Neural Computing and Applications

, Volume 31, Supplement 2, pp 931–945 | Cite as

A chance-constrained portfolio selection model with random-rough variables

  • Madjid TavanaEmail author
  • Rashed Khanjani Shiraz
  • Debora Di Caprio
Original Article


Traditional portfolio selection (PS) models are based on the restrictive assumption that the investors have precise information necessary for decision-making. However, the information available in the financial markets is often uncertain. This uncertainty is primarily the result of unquantifiable, incomplete, imprecise, or vague information. The uncertainty associated with the returns in PS problems can be addressed using random-rough (Ra-Ro) variables. We propose a new PS model where the returns are stochastic variables with rough information. More precisely, we formulate a Ra-Ro mathematical programming model where the returns are represented by Ra-Ro variables and the expected future total return maximized against a given fractile probability level. The resulting change-constrained (CC) formulation of the PS optimization problem is a non-linear programming problem. The proposed solution method transforms the CC model in an equivalent deterministic quadratic programming problem using interval parameters based on optimistic and pessimistic trust levels. As an application of the proposed method and to show its flexibility, we consider a probability maximizing version of the PS problem where the goal is to maximize the probability that the total return is higher than a given reference value. Finally, a numerical example is provided to further elucidate how the solution method works.


Portfolio selection Chance-constrained programming Quadratic programming Random-rough data Rough set theory 



The authors would like to thank the anonymous reviewers and the editor for their insightful comments and suggestions. The second author, Dr. Khanjani Shiraz, is grateful to the Iran National Science Foundation for the support he received through grant number 93040776.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Markowitz H (1952) Portfolio selection. J Financ 7:77–91Google Scholar
  2. 2.
    Wang S, Zhu S (2002) On fuzzy portfolio selection problem. Fuzzy Optim Decis Making 1:361–377MathSciNetzbMATHGoogle Scholar
  3. 3.
    Charnes A, Cooper WW (1959) Chance-constrained programming. Manag Sci 6(1):73–79MathSciNetzbMATHGoogle Scholar
  4. 4.
    Liu B (2002) Theory and practice of uncertain programming, 1st edn. Physica-Verlag, HeidelbergzbMATHGoogle Scholar
  5. 5.
    Mehlawat MK, Gupta P (2014) Fuzzy chance-constrained multiobjective portfolio selection model. IEEE Trans Fuzzy Syst 22(3):653–671Google Scholar
  6. 6.
    Omidi F, Abbasi B, Nazemi A (2017) An efficient dynamic model for solving a portfolio selection with uncertain chance constraint models. J Comput Appl Math 319:43–55MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bahrammirzaee A (2010) A comparative survey of artificial intelligence applications in finance: artificial neural networks, expert system and hybrid intelligent systems. Neural Comput & Applic 19:1165–1195Google Scholar
  8. 8.
    Lintner BJ (1965) Valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47:13–37Google Scholar
  9. 9.
    Mossin J (1966) Equilibrium in capital asset markets. Econometrica 34(4):768–783Google Scholar
  10. 10.
    Sharpe WF (1964) Capital asset prices: a theory of market equivalent under conditions of risk. J Financ 19(3):425–442Google Scholar
  11. 11.
    Konno H (1990) Piecewise linear risk functions and portfolio optimization. Journal of Operations Research Society of Japan 33:139–159MathSciNetzbMATHGoogle Scholar
  12. 12.
    Konno H, Shirakawa H, Yamazaki H (1993) A mean-absolute deviation-skewness portfolio optimization model. Ann Oper Res 45:205–220MathSciNetzbMATHGoogle Scholar
  13. 13.
    Bawa VS, Lindenberg EB (1977) Capital market equilibrium in a mean-lower partial moment framework. J Financ Econ 5:189–200Google Scholar
  14. 14.
    Elton EJ, Gruber MJ (1955) Modern portfolio theory and investment analysis. Wiley, New YorkGoogle Scholar
  15. 15.
    Rockafellar RT, Uryasev S (2000a) Optimization of conditional value-at-risk. Journal of Risk 2(3):1–21Google Scholar
  16. 16.
    Rockafellar RT, Uryasev S (2000b) Optimization of conditional value-at-risk. Journal of Risk 2(3):1–21Google Scholar
  17. 17.
    Best MJ, Hlouskova J (2000) The efficient frontier for bounded assets. Mathematical Methods of Operations Research 52:195–212MathSciNetzbMATHGoogle Scholar
  18. 18.
    Merton R (1972) An analytic derivation of the efficient frontier. J Financ Quant Anal 7:1851–1872Google Scholar
  19. 19.
    Perold AF (1984) Large-scale portfolio optimization. Manag Sci 30:1143–1160MathSciNetzbMATHGoogle Scholar
  20. 20.
    Sharp WF (1963) A simplified model for portfolio analysis. Manag Sci 9:277–293Google Scholar
  21. 21.
    Vörös J (1986) Portfolio analysis - an analytic derivation of the efficient portfolio frontier. Eur J Oper Res 23:294–300MathSciNetzbMATHGoogle Scholar
  22. 22.
    Yoshimoto A (1996) The mean-variance approach to portfolio optimization subject to transaction costs. Journal of Operations Research Society of Japan 39:99–117MathSciNetzbMATHGoogle Scholar
  23. 23.
    Chang J, Shi P (2011) Using investment satisfaction capability index based particle swarm optimization to construct a stock portfolio. Inf Sci 181(14):2989–2999MathSciNetGoogle Scholar
  24. 24.
    Chen L, He S, Zhang S (2011) Tight bounds for some risk measures, with applications to robust portfolio selection. Oper Res 59(4):847–855MathSciNetzbMATHGoogle Scholar
  25. 25.
    Fatah K, Shi P, Ameen J, Wiltshire R (2009) A preference ranking model based on both mean–variance analysis and cumulative distribution function using simulation. International Journal of Operational Research 5(3):311–327MathSciNetzbMATHGoogle Scholar
  26. 26.
    Fatah K, Shi P, Ameen J, Wiltshire R (2010) Risk averse preference models for normalised lotteries based on simulation. International Journal of Operational Research 8(2):189–207zbMATHGoogle Scholar
  27. 27.
    Gregory C, Darby-Dowman K, Mitra G (2011) Robust optimization and portfolio selection: the cost of robustness. Eur J Oper Res 212(2):417–428MathSciNetzbMATHGoogle Scholar
  28. 28.
    Huang D, Zhu S, Fabozzi F, Fukushima M (2010) Portfolio selection under distributional uncertainty: a relative robust CvaR approach. Eur J Oper Res 203:185–194zbMATHGoogle Scholar
  29. 29.
    Li X, Shou B, Qin X (2012) An expected regret minimization portfolio selection model. Eur J Oper Res 218(2):484–492MathSciNetzbMATHGoogle Scholar
  30. 30.
    Yu M, Takahashi S, Inoue H, Wang S (2010) Dynamic portfolio optimization with risk control for absolute deviation model. Eur J Oper Res 201:349–364MathSciNetzbMATHGoogle Scholar
  31. 31.
    Brockett PL, Charnes A, Cooper WW, Kwon KH, Ruefli TW (1992) Chance constrained programming approach to empirical analysis of mutual fund investment strategies. Decis Sci 23:385–408Google Scholar
  32. 32.
    Li SX (1995) An insurance and investment portfolio model using chance constrained programming. Omega 23:577–585Google Scholar
  33. 33.
    Li SX, Huang Z (1996) Determination of the portfolio selection for a property-liability insurance company. Eur J Oper Res 88:257–268zbMATHGoogle Scholar
  34. 34.
    Williams JO (1997) Maximizing the probability of achieving investment goals. J Portf Manag 24:77–81Google Scholar
  35. 35.
    Gupta P, Inuiguchi M, Mehlawat MK, Mittal G (2013) Multiobjective credibilistic portfolio selection model with fuzzy chance-constraints. Inf Sci 229:1–17MathSciNetzbMATHGoogle Scholar
  36. 36.
    Hasuike T, Katagiri H, Ishii H (2009) Portfolio selection problems with random fuzzy variable returns. Fuzzy Sets Syst 160(18):2579–2596MathSciNetzbMATHGoogle Scholar
  37. 37.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353Google Scholar
  38. 38.
    Tanaka H, Guo P (1999) Portfolio selection based on upper and lower exponential possibility distributions. Eur J Oper Res 114(1):115–126Google Scholar
  39. 39.
    Tanaka H, Guo P, Zimmermann HJ (2000) Possibility distributions of fuzzy decision variables obtained from possibilistic linear programming problems. Fuzzy Sets Syst 113(2):323–332Google Scholar
  40. 40.
    Parra MA, Terol AB, Urıa MR (2001) A fuzzy goal programming approach to portfolio selection. Eur J Oper Res 133(2):287–297Google Scholar
  41. 41.
    Carlsson C, Fullér R, Majlender P (2002) A possibilistic approach to selecting portfolios with highest utility score. Fuzzy Sets Syst 131(1):13–21Google Scholar
  42. 42.
    Ida M (2003) Portfolio selection problem with interval coefficients. Appl Math Lett 16:709–713MathSciNetzbMATHGoogle Scholar
  43. 43.
    Ammar E, Khalifa HA (2003) Fuzzy portfolio optimization: a quadratic programming approach. Chaos, Solutions & Fractals 18:1045–1054MathSciNetzbMATHGoogle Scholar
  44. 44.
    Giove S, Funari S, Nardelli C (2006) An interval portfolio selection problem based on regret function. Eur J Oper Res 170:253–264zbMATHGoogle Scholar
  45. 45.
    Gupta P, Mehlawat MK, Saxena A (2008) Asset portfolio optimization using fuzzy mathematical programming. Inf Sci 178(6):1734–1755MathSciNetzbMATHGoogle Scholar
  46. 46.
    Liu ST (2011a) The mean-absolute deviation portfolio selection problem with interval-valued returns. J Comput Appl Math 235:4149–4157MathSciNetzbMATHGoogle Scholar
  47. 47.
    Liu ST (2011b) A fuzzy modeling for fuzzy portfolio optimization. Expert Syst Appl 38:13803–13809Google Scholar
  48. 48.
    Khanjani Shiraz R, Charles V, Jalalzadeh L (2014) Fuzzy rough DEA model: a possibility and expected value approaches. Expert System with Applications 41(2):434–444Google Scholar
  49. 49.
    Tavana M, Khanjani Shiraz R, Hatami-Marbini A, Agrell P, Paryab K (2012) Fuzzy stochastic data envelopment analysis with application to base realignment and closure (BRAC). Expert Syst Appl 39(15):12247–12259Google Scholar
  50. 50.
    Tavana M, Khanjani Shiraz R, Hatami-Marbini A, Agrell P, Paryab K (2013) Chance-constrained DEA models with random fuzzy inputs and outputs. Knowl-Based Syst 52(2013):32–52Google Scholar
  51. 51.
    Tavana M, Khanjani Shiraz R, Hatami-Marbini A (2014) A new chance-constrained DEA model with Birandom input and output data. J Oper Res Soc. doi: 10.1057/jors.2013.157 Google Scholar
  52. 52.
    Mohagheghi V, Mousavi SM, Vahdani B, Shahriari MR (2016) R&D project evaluation and project portfolio selection by a new interval type-2 fuzzy optimization approach. Neural Comput & Applic. doi: 10.1007/s00521-016-2262-3 Google Scholar
  53. 53.
    Pawlak Z (1982) Rough sets. International Journal of Information and Computer Science 11:341–356zbMATHGoogle Scholar
  54. 54.
    Liu B (1997) Dependent-chance programming: a class of stochastic programming. Computers and Mathematics with Applications 34(12):89–104MathSciNetzbMATHGoogle Scholar
  55. 55.
    Liu B (2004) Uncertainty theory: an introduction to its axiomatic foundations. Springer Verlag, BerlinzbMATHGoogle Scholar
  56. 56.
    Xu J, Yao L (2009a) A class of expected value multi-objective programming problems with random rough coefficients. Mathematical and Computer Modeling 50(1–2):141–158MathSciNetzbMATHGoogle Scholar
  57. 57.
    Xu J, Yao L (2009b) A class of multiobjective linear programming models with random rough coefficients. Mathematical and Computer Modeling 49(1–2):189–206MathSciNetzbMATHGoogle Scholar
  58. 58.
    Lam M (2004) Neural network techniques for financial performance prediction, integrating fundamental and technical analysis. Decision Support System 37:567–581Google Scholar
  59. 59.
    West D, Scott D, Qian J (2005) Neural network ensemble strategies for financial decision applications. Comput Oper Res 32(10):2543–2559zbMATHGoogle Scholar
  60. 60.
    Fernandez A, Gomez S (2007) Portfolio selection using neural networks. Comput Oper Res 34:1177–1191zbMATHGoogle Scholar
  61. 61.
    Ko PC, Lin PC (2008) Resource allocation neural network in portfolio selection. Expert Syst Appl 35(1–2):330–337Google Scholar
  62. 62.
    Freitas FD, De Souza AF, de Almeida AR (2009) Prediction-based portfolio optimization model using neural networks. Neurocomputing 72(10–12):2155–2170Google Scholar
  63. 63.
    Hsu VM (2013) A hybrid procedure with feature selection for resolving stock/futures price forecasting problems. Neural Comput & Applic 22:651–671Google Scholar
  64. 64.
    Huang X (2007a) Two new models for portfolio selection with stochastic returns taking fuzzy information. Eur J Oper Res 180(1):396–405MathSciNetzbMATHGoogle Scholar
  65. 65.
    Huang X (2007b) A new perspective for optimal portfolio selection with random fuzzy returns. Inf Sci 177(23):5404–5414MathSciNetzbMATHGoogle Scholar
  66. 66.
    Li X, Zhang Y, Wong HS, Qin Z (2009) A hybrid intelligent algorithm for portfolio selection problem with fuzzy returns. J Comput Appl Math 233(2):264–278MathSciNetzbMATHGoogle Scholar
  67. 67.
    Halmos P (1974) Naive set theory. Springer, New YorkzbMATHGoogle Scholar
  68. 68.
    Charnes A, Cooper WW (1962) Programming with linear fractional functionals. Naval Research Logistics Quarterly 9:181–186MathSciNetzbMATHGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2017

Authors and Affiliations

  • Madjid Tavana
    • 1
    • 2
    Email author
  • Rashed Khanjani Shiraz
    • 3
  • Debora Di Caprio
    • 4
    • 5
  1. 1.Business Systems and Analytics Department, Lindback Distinguished Chair of Information Systems and Decision SciencesLa Salle UniversityPhiladelphiaUSA
  2. 2.Business Information Systems Department, Faculty of Business Administration and EconomicsUniversity of PaderbornPaderbornGermany
  3. 3.School of Mathematics ScienceUniversity of TabrizTabrizIran
  4. 4.Department of Mathematics and StatisticsYork UniversityTorontoCanada
  5. 5.Polo Tecnologico IISS G. GalileiBolzanoItaly

Personalised recommendations