A portfolio selection model based on the interval number. (English) Zbl 1512.91122
Summary: Traditional portfolio theory uses probability theory to analyze the uncertainty of financial market. The assets’ return in a portfolio is regarded as a random variable which follows a certain probability distribution. However, it is difficult to estimate the assets return in the real financial market, so the interval distribution of asset return can be estimated according to the relevant suggestions of experts and decision makers, that is, the interval number is used to describe the distribution of asset return. Therefore, this paper establishes a portfolio selection model based on the interval number. In this model, the semiabsolute deviation risk function is used to measure the portfolio’s risk, and the solution of the model is obtained by using the order relation of the interval number. At the same time, a satisfactory solution of the model is obtained by using the concept of acceptability of the interval number. Finally, an example is given to illustrate the practicability of the model.
MSC:
| 91G10 | Portfolio theory |
| 90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |
References:
| [1] | Markowitz, H., Portfolio selection, The Journal of Finance, 7, 1, 77-91 (1952) · doi:10.1111/j.1540-6261.1952.tb01525.x |
| [2] | Sharpe, W. F., Portfolio Theory and Capital Markets (1970), New York, NY, USA: McGraw-Hill, New York, NY, USA |
| [3] | Markowitz, H., Portfolio Selection: Efficient Diversification of Investments (1959), New York, NY, USA: Yale University Press, New York, NY, USA |
| [4] | Mao, J. C. T., Models of capital budgeting, E-V vs E-S, The Journal of Financial and Quantitative Analysis, 4, 5, 657-675 (1970) · doi:10.2307/2330119 |
| [5] | Swalm, R. O., Utility theory-insights into risk taking, Harvard Business Review, 44, 123-136 (1966) |
| [6] | Konno, H.; Yamazaki, H., Mean-absolute deviation portfolio optimization model and its applications to tokyo stock market, Management Science, 37, 5, 519-531 (1991) · doi:10.1287/mnsc.37.5.519 |
| [7] | Speranza, M. G., Linear programming model for portfolio optimization, Finance, 14, 107-123 (1993) |
| [8] | Jorion, P., Risk2: measuring the risk in value at risk, Financial Analysts Journal, 52, 6, 47-56 (1996) · doi:10.2469/faj.v52.n6.2039 |
| [9] | Basak, S.; Shapiro, A., Value-at-Risk-Based risk management: optimal policies and asset prices, Review of Financial Studies, 14, 2, 371-405 (2001) · doi:10.1093/rfs/14.2.371 |
| [10] | Chen, S. X.; Tang, C. Y., Nonparametric inference of value-at-risk for dependent financial returns, Journal of Financial Econometrics, 3, 2, 227-255 (2005) · doi:10.1093/jjfinec/nbi012 |
| [11] | Chen, F.-Y., Analytical VaR for international portfolios with common jumps, Computers & Mathematics with Applications, 62, 8, 3066-3076 (2011) · Zbl 1232.91613 · doi:10.1016/j.camwa.2011.08.018 |
| [12] | Mansini, R.; Ogryczak, W.; Speranza, M. G., Conditional value at risk and related linear programming models for portfolio optimization, Annals of Operations Research, 152, 1, 227-256 (2007) · Zbl 1132.91497 · doi:10.1007/s10479-006-0142-4 |
| [13] | Byrne, P.; Lee, S., Different risk measures: different portfolio compositions?, Journal of Property Investment & Finance, 22, 6, 501-511 (2004) · doi:10.1108/14635780410569489 |
| [14] | Kandasamy, H., Portfolio Selection under Various Risk Measures (2008), Clemson, South Carolina: Clemson University, Clemson, South Carolina |
| [15] | Yi, X.; Guo, R.; Qi, Y., Stabilization of chaotic systems with both uncertainty and disturbance by the UDE-based control method, IEEE Access, 8, 62471-62477 (2020) · doi:10.1109/access.2020.2983674 |
| [16] | Zhao, D.; Liu, Y.; Liu, Y.; Li, X., Controllability for a class of semilinear fractional evolution systems via resolvent operators, Communications on Pure & Applied Analysis, 18, 1, 455-478 (2019) · Zbl 06969373 · doi:10.3934/cpaa.2019023 |
| [17] | Jiang, C.; Akbar Zada; Tamer Senel, M.; Li, T., Synchronization of bidirectional N-coupled fractional-order chaotic systems with ring connection based on antisymmetric structure, Advances in Difference Equations, 16 (2019) · Zbl 1485.34039 |
| [18] | Liu, L.; Li, B.; Guo, R., Consensus control for networked manipulators with switched parameters and topologies, IEEE Access, 9, 9209-9217 (2021) · doi:10.1109/ACCESS.2021.3049261 |
| [19] | Jiang, C.; Zhang, F.; Li, T., Synchronization and antisynchronization ofN‐coupled fractional‐order complex chaotic systems with ring connection, Mathematical Methods in the Applied Sciences, 41, 7, 2625-2638 (2018) · Zbl 1391.34087 · doi:10.1002/mma.4765 |
| [20] | Zhao, D.; Mao, J., New controllability results of fractional nonlocal semilinear evolution systems with finite delay, Complexity, 2020 (2020) · Zbl 1467.34080 · doi:10.1155/2020/7652648 |
| [21] | Hou, T.; Liu, Y.; Deng, F., Stability for discrete-time uncertain systems with infinite Markov jump and time-delay, Science China Information Sciences, 64, 5, 1-11 (2021) · doi:10.1007/s11432-019-2897-9 |
| [22] | Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 1, 3-28 (1978) · Zbl 0377.04002 · doi:10.1016/0165-0114(78)90029-5 |
| [23] | Dubois, D.; Prade, H., The mean value of a fuzzy number, Fuzzy Sets and Systems, 24, 279-300 (1978) · Zbl 0634.94026 |
| [24] | Carlsson, C.; Fullér, R., On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems, 122, 2, 315-326 (2001) · Zbl 1016.94047 · doi:10.1016/s0165-0114(00)00043-9 |
| [25] | Zhang, W. G.; Nie, Z. K., On possibilistic variance of fuzzy numbers, Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, 2639, 398-402 (2003) · Zbl 1026.68672 |
| [26] | Carlsson, C.; Fullér, R., Possibility for Decision: A Possibilistic Approach to Real Life Decisions (2011), Berlin, Germany: Springer, Berlin, Germany · Zbl 1227.91002 |
| [27] | Georgescu, I., Possibility Theory and the Risk (2012), Berlin, Germany: Springer, Berlin, Germany · Zbl 1239.91002 |
| [28] | Tanaka, H.; Nakayama, H.; Yanagimoto, A., Possibility portfolio selection, Proceedings of the 1995 IEEE International Conference on Fuzzy Systems |
| [29] | Carlsson, C.; Fullér, R.; Majlender, P., A possibilistic approach to selecting portfolios with highest utility score, Fuzzy Sets and Systems, 131, 1, 13-21 (2002) · Zbl 1027.91038 · doi:10.1016/s0165-0114(01)00251-2 |
| [30] | Fullér, R.; Majlender, P., On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems, 136, 3, 363-374 (2003) · Zbl 1022.94032 · doi:10.1016/s0165-0114(02)00216-6 |
| [31] | Zhang, W.-G.; Wang, Y.-L.; Chen, Z.-P.; Nie, Z.-K., Possibilistic mean-variance models and efficient frontiers for portfolio selection problem, Information Sciences, 177, 13, 2787-2801 (2007) · Zbl 1286.91131 · doi:10.1016/j.ins.2007.01.030 |
| [32] | Zhang, W. G.; Wang, Y. L., Using fuzzy possibilistic mean and variance in portfolio selection model, Computational Intelligence and Security, 3801, 291-296 (2003) |
| [33] | Zhang, W.-G., Possibilistic mean-standard deviation models to portfolio selection for bounded assets, Applied Mathematics and Computation, 189, 2, 1614-1623 (2007) · Zbl 1243.91097 · doi:10.1016/j.amc.2006.12.080 |
| [34] | Zhang, W.-G.; Xiao, W.-L.; Wang, Y.-L., A fuzzy portfolio selection method based on possibilistic mean and variance, Soft Computing, 13, 6, 627-633 (2009) · Zbl 1175.91168 · doi:10.1007/s00500-008-0335-7 |
| [35] | Sui, Y.; Hu, J.; Ma, F., A possibilistic portfolio model with fuzzy liquidity constraint, Complexity, 2020 (2020) · Zbl 1445.91060 · doi:10.1155/2020/3703017 |
| [36] | Parra, M. A.; Terol, A. B.; Uria, M. V. R., A fuzzy goal programming approach to portfolio selection, European Journal of Operational Research, 133, 287-297 (2001) · Zbl 0992.90085 |
| [37] | Lai, K. K.; Wang, S. Y.; Xu, J. P.; Zhu, S. S.; Fang, Y., A class of linear interval programming problems and its application to portfolio selection, IEEE Transactions on Fuzzy Systems, 10, 6, 698-704 (2002) · doi:10.1109/tfuzz.2002.805902 |
| [38] | Ida, M., Solutions for the portfolio selection problem with interval and fuzzy coefficients, Reliable Computing, 10, 5, 389-400 (2004) · Zbl 1087.91026 · doi:10.1023/b:reom.0000032120.83979.d4 |
| [39] | Giove, S.; Funari, S.; Nardelli, C., An interval portfolio selection problem based on regret function, European Journal of Operational Research, 170, 253-264 (2005) · Zbl 1079.91030 |
| [40] | Bhattacharyya, R.; Kar, S.; Kar, S.; Majumder, D. D., Fuzzy mean-variance-skewness portfolio selection models by interval analysis, Computers & Mathematics with Applications, 61, 1, 126-137 (2011) · Zbl 1207.91059 · doi:10.1016/j.camwa.2010.10.039 |
| [41] | Liu, S.-T., The mean-absolute deviation portfolio selection problem with interval-valued returns, Journal of Computational and Applied Mathematics, 235, 14, 4149-4157 (2011) · Zbl 1231.91409 · doi:10.1016/j.cam.2011.03.008 |
| [42] | Sui, Y.; Hu, J.; Ma, F., A mean-variance portfolio selection model with interval-valued possibility measures, Mathematical Problems in Engineering, 2020, 1-12 (2020) · Zbl 1459.91184 · doi:10.1155/2020/4135740 |
| [43] | Mansini, R.; Speranza, M. G., Heuristic algorithms for the portfolio selection problem with minimum transaction lots, European Journal of Operational Research, 114, 2, 219-233 (1999) · Zbl 0935.91022 · doi:10.1016/s0377-2217(98)00252-5 |
| [44] | Ishibuchi, H.; Tanaka, H., Multiobjective programming in optimization of the interval objective function, European Journal of Operational Research, 48, 2, 219-225 (1990) · Zbl 0718.90079 · doi:10.1016/0377-2217(90)90375-l |
| [45] | Sengupta, A.; Pal, T. K.; Chakraborty, D., Interpretation of inequality constraints involving interval coefficients and a solution to interval linear programming, Fuzzy Sets and Systems, 119, 1, 129-138 (2001) · Zbl 1044.90534 · doi:10.1016/s0165-0114(98)00407-2 |
| [46] | Alefeld, G.; Herzberger, J., Introduction to Interval Computations (1983), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0552.65041 |
| [47] | Hansen, E., Global Optimization Using Interval Analysis (1992), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0762.90069 |
| [48] | Young, R. C., The algebra of many-valued quantities, Mathematische Annalen, 104, 1, 260-290 (1931) · Zbl 0001.01102 · doi:10.1007/bf01457934 |
| [49] | Moore, E. R., Methods and Applications of Interval Analysis (1979), London, UK: Prentice-Hall, London, UK · Zbl 0417.65022 |
| [50] | Moore, R.; Lodwick, W., Interval analysis and fuzzy set theory, Fuzzy Sets and Systems, 135, 1, 5-9 (2003) · Zbl 1015.03513 · doi:10.1016/s0165-0114(02)00246-4 |
| [51] | Sengupta, A.; Pal, T. K., On comparing interval numbers, European Journal of Operational Research, 127, 1, 28-43 (2000) · Zbl 0991.90080 · doi:10.1016/s0377-2217(99)00319-7 |
| [52] | Yoon, K., The propagation of errors in multiple-attribute decision analysis: a practical approach, Journal of the Operational Research Society, 40, 7, 681-686 (1989) · doi:10.1057/jors.1989.111 |
| [53] | Bryson, N.; Mobolurin, A., An action learning evaluation procedure for multiple criteria decision making problems, European Journal of Operational Research, 96, 2, 379-386 (1997) · Zbl 0917.90005 · doi:10.1016/0377-2217(94)00368-8 |
| [54] | Rommelfanger, H.; Hanuscheck, R.; Wolf, J., Linear programming with fuzzy objectives, Fuzzy Sets and Systems, 29, 1, 31-48 (1989) · Zbl 0662.90045 · doi:10.1016/0165-0114(89)90134-6 |
| [55] | Baoding Liu, B.; Kakuzo Iwamura, K., A note on chance constrained programming with fuzzy coefficients, Fuzzy Sets and Systems, 100, 1-3, 229-233 (1998) · Zbl 0948.90156 · doi:10.1016/s0165-0114(97)00291-1 |
| [56] | Chankong, V.; Haimes, Y. Y., Multiobjective Decision Making: Theory and Methodology (1983), North Holland: Amsterdam, Netherlands · Zbl 0622.90002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
