Fuzzy Optimization and Decision Making

, Volume 12, Issue 3, pp 289–304 | Cite as

On interval portfolio selection problem

  • Meng Wu
  • De-wang Kong
  • Jiu-ping Xu
  • Nan-jing Huang


The future returns of each securities cannot be correctly reflected by the data in the past, therefore the expert’s judgements and experiences should be considered to estimate the security returns for the future. In this paper, we propose an interval portfolio selection model in which both the returns and the risks of assets are defined as intervals. By using interval and convex analysis, we solve this model and get the noninferior solution. Finally, an example is given to illustrate our results. The interval portfolio selection model improves and generalizes the Markowitz’s mean-variance model and the results of Deng et al. (Eur J Oper Res 166(1):278–292, 2005).


Portfolio selection Interval Interval analysis  Inverse interval matrix 



This work was supported by the National Natural Science Foundation of China (71101099, 70831005) and the Special Funds of Sichuan University of the Fundamental Research Funds for the Central Universities (SKQY201330).


  1. Bertsimas, D., & Thiele, A. (2006). Robust and data-driven optimization: Modern decision making under uncertainty. In Tutorials in operations research (pp. 95–122). INFORMS.Google Scholar
  2. 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.CrossRefGoogle Scholar
  3. Best, M. J., & Grauer, R. R. (1991). Sensitivity analysis for mean-variance portfolio problems. Management Science, 37(8), 980–989.MATHCrossRefGoogle Scholar
  4. Deng, X. T., Li, Z. F., & Wang, S. Y. (2005). A minimax portfolio selection strategy with equilibrium. European Journal of Operational Research, 166(1), 278–292.MathSciNetMATHCrossRefGoogle Scholar
  5. Elton, E. J., Gruber, M. J., & Urich, T. J. (1978). Are betas best? Journal of Finance, 33(5), 1375–1384.Google Scholar
  6. Fan, K. (1953). Minimax theorems. Proceedings of the National Academy of Sciences of the United States of America, 39(1), 42–47.MathSciNetMATHCrossRefGoogle Scholar
  7. Giove, S., Funari, S., & Nardelli, C. (2006). An interval portfolio selection problem based on regret function. European Journal of Operational Research, 170(1), 253–264.CrossRefGoogle Scholar
  8. Inuiguchi, M., & Ramík, J. (2000). Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets and Systems, 111(1), 3–28.MathSciNetMATHCrossRefGoogle Scholar
  9. Inuiguchi, M., & Sakawa, M. (1995). Minimax regret solution to linear programming problems with an interval objective function. European Journal of Operational Research, 86(3), 526–536.MathSciNetMATHCrossRefGoogle Scholar
  10. Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management Science, 37(5), 519–531.CrossRefGoogle Scholar
  11. 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.CrossRefGoogle Scholar
  12. Lin, C. C., & Liu, Y. T. (2008). Genetic algorithms for portfolio selection problems with minimum transaction lots. European Journal of Operational Research, 185(1), 393–404.MATHCrossRefGoogle Scholar
  13. Mao, J. C. T. (1970). Models of capital budgeting, E-V VS E-S. Journal of Financial and Quantitative Analysis, 4(5), 657–675.CrossRefGoogle Scholar
  14. Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91.Google Scholar
  15. Markowitz, H. M. (1987). Mean-variance analysis in portfolio choice and capital markets. Oxford: Basil Blackwell.MATHGoogle Scholar
  16. Markowitz, H. M. (1991). Portfolio selection: Efficient diversification of investments (2nd ed.). Oxford: Basil Blackwell.Google Scholar
  17. Neumaier, A. (1990). Interval methods for systems of equations. Cambridge: Cambridge University Press.MATHGoogle Scholar
  18. Rohn, J. (1989). Systems of linear interval equations. Linear algebra and its applications, 126, 39–78.MathSciNetMATHCrossRefGoogle Scholar
  19. Rohn, J. (1993). Inverse interval matrix. SIAM Journal on Numerical Analysis, 30(3), 864–870.MathSciNetMATHCrossRefGoogle Scholar
  20. Sengupta, A., & Pal, T. K. (2000). On comparing interval numbers. European Journal of Operational Research, 127(1), 28–43.MathSciNetMATHCrossRefGoogle Scholar
  21. Wang, S. Y., & Zhu, S. S. (2002). On fuzzy portfolio selection problems. Fuzzy Optimization and Decision Making, 1(4), 361–377.MathSciNetMATHCrossRefGoogle Scholar
  22. Xia, Y. S., Liu, B. D., Wang, S. Y., & Lai, K. K. (2000). A model for portfolio selection with order of expected returns. Computers and Operations Research, 27(5), 409–422.MATHCrossRefGoogle Scholar
  23. 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.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Meng Wu
    • 1
  • De-wang Kong
    • 2
  • Jiu-ping Xu
    • 1
  • Nan-jing Huang
    • 3
  1. 1.College of Business AdministrationSichuan UniversityChengduPeople’s Republic of China
  2. 2.The Wang Yanan Institute for Studies in EconomicsXiamen UniversityXiamenChina
  3. 3.Department of MathematicsSichuan UniversityChengduChina

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