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Mean-variance models for portfolio selection with fuzzy random returns

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Abstract

This paper develops two novel types of mean-variance models for portfolio selection problems, in which the security returns are assumed to be characterized by fuzzy random variables with known possibility and probability distributions. In the proposed models, we take the expected return of a portfolio as the investment return and the variance of the expected return of a portfolio as the investment risk. We assume that the security returns are triangular fuzzy random variables. To solve the proposed portfolio problems, this paper first presents the variance formulas for triangular fuzzy random variables. Then this paper applies the variance formulas to the proposed models so that the original portfolio problems can be reduced to nonlinear programming ones. Due to the reduced programming problems include standard normal distribution in the objective functions, we cannot employ the conventional solution methods to solve them. To overcome this difficulty, this paper employs genetic algorithm (GA) to solve them, and verify the obtained optimal solutions via Kuhn-Tucker (K-T) conditions. Finally, two numerical examples are presented to demonstrate the effectiveness of the proposed models and methods.

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References

  1. Markowitz, H.: Portfolio selection. J. Finance 7, 77–91 (1952)

    Article  Google Scholar 

  2. Li, S.X.: An insurance and investment portfolio model using chance constrained programming. Omega 23, 577–585 (1995)

    Article  Google Scholar 

  3. Williams, J.O.: Maximizing the probability of achieving investment goals. J. Portfolio Manag. 24, 77–81 (1997)

    Google Scholar 

  4. Konno, H., Yamakazi, H.: Mean-absolute deviation portfolio optimization model and its applications to Tokio stock market. Manag. Sci. 37, 519–531 (1991)

    Article  Google Scholar 

  5. Simaan, Y.: Estimation risk in portfolio selection: the mean variance model versus the mean absolute deviation model. Manag. Sci. 4, 1437–1446 (1997)

    Article  Google Scholar 

  6. Harlow, W.V., Rao, R.K.S.: Asset pricing in a generalized mean-lower partial moment framework: Theory and evidence. J. Financ. Quant. Anal. 3, 285–311 (1989)

    Article  Google Scholar 

  7. Cai, X., Teo, K.L., Yang, X., Zhou, X.: Portfolio optimization under a minimax rule. Manag. Sci. 46, 957–972 (2000)

    Article  Google Scholar 

  8. Deng, X.T., Li, Z.F., Wang, S.Y.: A minimax portfolio selection strategy with equilibrium. Eur. J. Oper. Res. 166, 278–292 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Abdelaziz, F.B., Aouni, B., Fayedh, R.E.: Multi-objective stochastic programming for portfolio selection. Eur. J. Oper. Res. 177, 1811–1823 (2007)

    Article  MATH  Google Scholar 

  10. Corazza, M., Favaretto, D.: On the existence of solutions to the quadratic mixed-integer mean-variance portfolio selection problem. Eur. J. Oper. Res. 176, 1947–1960 (2007)

    Article  MATH  Google Scholar 

  11. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  12. Tanaka, H., Guo, P.: Portfolio selection based on fuzzy probabilities and possibility distributions. Fuzzy Sets Syst. 111, 387–397 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Parra, M.A., Terol, A.B.: A fuzzy goal programming approach to portfolio selection. Eur. J. Oper. Res. 133, 287–297 (2001)

    Article  MATH  Google Scholar 

  14. Carlsson, C., Fullér, R., Majlender, P.: A possibilistic approach to selecting portfolios with highest utility score. Fuzzy Sets Syst. 131, 13–21 (2002)

    Article  MATH  Google Scholar 

  15. Zhang, W.G., Nie, Z.K.: On admissible efficient portfolio selection problem. Appl. Math. Comput. 159, 357–371 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lin, C., Tan, B., Hsieh, P.J.: Application of the fuzzy weighted average in strategic portfolio management. Decis. Sci. 36, 489–511 (2005)

    Article  Google Scholar 

  17. Bilbao-Terol, A., Pérez-Gladish, B., Arenas-Parra, M.: Fuzzy compromise programming for portfolio selection. Appl. Math. Comput. 173, 251–264 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Sharpe, W.: A simplified model for portfolio analysis. Manag. Sci. 9, 277–293 (1963)

    Article  Google Scholar 

  19. Liu, B.: Uncertainty Theory: An Introduction to Its Axiomatic Foundations. Springer, Berlin (2004)

    MATH  Google Scholar 

  20. Liu, B.: A survey of credibility theory. Fuzzy Optim. Decis. Making 5, 387–408 (2006)

    Article  Google Scholar 

  21. Kwakernaak, H.: Fuzzy random variables I: Definitions and theorems. Inf. Sci. 15, 1–29 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  22. Feng, X., Liu, Y.-K.: Measurability criteria for fuzzy random vectors. Fuzzy Optim. Decis. Making 5, 245–253 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Liu, Y.-K., Liu, B.: Fuzzy random variable: A scalar expected value operator. Fuzzy Optim. Decis. Making 2, 143–160 (2003)

    Article  Google Scholar 

  24. Liu, Y.-K., Gao, J.: The independence of fuzzy variables with applications to fuzzy random optimization. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 15, 1–20 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Wang, G., Qiao, Z.: Linear programming with fuzzy random variable coefficients. Fuzzy Sets Syst. 57, 295–311 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  26. Luhandjula, M.K.: Fuzziness and randomness in an optimization framework. Fuzzy Sets Syst. 77, 291–297 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. Liu, Y.-K., Liu, B.: A class of fuzzy random optimization: expected value models. Inf. Sci. 155, 89–102 (2003)

    Article  MATH  Google Scholar 

  28. Liu, Y.-K., Liu, B.: On minimum-risk problems in fuzzy random decision systems. Comput. Oper. Res. 32, 257–283 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Liu, Y.-K.: The approximation method for two-stage fuzzy random programming with recourse. IEEE Trans. Fuzzy Syst. 15, 1197–1208 (2007)

    Article  Google Scholar 

  30. Liu, B.: Theory and Practice of Uncertain Programming. Physica, Heidelberg (2002)

    MATH  Google Scholar 

  31. Liu, Y.-K., Wang, S.: Theory of Fuzzy Random Optimization. China Agricultural University Press, Beijing (2006)

    Google Scholar 

  32. Bazaraa, M.S., Shetty, C.M.: Nonlinear Programming. Wiley, New York (1979)

    MATH  Google Scholar 

  33. Liu, B., Liu, Y.-K.: Expected value of fuzzy variable and fuzzy expected value models. IEEE Trans. Fuzzy Syst. 10, 445–450 (2002)

    Article  Google Scholar 

  34. Li, S.X., Huang, Z.: Determination of the portfolio selection for a property-liability insurance company. Eur. J. Oper. Res. 88, 257–268 (1996)

    Article  MATH  Google Scholar 

  35. Holland, J.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)

    Google Scholar 

  36. Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading (1989)

    MATH  Google Scholar 

  37. Koza, J.: Genetic Programming II. MIT Press, Cambridge (1994)

    MATH  Google Scholar 

  38. Gen, M., Cheng, R.W.: Genetic Algorithms and Engineering Optimization. Wiley, New York (2000)

    Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Hebei University, Baoding 071002, Hebei, China

    Fang-Fang Hao & Yan-Kui Liu

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  1. Fang-Fang Hao
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  2. Yan-Kui Liu
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Correspondence to Yan-Kui Liu.

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Hao, FF., Liu, YK. Mean-variance models for portfolio selection with fuzzy random returns. J. Appl. Math. Comput. 30, 9–38 (2009). https://doi.org/10.1007/s12190-008-0154-0

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  • DOI: https://doi.org/10.1007/s12190-008-0154-0

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