American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jimo.2020163

The skewness for uncertain random variable and application to portfolio selection problem

Bo Li 1,, and  Yadong Shu 2,

1. 

School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China
2. 

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author: Bo Li

Received  June 2020 Revised  August 2020 Published  November 2020

Uncertainty and randomness are two basic types of indeterminacy, where uncertain variable is used to represent quantities with human uncertainty and random variable is applied for modeling quantities with objective randomness. In many real systems, uncertainty and randomness often exist simultaneously. Then uncertain random variable and chance measure can be used to handle such cases. We know that the skewness is a measure of distributional asymmetry. However, the concept of skewness for uncertain random variable has not been clearly defined. In this paper, we first propose a concept of skewness for uncertain random variable and then present a formula for calculating the skewness via chance distribution. Applying the presented formula, the skewnesses of three special uncertain random variables are derived. Finally, a portfolio selection problem is carried out for showing the efficiency and applicability of skewness and presented formula.

Keywords: uncertainty, randomness, skewness, uncertain random variable, chance distribution.
Mathematics Subject Classification: 91B06(Decision theory).
Citation: Bo Li, Yadong Shu. The skewness for uncertain random variable and application to portfolio selection problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020163
References:
[1]

H. Ahmadzade, Y. Sheng and F. Hassantabar Darzi, Some results of moments of uncertain random variables, Iran. J. Fuzzy Syst., 14 (2017), 1-21.   Google Scholar

[2]

R. Bhattacharyya, A. Chatterjee and S. Kar, Mean-variance-skewness portfolio selection model in general uncertain environment, Indian J. Ind. Appl. Math., 3 (2012), 45-61.   Google Scholar

[3]

W. Briec, K. Kerstens and I. Van de Woestyne, Portfolio selection with skewness: A comparison of methods and a generalized one fund result, Eur. J. Oper. Res., 230 (2013), 412-421.  doi: 10.1016/j.ejor.2013.04.021.  Google Scholar

[4]

A. Chatterjee, R. Bhattacharyya, S. Mukherjee and S. Kar, Optimization of mean-semivariance-skewness portfolio selection model in fuzzy random environment, ICOMOS 2010, American Institute of Physics conference proceedings, 1298 (2010), 516-521.  doi: 10.1063/1.3516359.  Google Scholar

[5]

W. Chen, Y. Wang, P. Gupta and M. K. Mehlawat, A novel hybrid heuristic algorithm for a new uncertain mean-variance-skewness portfolio selection model with real constraints, Appl. Intell., 48 (2018), 2996-3018.  doi: 10.1007/s10489-017-1124-8.  Google Scholar

[6]

Y. Chen and Y. Zhu, Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems, J. Ind. Manag. Optim., 14 (2018), 913-930.  doi: 10.3934/jimo.2017082.  Google Scholar

[7]

A. Fernandez-Perez, B. Frijns, A. M. Fuertes and J. Miffre, The skewness of commodity futures returns, J. Bank. Financ., 86 (2018), 143-158.   Google Scholar

[8]

R. Gao and D. A. Ralescu, Elliptic entropy of uncertain set and its applications, Int. J. Intell. Syst., 33 (2018), 836-857.  doi: 10.1002/int.21970.  Google Scholar

[9]

X. Huang and H. Ying, Risk index based models for portfolio adjusting problem with returns subject to experts' evaluations, Econ. Model., 30 (2013), 61-66.   Google Scholar

[10]

R. G. Ibbotson, Price performance of common stock new issues, J. Financ. Econ., 2 (1975), 235-272.  doi: 10.1016/0304-405X(75)90015-X.  Google Scholar

[11]

A. Kolmogorov, Grundbegriffe Der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin, 1933.  Google Scholar

[12]

H. Kwakernaak, Fuzzy random variables-Ⅰ: Definitions and theorems, Inform. Sciences, 15 (1978), 1-29.  doi: 10.1016/0020-0255(78)90019-1.  Google Scholar

[13]

H. Kwakernaak, Fuzzy random variables-Ⅱ: Algorithms and examples for the discrete case, Inform. Sciences, 17 (1979), 253-278.  doi: 10.1016/0020-0255(79)90020-3.  Google Scholar

[14]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-292.  doi: 10.2307/1914185.  Google Scholar

[15]

X. Li, Z. Qin and K. Kar, Mean-variance-skewness model for portfolio selection with fuzzy returns, Eur. J. Oper. Res., 202 (2010), 239-247.   Google Scholar

[16]

B. Liu, Uncertainty Theory, Second ed., Springer-Verlag, Berlin, 2007.  Google Scholar

[17]

B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-39987-2.  Google Scholar

[18]

B. Li, Y. Sun, G. Aw and K. L. Teo, Uncertain portfolio optimization problem under a minimax risk measure, Appl. Math. Model., 76 (2019), 274-281.  doi: 10.1016/j.apm.2019.06.019.  Google Scholar

[19]

Y. Liu, Uncertain random variables: A mixture of uncertainty and randomness, Soft Comput., 17 (2013), 625-634.  doi: 10.1007/s00500-012-0935-0.  Google Scholar

[20]

B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst., 3 (2009), 3-10.   Google Scholar

[21]

Y. Liu, Uncertain random programming with applications, Fuzzy Optim. Decis. Ma., 12 (2013), 153-169.  doi: 10.1007/s10700-012-9149-2.  Google Scholar

[22]

H. M. Markowitz, Portfolio selection, J. Financ., 7 (1952), 77-91.   Google Scholar

[23]

A. J. Prakash, C. H. Chang and T. E. Pactwa, Selecting a portfolio with skewness: Recent evidence from US, European and Latin American equity markets, J. Bank. Financ., 27 (2003), 1375-1390.   Google Scholar

[24]

Z. Qin, Mean-variance model for portfolio optimization problem in the simultaneous presence of random and uncertain returns, Eur. J. Oper. Res., 245 (2015), 480-488.  doi: 10.1016/j.ejor.2015.03.017.  Google Scholar

[25]

W. Xu, G. Liu, H. Li and W. Luo, A study on project portfolio models with skewness risk and staffing, Int. J. Fuzzy Syst., 19 (2017), 2033-2047.  doi: 10.1007/s40815-017-0295-0.  Google Scholar

[26]

H. Yan, Y. Sun and Y. Zhu, A linear-quadratic control problem of uncertain discrete-time switched systems, J. Ind. Manag. Optim., 13 (2017), 267-282.  doi: 10.3934/jimo.2016016.  Google Scholar

[27]

X. Yang and J. Gao, Linear-quadratic uncertain differential games with application to resource extraction problem, IEEE Trans. Fuzzy Syst., 24 (2016), 819-826.  doi: 10.1109/TFUZZ.2015.2486809.  Google Scholar

[28]

T. Ye and Y. Zhu, A metric on uncertain variables, Int. J. Uncertain. Quan., 8 (2018), 251-266.  doi: 10.1615/Int.J.UncertaintyQuantification.2018020455.  Google Scholar

[29]

J. Zhai, M. Bai and H. Wu, Mean-risk-skewness models for portfolio optimization based on uncertain measure, Optimization, 67 (2018), 701-714.  doi: 10.1080/02331934.2018.1426577.  Google Scholar

[30]

L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

show all references

References:
[1]

H. Ahmadzade, Y. Sheng and F. Hassantabar Darzi, Some results of moments of uncertain random variables, Iran. J. Fuzzy Syst., 14 (2017), 1-21.   Google Scholar

[2]

R. Bhattacharyya, A. Chatterjee and S. Kar, Mean-variance-skewness portfolio selection model in general uncertain environment, Indian J. Ind. Appl. Math., 3 (2012), 45-61.   Google Scholar

[3]

W. Briec, K. Kerstens and I. Van de Woestyne, Portfolio selection with skewness: A comparison of methods and a generalized one fund result, Eur. J. Oper. Res., 230 (2013), 412-421.  doi: 10.1016/j.ejor.2013.04.021.  Google Scholar

[4]

A. Chatterjee, R. Bhattacharyya, S. Mukherjee and S. Kar, Optimization of mean-semivariance-skewness portfolio selection model in fuzzy random environment, ICOMOS 2010, American Institute of Physics conference proceedings, 1298 (2010), 516-521.  doi: 10.1063/1.3516359.  Google Scholar

[5]

W. Chen, Y. Wang, P. Gupta and M. K. Mehlawat, A novel hybrid heuristic algorithm for a new uncertain mean-variance-skewness portfolio selection model with real constraints, Appl. Intell., 48 (2018), 2996-3018.  doi: 10.1007/s10489-017-1124-8.  Google Scholar

[6]

Y. Chen and Y. Zhu, Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems, J. Ind. Manag. Optim., 14 (2018), 913-930.  doi: 10.3934/jimo.2017082.  Google Scholar

[7]

A. Fernandez-Perez, B. Frijns, A. M. Fuertes and J. Miffre, The skewness of commodity futures returns, J. Bank. Financ., 86 (2018), 143-158.   Google Scholar

[8]

R. Gao and D. A. Ralescu, Elliptic entropy of uncertain set and its applications, Int. J. Intell. Syst., 33 (2018), 836-857.  doi: 10.1002/int.21970.  Google Scholar

[9]

X. Huang and H. Ying, Risk index based models for portfolio adjusting problem with returns subject to experts' evaluations, Econ. Model., 30 (2013), 61-66.   Google Scholar

[10]

R. G. Ibbotson, Price performance of common stock new issues, J. Financ. Econ., 2 (1975), 235-272.  doi: 10.1016/0304-405X(75)90015-X.  Google Scholar

[11]

A. Kolmogorov, Grundbegriffe Der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin, 1933.  Google Scholar

[12]

H. Kwakernaak, Fuzzy random variables-Ⅰ: Definitions and theorems, Inform. Sciences, 15 (1978), 1-29.  doi: 10.1016/0020-0255(78)90019-1.  Google Scholar

[13]

H. Kwakernaak, Fuzzy random variables-Ⅱ: Algorithms and examples for the discrete case, Inform. Sciences, 17 (1979), 253-278.  doi: 10.1016/0020-0255(79)90020-3.  Google Scholar

[14]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-292.  doi: 10.2307/1914185.  Google Scholar

[15]

X. Li, Z. Qin and K. Kar, Mean-variance-skewness model for portfolio selection with fuzzy returns, Eur. J. Oper. Res., 202 (2010), 239-247.   Google Scholar

[16]

B. Liu, Uncertainty Theory, Second ed., Springer-Verlag, Berlin, 2007.  Google Scholar

[17]

B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-39987-2.  Google Scholar

[18]

B. Li, Y. Sun, G. Aw and K. L. Teo, Uncertain portfolio optimization problem under a minimax risk measure, Appl. Math. Model., 76 (2019), 274-281.  doi: 10.1016/j.apm.2019.06.019.  Google Scholar

[19]

Y. Liu, Uncertain random variables: A mixture of uncertainty and randomness, Soft Comput., 17 (2013), 625-634.  doi: 10.1007/s00500-012-0935-0.  Google Scholar

[20]

B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst., 3 (2009), 3-10.   Google Scholar

[21]

Y. Liu, Uncertain random programming with applications, Fuzzy Optim. Decis. Ma., 12 (2013), 153-169.  doi: 10.1007/s10700-012-9149-2.  Google Scholar

[22]

H. M. Markowitz, Portfolio selection, J. Financ., 7 (1952), 77-91.   Google Scholar

[23]

A. J. Prakash, C. H. Chang and T. E. Pactwa, Selecting a portfolio with skewness: Recent evidence from US, European and Latin American equity markets, J. Bank. Financ., 27 (2003), 1375-1390.   Google Scholar

[24]

Z. Qin, Mean-variance model for portfolio optimization problem in the simultaneous presence of random and uncertain returns, Eur. J. Oper. Res., 245 (2015), 480-488.  doi: 10.1016/j.ejor.2015.03.017.  Google Scholar

[25]

W. Xu, G. Liu, H. Li and W. Luo, A study on project portfolio models with skewness risk and staffing, Int. J. Fuzzy Syst., 19 (2017), 2033-2047.  doi: 10.1007/s40815-017-0295-0.  Google Scholar

[26]

H. Yan, Y. Sun and Y. Zhu, A linear-quadratic control problem of uncertain discrete-time switched systems, J. Ind. Manag. Optim., 13 (2017), 267-282.  doi: 10.3934/jimo.2016016.  Google Scholar

[27]

X. Yang and J. Gao, Linear-quadratic uncertain differential games with application to resource extraction problem, IEEE Trans. Fuzzy Syst., 24 (2016), 819-826.  doi: 10.1109/TFUZZ.2015.2486809.  Google Scholar

[28]

T. Ye and Y. Zhu, A metric on uncertain variables, Int. J. Uncertain. Quan., 8 (2018), 251-266.  doi: 10.1615/Int.J.UncertaintyQuantification.2018020455.  Google Scholar

[29]

J. Zhai, M. Bai and H. Wu, Mean-risk-skewness models for portfolio optimization based on uncertain measure, Optimization, 67 (2018), 701-714.  doi: 10.1080/02331934.2018.1426577.  Google Scholar

[30]

L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

Table 1.  The computational results for different $ p $ and $ q $
$ (p, q) $ $ (x_{1}^{*}, x_{2}^{*}) $ Expected value Variance Skewness
$ (0.04, 0.2) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.04, 0.5) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.04, 0.8) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.02, 0.2) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.02, 0.5) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.02, 0.8) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.01, 0.2) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.01, 0.5) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.01, 0.8) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
Table Options

Download as excel

$ (p, q) $ $ (x_{1}^{*}, x_{2}^{*}) $ Expected value Variance Skewness
$ (0.04, 0.2) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.04, 0.5) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.04, 0.8) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.02, 0.2) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.02, 0.5) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.02, 0.8) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.01, 0.2) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.01, 0.5) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
$ (0.01, 0.8) $ $ (0, 1) $ $ 0.04 $ 0.0075 $ 1.125\times10^{-4} $
[1]

Editorial Office. RETRACTION: Peng Zhang, Chance-constrained multiperiod mean absolute deviation uncertain portfolio selection. Journal of Industrial & Management Optimization, 2019, 15 (2) : 537-564. doi: 10.3934/jimo.2018056
[2]

Ying Jiao, Idris Kharroubi. Information uncertainty related to marked random times and optimal investment. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 3-. doi: 10.1186/s41546-018-0029-8
[3]

Hans Henrik Rugh. On dimensions of conformal repellers. Randomness and parameter dependency. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2553-2564. doi: 10.3934/dcds.2012.32.2553
[4]

Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111
[5]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221
[6]

Jiuping Xu, Pei Wei. Production-distribution planning of construction supply chain management under fuzzy random environment for large-scale construction projects. Journal of Industrial & Management Optimization, 2013, 9 (1) : 31-56. doi: 10.3934/jimo.2013.9.31
[7]

Lorenzo Mari, Marino Gatto, Renato Casagrandi. Local resource competition and the skewness of the sex ratio: a demographic model. Mathematical Biosciences & Engineering, 2008, 5 (4) : 813-830. doi: 10.3934/mbe.2008.5.813
[8]

Ke-Wei Ding, Nan-Jing Huang, Yi-Bin Xiao. Distributionally robust chance constrained problems under general moments information. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2923-2942. doi: 10.3934/jimo.2019087
[9]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099
[10]

Xiantao Xiao, Jian Gu, Liwei Zhang, Shaowu Zhang. A sequential convex program method to DC program with joint chance constraints. Journal of Industrial & Management Optimization, 2012, 8 (3) : 733-747. doi: 10.3934/jimo.2012.8.733
[11]

Kevin Ford. The distribution of totients. Electronic Research Announcements, 1998, 4: 27-34.

[12]

Lifen Jia, Wei Dai. Uncertain spring vibration equation. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021073
[13]

Ioannis D. Baltas, Athanasios N. Yannacopoulos. Uncertainty and inside information. Journal of Dynamics & Games, 2016, 3 (1) : 1-24. doi: 10.3934/jdg.2016001

[14]

Jinqiao Duan, Vincent J. Ervin, Daniel Schertzer. Dispersion in flows with obstacles and uncertainty. Conference Publications, 2001, 2001 (Special) : 131-136. doi: 10.3934/proc.2001.2001.131
[15]

Xiangfeng Yang. Stability in measure for uncertain heat equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6533-6540. doi: 10.3934/dcdsb.2019152
[16]

King-Yeung Lam, Daniel Munther. Invading the ideal free distribution. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3219-3244. doi: 10.3934/dcdsb.2014.19.3219
[17]

Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949
[18]

H.T. Banks, Jimena L. Davis. Quantifying uncertainty in the estimation of probability distributions. Mathematical Biosciences & Engineering, 2008, 5 (4) : 647-667. doi: 10.3934/mbe.2008.5.647
[19]

Hyeng Keun Koo, Shanjian Tang, Zhou Yang. A Dynkin game under Knightian uncertainty. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5467-5498. doi: 10.3934/dcds.2015.35.5467
[20]

François Gay-Balmaz, Darryl D. Holm. Predicting uncertainty in geometric fluid mechanics. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1229-1242. doi: 10.3934/dcdss.2020071

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (55)
  • HTML views (250)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences