Abstract
In classical Markowitz’s Mean-Variance model, parameters such as the mean and covariance of the underlying assets’ future return are assumed to be known exactly. However, this is not always the case. The parameters often correspond to quantities that fall within a range, or can be known ambiguously at the time when investment decision must be made. In such situations, investors determine returns on investment and risks etc. and make portfolio decisions based on experience and economic wisdom. This paper tries to use the concept of interval numbers in the fuzzy set theory to extend the classical mean-variance portfolio selection model to a mean-downside semi-variance model with consideration of liquidity requirements of a bank. The semi-variance constraint is employed to control the downside risk, filling in the existing interval portfolio optimization model based on the linear semi-absolute deviation to depict the downside risk. Simulation results show that the model behaves robustly for risky assets with highest or lowest mean historical rate of return and the optimal investment proportions have good stability. This suggests that for these kinds of assets the model can reduce the risk of high deviation caused by the deviation in the decision maker’s experience and economic wisdom.
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References
Markowitz H, Portfolio selection, The Journal of Finance, 1952, 7(1): 77–91.
Kolm P N, Tütüncü R, and Fabozzi F J, 60 Years of portfolio optimization: Practical challenges and current trends, European Journal of Operational Research, 2014, 234(2): 356–371.
Konno H and Yamazaki H, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Science, 1991, 37(5): 519–531.
Feinstein C D and Thapa M N, Notes: A reformulation of a mean-absolute deviation portfolio optimization model, Management Science, 1993, 39(39): 1552–1553.
Grootveld H and Hallerbach W, Variance vs downside risk: Is there really that much difference? European Journal of operational research, 1999, 114(2): 304–319.
Chiodi L, Mansini R, and Speranza M G, Semi-absolute deviation rule for mutual funds portfolio selection, Annals of Operations Research, 2003, 124(1–4): 245–265.
Young M R, A minimax portfolio selection rule with linear programming solution, Management science, 1998, 44(5): 673–683.
Cai X, Teo K L, Yang X, et al., Portfolio optimization under a minimax rule, Management Science, 2000, 46(7): 957–972.
Chi G T, Chi F, and Yan DW, The Three factors optimization model of mean-deviation-skewness on loans portfolio, Operations Research & Management Science, 2009, 18(4): 98–111.
Wu H L and Li Z F, Multi-period mean-variance portfolio selection with Markov regime switching and uncertain time-horizon, Journal of Systems Science and Complexity, 2011, 24(1): 140–155.
Bi J N, Guo J Y, and Bai L H, Optimal multi-asset investment with no-shorting constraint under mean-variance criterion for an insurer, Journal of Systems Science and Complexity, 2011, 24(2): 291–307.
Muller G E and Witbooi P J, An optimal portfolio and capital management strategy for Basel III compliant commercial banks, Journal of Applied Mathematics, 2014, 130(3): 343–376.
Li X, Qin Z, and Kar S, Mean-variance-skewness model for portfolio selection with fuzzy returns, European Journal of Operational Research, 2010, 202(1): 239–247.
Liu S T, A fuzzy modeling for fuzzy portfolio optimization, Expert Systems with Applications, 2011, 38(11): 13803–13809.
Gupta P, Inuiguchi M, Mehlawat M K, et al., Multiobjective credibilistic portfolio selection model with fuzzy chance-constraints, Information Sciences, 2013, 229(229): 1–17.
Huang X, A new perspective for optimal portfolio selection with random fuzzy returns, Information Sciences, 2007, 177(23): 5404–5414.
Hao F F and Liu Y K, Mean-variance models for portfolio selection with fuzzy random returns, Journal of Applied Mathematics & Computing, 2009, 30(1): 9–38.
Qin Z, Wang D ZW, and Li X, Mean-semivariance models for portfolio optimization problem with mixed uncertainty of fuzziness and randomness, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 2013, 21(1): 127–139.
Li Y F, Huang G H, Li Y P, et al., Regional-scale electric power system planning under uncertainty-A multistage interval-stochastic integer linear programming approach, Energy Policy, 2010, 38(1): 475–490.
Ji X D and Zhu S S, The convergence of set-valued scenario approach for downside risk minimization, Journal of Systems Science and Complexity, 2016, 29(3): 722–735.
Jobson J D and Korkie B, Estimation for Markowitz efficient portfolios, Journal of the American Statistical Association, 1980, 75(371): 544–554.
Tu J and Zhou G, Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies, Journal of Financial Economics, 2011, 99(1): 204–215.
Yu V F, Hu K J, and Chang A Y, An interactive approach for the multi-objective transportation problem with interval parameters, International Journal of Production Research, 2015, 53(4): 1051–1064.
Fu S, Chen J, Zhou H, et al., Application of multiple attribute decision-making approaches with interval numbers in fields of investment decision, Information Technology Journal, 2014, 13(5): 853–858.
Nikoo M R, Kerachian R, and Poorsepahy-Samian H, An interval parameter model for cooperative inter-basin water resources allocation considering the water quality issues, Water Resources Management, 2012, 26(11): 3329–3343.
Zhang W G, Zhang X L, and Xiao W L, Portfolio selection under possibilistic mean-variance utility and a SMO algorithm, European Journal of Operational Research, 2009, 197(2): 693–700.
Deng X T, Li Z F, and Wang S Y, A minimax portfolio selection strategy with equilibrium, European Journal of Operational Research, 2005, 166(1): 278–292.
Wu M, Kong D W, Xu J P, et al., On interval portfolio selection problem, Fuzzy Optimization & Decision Making, 2013, 12(3): 289–304.
Ida M, Portfolio selection problem with interval coefficients, Applied Mathematics Letters, 2003, 16(5): 709–713.
Bhattacharyya R, Kar S, and Majumder D D, Fuzzy mean-variance-skewness portfolio selection models by interval analysis, Computers & Mathematics with Applications, 2011, 61(1): 126–137.
Lai K K, Wang S Y, Xu J P, et al., A class of linear interval programming problems and its application to portfolio selection, IEEE Transactions on Fuzzy Systems, 2002, 10(6): 698–704.
Li X and Qin Z, Interval portfolio selection models within the framework of uncertainty theory, Economic Modeling, 2014, 41: 338–344.
Tien F and Seow E, Asset allocation in a downside risk framework, Journal of Real Estate Portfolio Management, 2000, 6(3): 213–223.
Estrada J, The cost of equity of internet stocks: A downside risk approach, European Journal of Finance, 2004, 10(4): 239–254.
Pla-Santamaria D and Bravo M, Portfolio optimization based on downside risk: A meansemivariance efficient frontier from Dow Jones blue chips, Annals of Operations Research, 2013, 205(1): 189–201.
Ishibuchi H and Tanaka H, Multiobjective programming in optimization of the interval objective function, European Journal of Operational Research, 1990, 48(2): 219–225.
Sengupta A and Pal T K, On comparing interval numbers, European Journal of Operational Research, 2000, 127(1): 28–43.
Nemirovski A and Shapiro A, Convex approximations of chance constrained programs, SIAM Journal on Optimization, 2006, 17(4): 969–996.
Ballestero E, Mean-semivariance efficient frontier: A downside risk model for portfolio selection, Applied Mathematical Finance, 2005, 12(1): 1–15.
Dorfleitner G and Pfister T, Capital allocation and per-unit risk in in homogeneous and stressed credit portfolios, The Journal of Fixed Income, 2013, 22(3): 64–78.
Hanna S D, Gutter M S, and Fan J X, A measure of risk tolerance based on economic theory, Journal of Financial Counseling and Planning, 2001, 12(2): 53–60.
Borio C and Zhu H, Capital regulation, risk-taking and monetary policy: A missing link in the transmission mechanism, Journal of Financial Stability, 2012, 8(4): 236–251.
Cooper W W, Kingyens A T, and Paradi J C, Two-stage financial risk tolerance assessment using data envelopment analysis, European Journal of Operational Research, 2014, 233(1): 273–280.
Zhao Y M and Chen H Y, Interval number linear programming model of portfolio investment, Operations Research & Management Science, 2006, 15(2): 124–127.
Chi G T, Sun X Y, and Dong H C, A portfolio optimization model of banking asset based on the adjusted credit grade and the Semivariance absolute deviation, Systems Engineering — Theory & Practice, 2006, 26(8): 1–16.
Rose P S and Hudgins S C, Bank Management & Financial Services, Beijing, China Machine Press, 2008.
El-Alem M M, El-Sayed S, and El-Sobky B, Local convergence of the interior-Point Newton method for general nonlinear programming, Journal of Optimization Theory & Applications, 2004, 120(3): 487–502.
Wächter A and Biegler L T, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 2006, 106(1): 25–57.
Haeser and Gabriel, On the global convergence of interior-point nonlinear programming algorithms, Computational & Applied Mathematics, 2010, 29(2): 125–138.
Gould N and Toint P L, Global convergence of a hybrid trust-region SQP-filter algorithm for general nonlinear programming, SIAM Journal on Optimization, 1999, 13(3): 635–659.
Xu D C, Han J Y, and Chen Z W, Nonmonotone trust-region method for nonlinear programming with general constraints and simple bounds, Journal of Optimization Theory & Applications, 2004, 122(1): 185–206.
Huang M and Pu D, A trust-region SQP method without a penalty or a filter for nonlinear programming, Journal of Computational & Applied Mathematics, 2015, 281(C): 107–119.
Chter A and Biegler L T, Line search filter methods for nonlinear programming: Local convergence, SIAM Journal on Optimization, 2005, 16(1): 32–48.
Wächter A and Biegler L T, Line search filter methods for nonlinear programming: Motivation and global convergence, SIAM Journal on Optimization, 2005, 16(1): 1–31.
Birgin E G, Floudas C A, and Martínez J M, Global minimization using an augmented Lagrangian method with variable lower-level constraints, Mathematical Programming, 2010, 125(1): 139–162.
Androulakis I P, Maranas C D, and Floudas C A, αBB: A global optimization method for general constrained nonconvex problems, Journal of Global Optimization, 1995, 7(4): 337–363.
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This research was supported by the National Natural Science Foundation of China under Grant Nos. 71301017,71731003, 71671023, 11301050 and 51375067, the National Social Science Foundation of China under Grant No. 16BTJ017, China Postdoctoral Science Foundation Funded Project under Grant No. 2016M600207 and the Doctoral Fund of Liaoning Province under Grant No. 20131017.
This paper was recommended for publication by Editor WANG Shouyang.
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Yan, D., Hu, Y. & Lai, K. A Nonlinear Interval Portfolio Selection Model and Its Application in Banks. J Syst Sci Complex 31, 696–733 (2018). https://doi.org/10.1007/s11424-017-6070-3
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DOI: https://doi.org/10.1007/s11424-017-6070-3