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Multiperiod mean absolute deviation uncertain portfolio selection with real constraints

  • Peng Zhang
Methodologies and Application
  • 49 Downloads

Abstract

Absolute deviation is a commonly used risk measure, which has attracted more attentions in portfolio optimization. Most of existing mean–absolute deviation models are devoted to stochastic single-period portfolio optimization. However, practical investment decision problems often involve the uncertain dynamic information. Considering transaction costs, borrowing constraints, threshold constraints, cardinality constraints and risk control, we present a novel multiperiod mean absolute deviation uncertain portfolio selection model, which an optimal investment policy can be generated to help investors not only achieve an optimal return, but also have a good risk control. In proposed model, the return rate of asset and the risk are quantified by uncertain expected value and uncertain absolute deviation, respectively. Cardinality constraints limit the number of risky assets in the optimal portfolio. Threshold constraints limit the amount of capital to be invested in each asset and prevent very small investments in any asset. Based on uncertainty theories, the model is transformed into a crisp dynamic optimization problem. Because of the transaction costs and cardinality constraints, the multiperiod portfolio selection is a mix integer dynamic optimization problem with path dependence, which is “NP hard” problem that is very difficult to solve. The proposed model is approximated to a mix integer dynamic programming model. A novel discrete iteration method is designed to obtain the optimal portfolio strategy and is proved linearly convergent. Finally, an example is given to illustrate the behavior of the proposed model and the designed algorithm using real data from the Shanghai Stock Exchange.

Keywords

Uncertain modeling Multiperiod portfolio optimization Mean absolute deviation model Uncertainty theory The discrete iteration method 

Notes

Acknowledgements

This research was supported by the National Natural Science Foundation of China (No. 71271161).

Compliance with ethical standards

Conflict of interest

Peng Zhang declares that he/she has no conflict of interest.

Ethical approval

This article does not contain any studies with human participants performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.

References

  1. Anagnostopoulos KP, Mamanis G (2011) The mean-variance cardinality constrained portfolio optimization problem: an experimental evaluation of five multiobjective evolutionary algorithms. Expert Syst Appl 38:14208–14217Google Scholar
  2. Bertsimas D, Shioda R (2009) Algorithms for cardinality-constrained quadratic optimization. Comput Optim Appl 43:1–22MathSciNetCrossRefzbMATHGoogle Scholar
  3. Cesarone F, Scozzari A, Tardella F (2013) A new method for mean–variance portfolio optimization with cardinality constraints. Ann Oper Res 205:213–234MathSciNetCrossRefzbMATHGoogle Scholar
  4. Chen Z, Li G, Zhao Y (2014) Time-consistent investment policies in Markovian markets: a case of mean–variance analysis. J Econ Dyn Control 40(1):293–316MathSciNetCrossRefGoogle Scholar
  5. Chen Z, Liu J, Li G, Yan Z (2016) Composite time-consistent multi-period risk measure and its application in optimal portfolio selection. TOP 24(3):515–540MathSciNetCrossRefzbMATHGoogle Scholar
  6. Cui XY, Li D, Wang SY, Zhu SS (2012) Better than dynamic mean–variance: time inconsistency and free cash flow stream. Math Finance 22(2):346–378MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cui XT, Zheng XJ, Zhu SS, Sun XL (2013) Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems. J Global Optim 56:1409–1423MathSciNetCrossRefzbMATHGoogle Scholar
  8. Cui XY, Li X, Li D (2014) Unified framework of mean-field formulations for optimal multi-period mean–variance portfolio selection. IEEE Trans Autom Control 59(7):1833–1844MathSciNetCrossRefzbMATHGoogle Scholar
  9. Cui X, Li D, Li X (2017) Mean variance policy for discrete time cone-constrained markets: time consistency in efficiency and the minimum-variance signed supermartingale measure. Math Finance 27(2):471–504MathSciNetCrossRefGoogle Scholar
  10. Deng GF, Lin WT, Lo CC (2012) Markowitz-based portfolio selection with cardinality constraints using improved particle swarm optimization. Expert Syst Appl 39:4558–4566CrossRefGoogle Scholar
  11. Fernández A, Gómez S (2007) Portfolio selection using neural networks. Comput Oper Res 34:1177–1191CrossRefzbMATHGoogle Scholar
  12. Gao JJ, Li D, Cui XY, Wang SY (2015) Time cardinality constrained mean–variance dynamic portfolio selection and market timing: a stochastic control approach. Automatica 54(C):91–99MathSciNetCrossRefzbMATHGoogle Scholar
  13. Gülpınar N, Rustem B (2007) Worst-case robust decisions for multi-period mean–variance portfolio optimization. Eur J Oper Res 183(3):981–1000MathSciNetCrossRefzbMATHGoogle Scholar
  14. Huang X (2008) Mean–semivariance models for fuzzy portfolio selection. J Comput Appl Math 217:1–8MathSciNetCrossRefzbMATHGoogle Scholar
  15. Huang X (2012) A risk index model for portfolio selection with returns subject to experts’ estimations. Fuzzy Optim Decis Mak 11(4):451–463MathSciNetCrossRefzbMATHGoogle Scholar
  16. Huang X, Qiao L (2012) A risk index model for multi-period uncertain portfolio selection. Inf Sci 217:108–116MathSciNetCrossRefzbMATHGoogle Scholar
  17. Heidergott B, Olsder GJ, Woude JV (2006) Max plus at work-modeling and analysis of synchronized systems: a course on max-plus algebra and its applications. Princeton University Press, PrincetonzbMATHGoogle Scholar
  18. Konno H, Yamazaki H (1991) Mean absolute portfolio optimization model and its application to Tokyo stock market. Manag Sci 37(5):519–531CrossRefGoogle Scholar
  19. Köksalan M, Şakar CT (2016) An interactive approach to stochastic programming-based portfolio optimization. Ann Oper Res 245(2):47–66MathSciNetCrossRefzbMATHGoogle Scholar
  20. Le Thi HA, Moeini M (2014) Long-short portfolio optimization under cardinality constraints by difference of convex functions algorithm. J Optim Theory Appl 161:199–224MathSciNetCrossRefzbMATHGoogle Scholar
  21. Le Thi HA, Moeini M, Dinh TP (2009) Portfolio selection under downside risk measures and cardinality constraints based on DC programming and DCA. CMS 6:459–475MathSciNetCrossRefzbMATHGoogle Scholar
  22. Li CJ, Li ZF (2012) Multi-period portfolio optimization for asset-liability management with bankrupt control. Appl Math Comput 218:11196–11208MathSciNetzbMATHGoogle Scholar
  23. Li D, Ng WL (2000) Optimal dynamic portfolio selection: multiperiod mean–variance formulation. Math Finance 10(3):387–406MathSciNetCrossRefzbMATHGoogle Scholar
  24. Li D, Sun X, Wang J (2006) Optimal lot solution to cardinality constrained mean–variance formulation for portfolio selection. Math Finance 16:83–101MathSciNetCrossRefzbMATHGoogle Scholar
  25. Li X, Qin Z, Kar S (2010) Mean–variance–skewness model for portfolio selection with fuzzy returns. Eur J Oper Res 202:239–247CrossRefzbMATHGoogle Scholar
  26. Liu B (2007) Uncertainty theory, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  27. Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10Google Scholar
  28. Liu YJ, Zhang WG, Xu WJ (2012) Fuzzy multi-period portfolio selection optimization models using multiple criteria. Automatica 48:3042–3053MathSciNetCrossRefzbMATHGoogle Scholar
  29. Liu YJ, Zhang WG, Zhang P (2013) A multi-period portfolio selection optimization model by using interval analysis. Econ Model 33:113–119CrossRefGoogle Scholar
  30. Mansini R, Ogryczak W, Speranza MG (2007) Conditional value at risk and related linear programming models for portfolio optimization. Ann Oper Res 152:227–256MathSciNetCrossRefzbMATHGoogle Scholar
  31. Markowitz HM (1952) Portfolio selection. J Finance 7:77–91Google Scholar
  32. Markowitz HM (1959) Portfolio selection: efficient diversification of investments. Wiley, New YorkGoogle Scholar
  33. Mehlawat MK (2016) Credibilistic mean-entropy models for multi-period portfolio selection with multi-choice aspiration levels. Inf Sci 345:9–26CrossRefGoogle Scholar
  34. Murray W, Shek H (2012) A local relaxation method for the cardinality constrained portfolio optimization problem. Comput Optim Appl 53:681–709MathSciNetCrossRefzbMATHGoogle Scholar
  35. Qin Z (2017) Random fuzzy mean-absolute deviation models for portfolio optimization problem with hybrid uncertainty. Appl Soft Comput 56:597–603CrossRefGoogle Scholar
  36. Qin Z, Kar S (2013) Single-period inventory problem under uncertain environment. J Appl Math Comput 219(18):9630–9638MathSciNetCrossRefzbMATHGoogle Scholar
  37. Qin Z, Wen M, Gu C (2011) Mean-absolute deviation portfolio selection model with fuzzy returns. Iran J Fuzzy Syst 8:61–75MathSciNetzbMATHGoogle Scholar
  38. Ruiz-Torrubiano R, Suarez A (2010) Hybrid approaches and dimensionality reduction for portfolio selection with cardinality constrains. IEEE Comput Intell Mag 5:92–107CrossRefGoogle Scholar
  39. Sadjadi SJ, Seyedhosseini SM, Hassanlou K (2011) Fuzzy multi period portfolio selection with different rates for borrowing and lending. Appl Soft Comput 11:3821–3826CrossRefGoogle Scholar
  40. Shaw DX, Liu S, Kopman L (2008) Lagrangian relaxation procedure for cardinality-constrained portfolio optimization. Optim Methods Softw 23:411–420MathSciNetCrossRefzbMATHGoogle Scholar
  41. Speranza MG (1993) Linear programming model for portfolio optimization. Finance 14:107–123Google Scholar
  42. Sun XL, Zheng XJ, Li D (2013) Recent advances in mathematical programming with semi-continuous variables and cardinality constraint. J Oper Res Soc China 1:55–77CrossRefzbMATHGoogle Scholar
  43. van Binsbergen JH, Brandt M (2007) Solving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function? Comput Econ 29:355–367CrossRefzbMATHGoogle Scholar
  44. Vercher E, Bermudez J, Segura J (2007) Fuzzy portfolio optimization under downside risk measures. Fuzzy Sets Syst 158:769–782MathSciNetCrossRefzbMATHGoogle Scholar
  45. Woodside-Oriakhi M, Lucas C, Beasley JE (2011) Heuristic algorithms for the cardinality constrained efficient frontier. Eur J Oper Res 213:538–550MathSciNetCrossRefzbMATHGoogle Scholar
  46. Wu HL, Li ZF (2012) Multi-period mean–variance portfolio selection with regime switching and a stochastic cash flow. Insur Math Econ 50:371–384MathSciNetCrossRefzbMATHGoogle Scholar
  47. Wu H, Zeng Y (2015) Equilibrium investment strategy for defined-contribution pension schemes with generalized mean–variance criterion and mortality risk. Insur Math Econ 64:396–408MathSciNetCrossRefzbMATHGoogle Scholar
  48. Yan W, Li SR (2009) A class of multi-period semi-variance portfolio selection with a four-factor futures price model. J Appl Math Comput 29:19–34MathSciNetCrossRefzbMATHGoogle Scholar
  49. Yan W, Miao R, Li SR (2007) Multi-period semi-variance portfolio selection: model and numerical solution. Appl Math Comput 194:128–134MathSciNetzbMATHGoogle Scholar
  50. Yao K, Ji X (2014) Uncertain decision making and its application to portfolio selection problem. Int J Uncertain Fuzziness Knowl-Based Syst 22(1):113–123MathSciNetCrossRefzbMATHGoogle Scholar
  51. Yu M, Takahashi S, Inoue H, Wang SY (2010) Dynamic portfolio optimization with risk control for absolute deviation model. Eur J Oper Res 201(2):349–364MathSciNetCrossRefzbMATHGoogle Scholar
  52. Yu M, Wang SY (2012) Dynamic optimal portfolio with maximum absolute deviation model. J Global Optim 53:363–380MathSciNetCrossRefzbMATHGoogle Scholar
  53. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefzbMATHGoogle Scholar
  54. Zhang WG, Liu YJ (2014) Credibilitic mean-variance model for multi-period portfolio selection problem with risk control. OR Spectrum 36:113–132MathSciNetCrossRefzbMATHGoogle Scholar
  55. Zhang P, Zhang WG (2014) Multiperiod mean absolute deviation fuzzy portfolio selection model with risk control and cardinality constraints. Fuzzy Sets Syst 255:74–91MathSciNetCrossRefzbMATHGoogle Scholar
  56. Zhang WG, Liu YJ, Xu WJ (2012) A possibilistic mean–semivariance-entropy model for multi-period portfolio selection with transaction costs. Eur J Oper Res 222:41–349MathSciNetCrossRefzbMATHGoogle Scholar
  57. Zhang WG, Liu YJ, Xu WJ (2014) A new fuzzy programming approach for multi-period portfolio optimization with return demand and risk control. Fuzzy Sets Syst 246:107–126MathSciNetCrossRefzbMATHGoogle Scholar
  58. Zhou Z, Xiao H, Yin J, Zeng X, Lin L (2016) Pre-commitment vs. time-consistent strategies for the generalized multi-period portfolio optimization with stochastic cash flows. Insur Math Econ 68:187–202MathSciNetCrossRefzbMATHGoogle Scholar
  59. Zhu Y (2010) Uncertain optimal control with application to a portfolio selection model. Cybern Syst 41(7):535–547CrossRefzbMATHGoogle Scholar
  60. Zhu SS, Li D, Wang SY (2004) Risk control over bankruptcy in dynamic portfolio selection: a generalized mean–variance formulation. IEEE Trans Autom Control 49(3):447–457MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Economics and ManagementSouth China Normal UniversityGuangzhouPeople’s Republic of China

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