Dimensions Analysis to Excess Investment in Fuzzy Portfolio Model from the Threshold of Guaranteed Return Rates
Abstract
:1. Introduction
2. Preliminaries
- (i.)
- is normal, i.e., there is an x ∈ R such that ;
- (ii.)
- is convex, i.e., (λx + (1−λ)y) ≥ min{(x), (y)}, ∀ x, y∈ R and λ ∈ [0, 1];
- (iii.)
- is upper semicontinuous, i.e., { x ∈ R:(x) ≥ α} = is a closed subset of U for each α ∈ (0, 1];
- (iv.)
- the 0-level set is a compact subset of R.
3. Dimension Analysis to Guaranteed Rate of Return
4. Illustration of the Proposed Method
4.1. Data Description and Model Explanation
4.2. Results and Discussions
5. Conclusions
- (1)
- The numerical results. The dimensions of the excess investment are proposed in the fuzzy portfolio selection. The numerical results show that the selected dimension to the excess investment by an investor with a different degree of risk preference can be operated by one or two dimensions to the excess investment. For lower-risk preference investors, we suggest selecting two dimensions of excess investment to obtain the optimal portfolio with lower constrained risk; by contrast, one dimension of excess investment is adopted for obtaining a bigger expected return rate with higher constrained risk.
- (2)
- Limitations. Because different dimensions of excess investment affect the portfolio selection in our proposed model, it is important to formulate a process to evaluate the risk attitude of an investor. The limitation of this study should be considered for an excellent measure scale and being evaluated on the risk priority of past investment events, and then we can confirm the risk attitudes to make the portfolio selection.
- (3)
- Future research. In order to maximize the expected return rate under constrained risk, efforts to manage different types of risk attitudes play a major role in portfolio selection. This is because there are still numerous investors whose investment behaviors are not fully understood and cannot be applied to our proposed model.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Markowitz, H. Portfolio selection. J. Financ. 1952, 7, 77–91. [Google Scholar]
- Sharpe, W.F. Portfolio Theory and Capital Markets; McGraw-Hill: New York, NY, USA, 1970. [Google Scholar]
- Merton, R.C. An analytic derivation of the efficient frontier. J. Financ. Quant. Anal. 1972, 10, 1851–1872. [Google Scholar] [CrossRef] [Green Version]
- Pang, J.S. A new efficient algorithm for a class of portfolio selection problems. Oper. Res. 1980, 28, 754–767. [Google Scholar] [CrossRef]
- Perold, A.F. Large-scale portfolio optimization. Manage. Sci. 1984, 30, 1143–1160. [Google Scholar] [CrossRef] [Green Version]
- Vörös, J. Portfolio analysis—An analytic derivation of the efficient portfolio frontier. Eur. J. Oper. Res. 1990, 203, 294–300. [Google Scholar] [CrossRef]
- Best, M.J.; Grauer, R.R. The efficient set mathematics when mean–variance problems are subject to general linear constrains. J. Econ. Bus. 1990, 42, 105–120. [Google Scholar] [CrossRef] [Green Version]
- Best, M.J.; Hlouskova, J. The efficient frontier for bounded assets. Math. Method Oper. Res. 2000, 52, 195–212. [Google Scholar] [CrossRef]
- Tanaka, H.; Guo, P. Portfolio selection based on upper and lower exponential possibility distributions. Eur. J. Oper. Res. 1999, 114, 115–126. [Google Scholar] [CrossRef]
- Tanaka, H.; Guo, P.; Türksen, I. Portfolio selection based on fuzzy probabilities and possibility distributions. Fuzzy Sets Syst. 2000, 111, 387–397. [Google Scholar] [CrossRef]
- Yan, W.; Miao, R.; Li, S.R. Multi-period semi-variance portfolio selection: Model and numerical solution. Appl. Math. Comput. 2007, 194, 128–134. [Google Scholar] [CrossRef]
- Li, X.; Qin, Z.; Kar, S. Mean-variance-skewness model for portfolio selection with fuzzy returns. Eur. J. Oper. Res. 2010, 202, 239–247. [Google Scholar] [CrossRef]
- Mansour, N.; Cherif, M.S.; Abdelfattah, W. Multi-objective imprecise programming for financial portfolio selection with fuzzy returns. Expert Syst. Appl. 2019, 138, 112810. [Google Scholar] [CrossRef]
- Zhang, W.G.; Nie, Z.K. On possibilistic variance of fuzzy numbers. Lect. Notes Comput. Sci. 2003, 2639, 398–402. [Google Scholar]
- Huang, X. Mean-semivariance models for fuzzy portfolio selection. J. Comput. Appl. Math. 2008, 217, 1–8. [Google Scholar] [CrossRef] [Green Version]
- Liu, Y.-J.; Zhang, W.-G. A multi-period fuzzy portfolio optimization model with minimum transaction lots. Eur. J. Oper. Res. 2015, 242, 933–941. [Google Scholar] [CrossRef]
- Guo, S.; Yu, L.; Li, X.; Kar, S. Fuzzy multi-period portfolio selection with different investment horizons. Eur. J. Oper. Res. 2016, 254, 1026–1035. [Google Scholar] [CrossRef]
- Gupta, P.; Mehlawat, M.K.; Yadav, S.; Kumar, A. A polynomial goal programming approach for intuitionistic fuzzy portfolio optimization using entropy and higher moments. Appl. Soft Comput. 2019, 85, 105781. [Google Scholar] [CrossRef]
- Tsaur, R.-C.; Chiu, C.-L.; Huang, Y.-Y. Fuzzy Portfolio Selection in COVID-19 Spreading Period Using Fuzzy Goal Programming Model. Mathematics 2021, 9, 835. [Google Scholar] [CrossRef]
- Zhai, J.; Bai, M. Mean-risk model for uncertain portfolio selection with background risk. J. Comput. Appl. Math. 2018, 330, 59–69. [Google Scholar] [CrossRef]
- Gupta, P.; Mehlawat, M.K.; Kumar, A.; Yadav, S.; Aggarwal, A. A Credibilistic Fuzzy DEA Approach for Portfolio Efficiency Evaluation and Rebalancing Toward Benchmark Portfolios Using Positive and Negative Returns. Int. J. Fuzzy Syst. 2020, 22, 824–843. [Google Scholar] [CrossRef]
- Mehralizade, R.; Amini, M.; Gildeh, B.S.; Ahmadzade, H. Uncertain random portfolio selection based on risk curve. Soft Comput. 2020, 24, 13331–13345. [Google Scholar] [CrossRef]
- Zhang, W.-G.; Zhang, X.; Chen, Y. Portfolio adjusting optimization with added assets and transaction costs based on credibility measures. Insur. Math. Econ. 2011, 49, 353–360. [Google Scholar] [CrossRef]
- Mehlawat, M.K.; Gupta, P.; Kumar, A.; Yadav, S.; Aggarwal, A. Multi-objective fuzzy portfolio performance evaluation using data envelopment analysis under credibilistic framework. IEEE Trans. Fuzzy Syst. 2020, 11, 2726–2737. [Google Scholar] [CrossRef]
- Li, X.; Shou, B.Y.; Qin, Z.F. An expected regret minimization portfolio selection model. Eur. J. Oper. Res. 2012, 218, 484–492. [Google Scholar] [CrossRef]
- Yue, W.; Wang, Y. A new fuzzy multi-objective higher order moment portfolio selection model for diversified portfolios. Physica A. 2017, 465, 124–140. [Google Scholar] [CrossRef]
- Guo, S.; Ching, W.-K.; Li, W.-K.; Siu, T.-K.; Zhang, Z. Fuzzy hidden Markov-switching portfolio selection with capital gain tax. Expert Syst. Appl. 2020, 149, 113304. [Google Scholar] [CrossRef]
- Liagkouras, K.; Metaxiotis, K. Multi-period mean–variance fuzzy portfolio optimization model with transaction costs. Eng. Appl. Artif. Intell. 2018, 67, 260–269. [Google Scholar] [CrossRef]
- Zhou, X.; Wang, J.; Yang, X.; Lev, B.; Tu, Y.; Wang, S. Portfolio selection under different attitudes in fuzzy environment. Inf. Sci. 2018, 462, 278–289. [Google Scholar] [CrossRef]
- Zhang, W.-G.; Nie, Z.-K. On admissible efficient portfolio selection problem. Appl. Math. Comput. 2004, 159, 357–371. [Google Scholar] [CrossRef]
- Zhou, W.; Xu, Z. Portfolio selection and risk investment under the hesitant fuzzy environment. Knowl.-Based Syst. 2018, 144, 21–31. [Google Scholar] [CrossRef]
- Tsaur, R.C.; Chiu, C.-L.; Huang, Y.-Y. Guaranteed rate of return for excess investment in a fuzzy portfolio analysis. Int. J. Fuzzy Syst. 2021, 23, 94–106. [Google Scholar] [CrossRef]
- Huang, Y.-Y.; Chen, I.-F.; Chiu, C.-L.; Tsaur, R.-C. Adjustable Security Proportions in the Fuzzy Portfolio Selection under Guaranteed Return Rates. Mathematics 2021, 9, 3026. [Google Scholar] [CrossRef]
- Huang, Y.-Y.; Tsaur, R.-C.; Huang, N.-C. Sustainable Fuzzy Portfolio Selection Concerning Multi-Objective Risk Attitudes in Group Decision. Mathematics. 2022, 10, 3304. [Google Scholar] [CrossRef]
- Zimmermann, H.-J. Fuzzy Set Theory-and Its Applications; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Carlsson, C.; Fullér, R. On possibilistic mean value and variance of fuzzy numbers. Fuzzy Sets Syst. 2001, 122, 315–326. [Google Scholar] [CrossRef] [Green Version]
- Rao, P.P.B.; Shankar, N.R. Ranking fuzzy numbers with a distance method using circumcenter of centroids and an index of modality. Adv. Fuzzy Syst. 2011, 2011, 178308. [Google Scholar] [CrossRef] [Green Version]
- Zhang, W.G. Possibilistic mean–standard deviation models to portfolio selection for bounded assets. Comput. Appl. Math. 2007, 189, 1614–1623. [Google Scholar] [CrossRef]
Risk | 4.4% | 4.5% | 4.7% | 5% | 5.3% | 5.5% | 5.7% | 5.8% | |
---|---|---|---|---|---|---|---|---|---|
Proportion | |||||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.1236 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.4 | 0.265 | 0.12 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.2764 | 0.435 | 0.5 | 0.3584 | 0.2406 | 0.1229 | 0.1 | ||
x5 | 0.1 | 0.1 | 0.1791 | 0.3416 | 0.4594 | 0.5771 | 0.6 | ||
Expected Return Rate | 0.15211 | 0.15859 | 0.16769 | 0.17658 | 0.18249 | 0.18954 | 0.18954 |
Risk | 8.2% | 8.3% | 8.5% | 8.7% | 8.9% | 9% | 9.3% | |
---|---|---|---|---|---|---|---|---|
Proportion | ||||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.4569 | 0.3381 | 0.2193 | 0. 1005 | 0.1 | 0.1 | ||
x5 | 0.2431 | 0.3619 | 0.4807 | 0.5995 | 0.6 | 0.6 | ||
Expected Return Rate | 0.16901 | 0.17495 | 0.18081 | 0.18667 | 0.18669 | 0.18669 |
Risk | 11.0% | 11.1% | 11.5% | 12% | 12.5% | 13% | 13.5% | |
---|---|---|---|---|---|---|---|---|
Proportion | ||||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x5 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | ||
Expected Return Rate | 0.17977 | 0.17977 | 0.17977 | 0.17977 | 0.17977 | 0.17977 |
Risk | 7.2% | 7.3% | 7.7% | 8.3% | 8.7% | 9.3% | 9.5% | |
---|---|---|---|---|---|---|---|---|
Proportion | ||||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.3645 | 0.1715 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.3355 | 0.4 | 0.177 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.1 | 0.2285 | 0.5 | 0.4517 | 0.178 | 0.1 | ||
x5 | 0.1 | 0.1 | 0.123 | 0.2483 | 0.522 | 0.6 | ||
Expected Return Rate | 0.17570 | 0.19672 | 0.22237 | 0.23793 | 0.26106 | 0.26765 |
Risk | 12.2% | 12.3% | 12.5% | 12.7% | 13% | 13.3% | 14% | |
---|---|---|---|---|---|---|---|---|
Proportion | ||||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.459 | 0.3677 | 0.2765 | 0.396 | 0.1 | 0.1 | ||
x5 | 0.241 | 0.3323 | 0.4235 | 0.5604 | 0.6 | 0.6 | ||
Expected Return Rate | 0.19622 | 0.20393 | 0.21164 | 0.22320 | 0.22655 | 0.22655 |
Risk | 15.4% | 15.5% | 16% | 17% | 18% | 19% | 20% | |
---|---|---|---|---|---|---|---|---|
Proportion | ||||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x5 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | ||
Expected Return Rate | 0.18355 | 0.18355 | 0.18355 | 0.18355 | 0.18355 | 0.18355 |
Risk | 4.4% | 4.5% | 4.7% | 5% | 5.3% | 5.4% | |
---|---|---|---|---|---|---|---|
Proportion | |||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.1214 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.5 | 0.3739 | 0.1302 | 0.1 | 0.1 | ||
x5 | 0.1786 | 0.3261 | 0.5698 | 0.6 | 0.6 | ||
Expected Return Rate | 0.16216 | 0.16915 | 0.17959 | 0.18088 | 0.18088 |
Risk | 7.9% | 8% | 8.1% | 8.3% | 8.5% | 8.6% | |
---|---|---|---|---|---|---|---|
Proportion | |||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.4842 | 0.398 | 0.2255 | 0.1 | 0.1 | ||
x5 | 0.2158 | 0.302 | 0.4745 | 0.6 | 0.6 | ||
Expected Return Rate | 0.16389 | 0.16750 | 0.17471 | 0.17996 | 0.17996 |
Risk | 10.9% | 11% | 11.5% | 12% | 12.5% | 13% | |
---|---|---|---|---|---|---|---|
Proportion | |||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x5 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | ||
Expected Return Rate | 0.17975 | 0.17975 | 0.17975 | 0.17975 | 0.17975 |
Risk | 6.0% | 6.1% | 6.3% | 6.5% | 6.7% | 6.8% | |
---|---|---|---|---|---|---|---|
Proportion | |||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.2785 | 0.1299 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.4215 | 0.5 | 0.373 | 0.1 | 0.1 | ||
x5 | 0.1 | 0.1701 | 0.327 | 0.6 | 0.6 | ||
Expected Return Rate | 0.15342 | 0.16102 | 0.16851 | 0.17988 | 0.17988 |
Risk | 7.9% | 8% | 8.1% | 8.3% | 8.5% | 8.6% | |
---|---|---|---|---|---|---|---|
Proportion | |||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.4693 | 0.3785 | 0.1969 | 0.1 | 0.1 | ||
x5 | 0.2307 | 0.3215 | 0.5031 | 0.6 | 0.6 | ||
Expected Return Rate | 0.16443 | 0.16820 | 0.17574 | 0.17976 | 0.17976 |
Risk | 10.9% | 11% | 11.5% | 12% | 12.5% | 13% | |
---|---|---|---|---|---|---|---|
Proportion | |||||||
x1 | Infeasible Solution | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
x2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x3 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x4 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | ||
x5 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | ||
Expected Return Rate | 0.17975 | 0.17975 | 0.17975 | 0.17975 | 0.17975 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Chen, K.-S.; Tsaur, R.-C.; Lin, N.-C. Dimensions Analysis to Excess Investment in Fuzzy Portfolio Model from the Threshold of Guaranteed Return Rates. Mathematics 2023, 11, 44. https://doi.org/10.3390/math11010044
Chen K-S, Tsaur R-C, Lin N-C. Dimensions Analysis to Excess Investment in Fuzzy Portfolio Model from the Threshold of Guaranteed Return Rates. Mathematics. 2023; 11(1):44. https://doi.org/10.3390/math11010044
Chicago/Turabian StyleChen, Kuen-Suan, Ruey-Chyn Tsaur, and Nei-Chih Lin. 2023. "Dimensions Analysis to Excess Investment in Fuzzy Portfolio Model from the Threshold of Guaranteed Return Rates" Mathematics 11, no. 1: 44. https://doi.org/10.3390/math11010044
APA StyleChen, K. -S., Tsaur, R. -C., & Lin, N. -C. (2023). Dimensions Analysis to Excess Investment in Fuzzy Portfolio Model from the Threshold of Guaranteed Return Rates. Mathematics, 11(1), 44. https://doi.org/10.3390/math11010044