  MATH  Article  Google Scholar 

  • Huang, X. (2009). A review of credibilistic portfolio selection. Fuzzy Optimization and Decision Making, 8(3), 263–281.

    MathSciNet  MATH  Article  Google Scholar 

  • Huang, X. (2011). Minimax mean-variance models for fuzzy portfolio selection. Soft Computing, 15(2), 251–260.

    MathSciNet  MATH  Article  Google Scholar 

  • Huang, X. (2012). Mean-variance models for portfolio selection subject to experts estimations. Expert Systems with Applications, 39(5), 5887–5893.

    Article  Google Scholar 

  • Ida, M. (2003). Portfolio selection problem with interval coefficients. Applied Mathematics Letters, 16(5), 709–713.

    MathSciNet  MATH  Article  Google Scholar 

  • Jagannathan, R., & Ma, T. (2002). Risk reduction in large portfolios: Why imposing the wrong constraints helps (No. w8922). National Bureau of Economic Research.

  • James, W., & Stein, C. (1961). Estimation with quadratic loss. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (pp. 361–379).

  • Jin, H., & Zhou, X. Y. (2007). Continuous-time Markowitz’s problems in an incomplete market, with no-shorting portfolios. Stochastic Analysis and Applications (pp. 435–459). Berlin: Springer.

  • Jorion, P. (1986). Bayes-Stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis, 21(03), 279–292.

    Article  Google Scholar 

  • Kan, R., & Zhou, G. (2007). Optimal portfolio choice with parameter uncertainty. Journal of Financial and Quantitative Analysis, 42(03), 621–656.

    Article  Google Scholar 

  • Karatzas, I., Lehoczky, J. P., & Shreve, S. E. (1987). Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM Journal on Control and Optimization, 25(6), 1557–1586.

    MathSciNet  MATH  Article  Google Scholar 

  • Kim, J. H., Kim, W. C., & Fabozzi, F. J. (2014). Recent developments in robust portfolios with a worst-case approach. Journal of Optimization Theory and Applications, 161(1), 103–121.

    MathSciNet  MATH  Article  Google Scholar 

  • Klein, R. W., & Bawa, V. S. (1976). The effect of estimation risk on optimal portfolio choice. Journal of Financial Economics, 3(3), 215–231.

    Article  Google Scholar 

  • Kruse, R., & Meyer, K. D. (1987). Statistics with vague data. New York: Springer.

    MATH  Book  Google Scholar 

  • Kwarkernaak, H. (1978). Fuzzy random variables (I). Information Sciences, 15(1), 1–29.

    MathSciNet  Article  Google Scholar 

  • Kwarkernaak, H. (1979). Fuzzy random variables (II). Information Sciences, 17, 253–278.

    MathSciNet  Article  Google Scholar 

  • Lai, K. K., Wang, S. Y., Xu, J. P., Zhu, S. S., & Fang, Y. (2002). A class of linear interval programming problems and its application to portfolio selection. IEEE Transactions on Fuzzy Systems, 10(6), 698–704.

    Article  Google Scholar 

  • Ledoit, O., & Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10(5), 603–621.

    Article  Google Scholar 

  • Ledoit, O., & Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88(2), 365–411.

    MathSciNet  MATH  Article  Google Scholar 

  • Leippold, M., Trojani, F., & Vanini, P. (2004). A geometric approach to multiperiod mean variance optimization of assets and liabilities. Journal of Economic Dynamics and Control, 28(6), 1079–1113.

    MathSciNet  MATH  Article  Google Scholar 

  • Leippold, M., Trojani, F., & Vanini, P. (2011). Multiperiod mean-variance efficient portfolios with endogenous liabilities. Quantitative Finance, 11(10), 1535–1546.

    MathSciNet  MATH  Article  Google Scholar 

  • Liagkouras, K., & Metaxiotis, K. (2015). Efficient portfolio construction with the use of multiobjective evolutionary algorithms: Best practices and performance metrics. International Journal of Information Technology and Decision Making, 14(03), 535–564.

    Article  Google Scholar 

  • Li, C., & Li, Z. (2012). Multi-period portfolio optimization for asset-liability management with bankrupt control. Applied Mathematics and Computation, 218(22), 11196–11208.

    MathSciNet  MATH  Article  Google Scholar 

  • Li, X., & Liu, B. (2006a). A sufficient and necessary condition for credibility measures. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 14(05), 527–535.

    MathSciNet  MATH  Article  Google Scholar 

  • Li, X., & Liu, B. (2006b). New independence definition of fuzzy random variable and random fuzzy variable. World Journal of Modelling and Simulation, 2(5), 338–342.

    Google Scholar 

  • Li, X., & Liu, B. (2009). Chance measure for hybrid events with fuzziness and randomness. Soft Computing, 13(2), 105–115.

    MathSciNet  MATH  Article  Google Scholar 

  • Li, D., & Ng, W. L. (2000). Optimal dynamic portfolio selection: Multiperiod mean-variance formulation. Mathematical Finance, 10(3), 387–406.

    MathSciNet  MATH  Article  Google Scholar 

  • Li, X., Guo, S. N., & Yu, L. A. (2015a). Skewness of fuzzy numbers and its applications in portfolio selection. IEEE Transactions on Fuzzy Systems, 23(6), 2135–2143.

    Article  Google Scholar 

  • Li, Y., Qiao, H., Wang, S., & Zhang, L. (2015b). Time-consistent investment strategy under partial information. Insurance: Mathematics and Economics, 65, 187–197.

    MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Article  Google Scholar 

  • Li, Z. F., & Xie, S. X. (2010). Mean-variance portfolio optimization under stochastic income and uncertain exit time. Dynamics of Continuous, Discrete and Impulsive Systems B: Applications and Algorithms, 17, 131–147.

    MathSciNet  MATH  Google Scholar 

  • Li, J., & Xu, J. (2009). A novel portfolio selection model in a hybrid uncertain environment. Omega, 37(2), 439–449.

    Article  Google Scholar 

  • Li, J., & Xu, J. (2013). Multi-objective portfolio selection model with fuzzy random returns and a compromise approach-based genetic algorithm. Information Sciences, 220, 507–521.

    MathSciNet  MATH  Article  Google Scholar 

  • Li, X., Zhang, Y., Wong, H. S., & Qin, Z. (2009). A hybrid intelligent algorithm for portfolio selection problem with fuzzy returns. Journal of Computational and Applied Mathematics, 233(2), 264–278.

  • Li, T., Zhang, W., & Xu, W. (2015c). A fuzzy portfolio selection model with background risk. Applied Mathematics and Computation, 256, 505–513.

    MathSciNet  MATH  Article  Google Scholar 

  • Li, X., Zhou, X. Y., & Lim, A. E. (2002). Dynamic mean-variance portfolio selection with no-shorting constraints. SIAM Journal on Control and Optimization, 40(5), 1540–1555.

    MathSciNet  MATH  Article  Google Scholar 

  • Lim, A. E., & Zhou, X. Y. (2002). Mean-variance portfolio selection with random parameters in a complete market. Mathematics of Operations Research, 27(1), 101–120.

    MathSciNet  MATH  Article  Google Scholar 

  • Liu, B. (2002). Random fuzzy dependent-chance programming and its hybrid intelligent algorithm. Information Sciences, 141(3), 259–271.

    MATH  Article  Google Scholar 

  • Liu, B. (2007). Uncertainty theory. Berlin: Springer.

    MATH  Book  Google Scholar 

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

    Article  Google Scholar 

  • Liu, B., & Liu, Y. K. (2002). Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems, 10(4), 445–450.

    Article  Google Scholar 

  • Liu, Y. K., & Liu, B. (2003a). Fuzzy random variables: A scalar expected value operator. Fuzzy Optimization and Decision Making, 2(2), 143–160.

    MathSciNet  Article  Google Scholar 

  • Liu, Y. K., & Liu, B. (2003b). Expected value operator of random fuzzy variable and random fuzzy expected value models. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 11(02), 195–215.

    MathSciNet  MATH  Article  Google Scholar 

  • Liu, H., & Loewenstein, M. (2002). Optimal portfolio selection with transaction costs and finite horizons. Review of Financial Studies, 15(3), 805–835.

    Article  Google Scholar 

  • Liu, Y. K., Wu, X., & Hao, F. (2012). A new chance-variance optimization criterion for portfolio selection in uncertain decision systems. Expert Systems with Applications, 39(7), 6514–6526.

    Article  Google Scholar 

  • Liu, Y. J., Zhang, W. G., & Zhang, P. (2013). A multi-period portfolio selection optimization model by using interval analysis. Economic Modelling, 33, 113–119.

    Article  Google Scholar 

  • Lobo, M. S., Fazel, M., & Boyd, S. (2007). Portfolio optimization with linear and fixed transaction costs. Annals of Operations Research, 152(1), 341–365.

    MathSciNet  MATH  Article  Google Scholar 

  • Lu, Z. (2011). Robust portfolio selection based on a joint ellipsoidal uncertainty set. Optimization Methods and Software, 26(1), 89–104.

    MathSciNet  MATH  Article  Google Scholar 

  • Ma, H. Q., Wu, M., & Huang, N. J. (2015). A random parameter model for continuous-time mean–variance asset–liability management. Mathematical Problems in Engineering 2015.

  • Maillet, B., Tokpavi, S., & Vaucher, B. (2015). Global minimum variance portfolio optimisation under some model risk: A robust regression-based approach. European Journal of Operational Research, 244(1), 289–299.

    MathSciNet  MATH  Article  Google Scholar 

  • Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91.

  • Markowitz, H. (1959). Portfolio selection: Efficient diversfication of investments. New York: Wiley.

  • Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. The review of Economics and Statistics, 51(3), 247–257.

    Article  Google Scholar 

  • Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3(4), 373–413.

    MathSciNet  MATH  Article  Google Scholar 

  • Merton, R. C. (1972). An analytic derivation of the efficient portfolio frontier. Journal of Financial and Quantitative Analysis, 7(04), 1851–1872.

    Article  Google Scholar 

  • Metaxiotis, K., & Liagkouras, K. (2012). Multiobjective evolutionary algorithms for portfolio management: A comprehensive literature review. Expert Systems with Applications, 39(14), 11685–11698.

    Article  Google Scholar 

  • Meucci, A. (2010). The Black–Litterman approach: Original model and extensions. Encyclopedia of Quantitative Finance. Wiley. doi:10.2139/ssrn.1117574.

  • Michaud, R. O. (1989). The Markowitz optimization enigma: Is “optimized” optimal? Financial Analysts Journal, 45(1), 31–42.

    Article  Google Scholar 

  • Michaud, R. O., & Michaud, R. (1998). Efficient asset management. Boston: Harvard Business School Press.

    Google Scholar 

  • Morton, A. J., & Pliska, S. R. (1995). Optimal portfolio management with fixed transaction costs. Mathematical Finance, 5(4), 337–356.

    MATH  Article  Google Scholar 

  • Norman, A. S. (2011). Financial analysis as a consideration for stock exchange investment decisions in Tanzania. Journal of Accounting and Taxation, 3(4), 60–69.

    MathSciNet  Google Scholar 

  • Norman, A. S. (2012). The usefulness of financial information in capital markets investment decision making in Tanzania: A case of Iringa region. International Journals of Marketing and Technology, 2(8), 50–65.

    Google Scholar 

  • Oksendal, B., & Sulem, A. (2002). Optimal consumption and portfolio with both fixed and proportional transaction costs. SIAM Journal on Control and Optimization, 40(6), 1765–1790.

    MathSciNet  MATH  Article  Google Scholar 

  • Pedrycz, W., & Song, M. (2012). Granular fuzzy models: A study in knowledge management in fuzzy modeling. International Journal of Approximate Reasoning, 53(7), 1061–1079.

    MathSciNet  Article  Google Scholar 

  • Peng, H., Kitagawa, G., Gan, M., & Chen, X. (2011). A new optimal portfolio selection strategy based on a quadratic form mean-variance model with transaction costs. Optimal Control Applications and Methods, 32(2), 127–138.

    MathSciNet  MATH  Article  Google Scholar 

  • Pliska, S. (1997). Introduction to mathematical finance. Oxford: Blackwell publishers.

    Google Scholar 

  • Pogue, G. A. (1970). An extension of the Markowitz portfolio selection model to include variable transactions’ costs, short sales, leverage policies and taxes. The Journal of Finance, 25(5), 1005–1027.

    Article  Google Scholar 

  • Puri, M. L., & Ralescu, D. A. (1986). Fuzzy random variables. Journal of Mathematical Analysis and Applications, 114(2), 409–422.

    MathSciNet  MATH  Article  Google Scholar 

  • Qin, Z. (2015). Mean-variance model for portfolio optimization problem in the simultaneous presence of random and uncertain returns. European Journal of Operational Research, 245(2), 480–488.

    MathSciNet  MATH  Article  Google Scholar 

  • Ruiz-Torrubiano, R., & Suárez, A. (2015). A memetic algorithm for cardinality-constrained portfolio optimization with transaction costs. Applied Soft Computing, 36, 125–142.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Samuelson, P. A. (1969). Lifetime portfolio selection by dynamic stochastic programming. The Review of Economics and Statistics, 51(3), 239–246.

    Article  Google Scholar 

  • Sethi, S., & Sorger, G. (1991). A theory of rolling horizon decision making. Annals of Operations Research, 29(1), 387–415.

    MathSciNet  MATH  Article  Google Scholar 

  • Shaw, D. X., Liu, S., & Kopman, L. (2008). Lagrangian relaxation procedure for cardinality-constrained portfolio optimization. Optimisation Methods and Software, 23(3), 411–420.

    MathSciNet  MATH  Article  Google Scholar 

  • Shen, Y. (2015). Mean-variance portfolio selection in a complete market with unbounded random coefficients. Automatica, 55, 165–175.

    MathSciNet  MATH  Article  Google Scholar 

  • Soleimani, H., Golmakani, H. R., & Salimi, M. H. (2009). Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm. Expert Systems with Applications, 36(3), 5058–5063.

    Article  Google Scholar 

  • Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 399(1), 197–206.

    MathSciNet  MATH  Google Scholar 

  • Tanaka, H. (1995). Possibility portfolio selection. In Proceedings of 4th IEEE international conference on fuzzy systems (pp. 813–818).

  • Tanaka, H., & Guo, P. (1999). Portfolio selection based on upper and lower exponential possibility distributions. European Journal of Operational Research, 114(1), 115–126.

    MATH  Article  Google Scholar 

  • Tanaka, H., Guo, P., & Türksen, I. B. (2000). Portfolio selection based on fuzzy probabilities and possibility distributions. Fuzzy Sets and Systems, 111(3), 387–397.

    MathSciNet  MATH  Article  Google Scholar 

  • Tu, J., & Zhou, G. (2010). Incorporating economic objectives into Bayesian priors: Portfolio choice under parameter uncertainty. Journal of Financial and Quantitative Analysis, 45(4), 959–986.

    Article  Google Scholar 

  • Tuba, M., & Bacanin, N. (2014). Artificial bee colony algorithm hybridized with firefly algorithm for cardinality constrained mean-variance portfolio selection problem. Applied Mathematics and Information Sciences, 8(6), 2831–2844.

    MathSciNet  Article  Google Scholar 

  • Tütüncü, R. H., & Koenig, M. (2004). Robust asset allocation. Annals of Operations Research, 132(1–4), 157–187.

    MathSciNet  MATH  Article  Google Scholar 

  • Wang, J., & Forsyth, P. A. (2011). Continuous time mean variance asset allocation: A time-consistent strategy. European Journal of Operational Research, 209(2), 184–201.

    MathSciNet  MATH  Article  Google Scholar 

  • Wang, Z., & Liu, S. (2013). Multi-period mean-variance portfolio selection with fixed and proportional transaction costs. Journal of Industrial and Management Optimization, 9(3), 643–657.

    MathSciNet  MATH  Article  Google Scholar 

  • Wang, Z., Liu, S., & Kong, X. (2012). Artificial bee colony algorithm for portfolio optimization problems. International Journal of Advancements in Computing Technology, 4(4), 8–16.

    Article  Google Scholar 

  • Wei, J., Wong, K. C., Yam, S. C. P., & Yung, S. P. (2013). Markowitz’s mean-variance asset-liability management with regime switching: A time-consistent approach. Insurance: Mathematics and Economics, 53(1), 281–291.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  • 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.

    MathSciNet  MATH  Article  Google Scholar 

  • Wu, H. (2013). Time-consistent strategies for a multiperiod mean–variance portfolio selection problem. Journal of Applied Mathematics. doi:10.1155/2013/841627.

  • Wu, H., & Chen, H. (2015). Nash equilibrium strategy for a multi-period mean-variance portfolio selection problem with regime switching. Economic Modelling, 46, 79–90.

    Article  Google Scholar 

  • Wu, H., & Li, Z. (2011). Multi-period mean-variance portfolio selection with Markov regime switching and uncertain time-horizon. Journal of Systems Science and Complexity, 24(1), 140–155.

    MathSciNet  MATH  Article  Google Scholar 

  • Wu, H., & Li, Z. (2012). Multi-period mean-variance portfolio selection with regime switching and a stochastic cash flow. Insurance: Mathematics and Economics, 50(3), 371–384.

    MathSciNet  MATH  Google Scholar 

  • Wu, C., Luo, P., Li, Y., & Chen, K. (2015). Stock price forecasting: Hybrid model of artificial intelligent methods. Engineering Economics, 26(1), 40–48.

    Article  Google Scholar 

  • Wu, H., Zeng, Y., & Yao, H. (2014). Multi-period Markowitz’s mean-variance portfolio selection with state-dependent exit probability. Economic Modelling, 36, 69–78.

    Article  Google Scholar 

  • Xia, H., Min, X., & Deng, S. (2015). Effectiveness of earnings forecasts in efficient global portfolio construction. International Journal of Forecasting, 31(2), 568–574.

    Article  Google Scholar 

  • Xia, J., & Yan, J. A. (2006). Markowitz’s portfolio optimization in an incomplete market. Mathematical Finance, 16(1), 203–216.

    MathSciNet  MATH  Article  Google Scholar 

  • Xiao, Y., & Valdez, E. A. (2015). A Black-Litterman asset allocation model under Elliptical distributions. Quantitative Finance, 15(3), 509–519.

    MathSciNet  Article  MATH  Google Scholar 

  • Xie, S., Li, Z., & Wang, S. (2008). Continuous-time portfolio selection with liability: Mean-variance model and stochastic LQ approach. Insurance: Mathematics and Economics, 42(3), 943–953.

    MathSciNet  MATH  Google Scholar 

  • Xiong, J., & Zhou, X. Y. (2007). Mean-variance portfolio selection under partial information. SIAM Journal on Control and Optimization, 46(1), 156–175.

    MathSciNet  MATH  Article  Google Scholar 

  • Xu, Y., & Wu, Z. (2014). Continuous-time mean-variance portfolio selection with inflation in an incomplete market. Journal of Financial Risk Management, 3(02), 19.

    Article  Google Scholar 

  • Xue, H. G., Xu, C. X., & Feng, Z. X. (2006). Mean-variance portfolio optimal problem under concave transaction cost. Applied Mathematics and Computation, 174(1), 1–12.

    MathSciNet  MATH  Article  Google Scholar 

  • Yan, L. (2009). Optimal portfolio selection models with uncertain returns. Editorial Board, 3(8), 76.

    MATH  Google Scholar 

  • Yang, X. (2006). Improving portfolio efficiency: A genetic algorithm approach. Computational Economics, 28(1), 1–14.

    MathSciNet  MATH  Article  Google Scholar 

  • Yang, L., Couillet, R., & McKay, M. (2014). Minimum variance portfolio optimization with robust shrinkage covariance estimation. In Asilomar conference on signals, systems, and computers.

  • Yao, H., Lai, Y., & Hao, Z. (2013a). Uncertain exit time multi-period mean-variance portfolio selection with endogenous liabilities and Markov jumps. Automatica, 49(11), 3258–3269.

    MathSciNet  MATH  Article  Google Scholar 

  • Yao, H., Lai, Y., & Li, Y. (2013b). Continuous-time mean-variance asset-liability management with endogenous liabilities. Insurance: Mathematics and Economics, 52(1), 6–17.

    MathSciNet  MATH  Google Scholar 

  • Yao, H., Zeng, Y., & Chen, S. (2013c). Multi-period mean-variance asset-liability management with uncontrolled cash flow and uncertain time-horizon. Economic Modelling, 30, 492–500.

    Article  Google Scholar 

  • Ye, K., Parpas, P., & Rustem, B. (2012). Robust portfolio optimization: A conic programming approach. Computational Optimization and Applications, 52(2), 463–481.

    MathSciNet  MATH  Article  Google Scholar 

  • Yi, L., Li, Z. F., & Li, D. (2008). Multi-period portfolio selection for asset-liability management with uncertain investment horizon. Journal of Industrial and Management Optimization, 4(3), 535–552.

    MathSciNet  MATH  Article  Google Scholar 

  • Yi, L., Wu, X., Li, X., & Cui, X. (2014). A mean-field formulation for optimal multi-period mean-variance portfolio selection with an uncertain exit time. Operations Research Letters, 42(8), 489–494.

    MathSciNet  Article  MATH  Google Scholar 

  • Yin, G., & Zhou, X. Y. (2004). Markowitz’s mean-variance portfolio selection with regime switching: From discrete-time models to their continuous-time limits. IEEE Transactions on Automatic Control, 49(3), 349–360.

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  • Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353.

    MathSciNet  MATH  Article  Google Scholar 

  • Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1, 3–28.

    MathSciNet  MATH  Article  Google Scholar 

  • Zeng, Y., & Li, Z. (2011). Asset-liability management under benchmark and mean-variance criteria in a jump diffusion market. Journal of Systems Science and Complexity, 24(2), 317–327.

    MathSciNet  MATH  Article  Google Scholar 

  • Zhang, W. G., Liu, W. A., & Wang, Y. L. (2006). On admissible efficient portfolio selection: Models and algorithms. Applied Mathematics and Computation, 176(1), 208–218.

    MathSciNet  MATH  Article  Google Scholar 

  • Zhang, W. G., & Nie, Z. K. (2003). On possibilistic variance of fuzzy numbers. Lecture Notes in Computer Science (pp. 398–402).

  • Zhang, W. G., & Nie, Z. K. (2004). On admissible efficient portfolio selection problem. Applied Mathematics and Computation, 159(2), 357–371.

    MathSciNet  MATH  Article  Google Scholar 

  • Zhang, W. G., & Wang, Y. L. (2005a). Portfolio selection: Possibilistic mean–variance model and possibilistic efficient frontier. In Algorithmic applications in management (p. 203).

  • Zhang, W., & Wang, Y. (2005b). Using fuzzy possibilistic mean and variance in portfolio selection model. In Computational intelligence and security (pp. 291–296).

  • Zhang, W. G., & Wang, Y. L. (2008). An analytic derivation of admissible efficient frontier with borrowing. European Journal of Operational Research, 184(1), 229–243.

    MathSciNet  MATH  Article  Google Scholar 

  • Zhang, W. G., Wang, Y. L., Chen, Z. P., & Nie, Z. K. (2007). Possibilistic mean-variance models and efficient frontiers for portfolio selection problem. Information Sciences, 177(13), 2787–2801.

    MathSciNet  MATH  Article  Google Scholar 

  • Zhang, W. G., & Xiao, W. L. (2009). On weighted lower and upper possibilistic means and variances of fuzzy numbers and its application in decision. Knowledge and Information Systems, 18(3), 311–330.

    Article  Google Scholar 

  • Zhang, W. G., Xiao, W. L., & Wang, Y. L. (2009a). A fuzzy portfolio selection method based on possibilistic mean and variance. Soft Computing, 13(6), 627–633.

    MATH  Article  Google Scholar 

  • Zhang, W. G., Zhang, X. L., & Xiao, W. L. (2009b). Portfolio selection under possibilistic mean-variance utility and a SMO algorithm. European Journal of Operational Research, 197(2), 693–700.

    MATH  Article  Google Scholar 

  • Zhang, W. G., Zhang, X. L., & Xu, W. J. (2010). A risk tolerance model for portfolio adjusting problem with transaction costs based on possibilistic moments. Insurance: Mathematics and Economics, 46(3), 493–499.

    MathSciNet  MATH  Google Scholar 

  • Zhang, X., Zhang, W. G., & Xu, W. J. (2011). An optimization model of the portfolio adjusting problem with fuzzy return and a SMO algorithm. Expert Systems with Applications, 38(4), 3069–3074.

    Article  Google Scholar 

  • Zhou, X. Y., & Li, D. (2000). Continuous-time mean-variance portfolio selection: A stochastic LQ framework. Applied Mathematics and Optimization, 42(1), 19–33.

    MathSciNet  MATH  Article  Google Scholar 

  • Zhou, X. Y., & Yin, G. (2003). Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model. SIAM Journal on Control and Optimization, 42(4), 1466–1482.

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  • Zhu, H., Wang, Y., Wang, K., & Chen, Y. (2011). Particle Swarm Optimization (PSO) for the constrained portfolio optimization problem. Expert Systems with Applications, 38(8), 10161–10169.

    Article  Google Scholar 

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Acknowledgements

Funding was provided by National Natural Science Foundation of China (Grant No. 71371027).

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Authors and Affiliations

  1. School of Economics and Management, Beijing University of Chemical Technology, Beijing, 100029, China

    Yuanyuan Zhang, Xiang Li & Sini Guo

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  1. Yuanyuan Zhang
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  2. Xiang Li
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Correspondence to Xiang Li.

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Zhang, Y., Li, X. & Guo, S. Portfolio selection problems with Markowitz’s mean–variance framework: a review of literature. Fuzzy Optim Decis Making 17, 125–158 (2018). https://doi.org/10.1007/s10700-017-9266-z

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Keywords

  • Portfolio selection
  • Mean–variance model
  • Dynamic optimization
  • Fuzzy optimization
  • Robust optimization