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The Admissible Multiperiod Mean Variance Portfolio Selection Problem with Cardinality Constraints

  • Zhang, Peng (School of Management, Wuhan University of Science and Technology) ;
  • Li, Bing (School of Management, Wuhan University of Science and Technology)
  • Received : 2016.05.09
  • Accepted : 2016.11.04
  • Published : 2017.03.30

Abstract

Uncertain factors in finical markets make the prediction of future returns and risk of asset much difficult. In this paper, a model,assuming the admissible errors on expected returns and risks of assets, assisted in the multiperiod mean variance portfolio selection problem is built. The model considers transaction costs, upper bound on borrowing risk-free asset constraints, cardinality constraints and threshold constraints. Cardinality constraints limit the number of assets to be held in an efficient portfolio. At the same time, threshold constraints limit the amount of capital to be invested in each stock and prevent very small investments in any stock. Because of these limitations, the proposed model is a mix integer dynamic optimization problem with path dependence. The forward dynamic programming method is designed to obtain the optimal portfolio strategy. Finally, to evaluate the model, our result of a meaning example is compared to the terminal wealth under different constraints.

Keywords

Multiperiod Portfolio Selection;Admissible Return;Admissible Variance;Cardinality Constraints Constraints;The Forward Dynamic Programming Method

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. Anagnostopoulos, K. P. and Mamanis, G. (2011), The mean-variance cardinality constrained portfolio optimization problem: An experimental evaluation of five multiobjective evolutionary algorithms, Expert Systems with Applications, 38, 14208-14217.
  2. Bertsimas, D. and Shioda, R. (2009), Algorithms for cardinality-constrained quadratic optimization, Computational Optimization and Applications, 43, 1-22. https://doi.org/10.1007/s10589-007-9126-9
  3. Bienstock, D. (1996), Computational study of a family of mixed-integer quadratic programming problems, Mathematical Programming, 74, 121-140.
  4. Bodnar, T., Parolya, N. and Schmid, W. (2015), A closedform solution of the multi-period portfolio choice problem for a quadratic utility function, Annals of Operations Research, 229(1), 121-158. https://doi.org/10.1007/s10479-015-1802-z
  5. Brandt, M. and Santa-Clara, P. (2006), Dynamic portfolio selection by augmenting the asset space, The Journal of Finance, 61, 2187-2217. https://doi.org/10.1111/j.1540-6261.2006.01055.x
  6. Calafiore, G. C. (2008), Multi-period portfolio optimization with linear control policies, Automatica, 44(10), 2463-2473. https://doi.org/10.1016/j.automatica.2008.02.007
  7. Carlsson, C. and Fuller, R. (2001), On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems, 122, 315-326. https://doi.org/10.1016/S0165-0114(00)00043-9
  8. Carlsson, C., Fuller, R. and Majlender, P. (2002), A possibilistic approach to selecting portfolios with highest utility score, Fuzzy Sets and Systems, 131, 13-21. https://doi.org/10.1016/S0165-0114(01)00251-2
  9. Cesarone, F., Scozzari, A. and Tardella, F. (2013), A new method for mean-variance portfolio optimization with cardinality constraints, Annals of Operations Research, 205, 213-234. https://doi.org/10.1007/s10479-012-1165-7
  10. Cui, X. T., Zheng, X. J., Zhu, S. S., and Sun, X. L. (2013), Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems, Journal of Global Optimization, 56, 1409-1423. https://doi.org/10.1007/s10898-012-9842-2
  11. Clikyurt, U. and Oekici, S. (2007), Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach, European Journal of Operational Research, 179(1), 186-202. https://doi.org/10.1016/j.ejor.2005.02.079
  12. Deng, G. F., Lin, W. T., and Lo, C. C. (2012), Markowitz-based portfolio selection with cardinality constraints using improved particle swarm optimization, Expert Systems with Applications, 39, 4558-4566. https://doi.org/10.1016/j.eswa.2011.09.129
  13. Dubois, D. and Prade, H. (1988), Possibility Theory, Plenum Perss, New York.
  14. Fernandez, A. and Gomez, S. (2007), Portfolio selection using neural networks, Computers & Operations Research, 34, 1177-1191. https://doi.org/10.1016/j.cor.2005.06.017
  15. Gulpinar, N. and Rustem, B. (2007), Worst-case robust decisions for multi-period mean-variance portfolio optimization, European Journal of Operational Research, 183(3), 981-1000. https://doi.org/10.1016/j.ejor.2006.02.046
  16. Huang, X. (2008), Risk Curve and Fuzzy Portfolio Selection, Computers and Mathematics with Applications, 55, 1102-1112. https://doi.org/10.1016/j.camwa.2007.06.019
  17. Koksalan, M. and Sakar, C. T. (2014), An interactive approach to stochastic programming-based portfolio optimization, To Appear in Annals of Operations Research.
  18. Le Thi, H. A., Moeini, M., and Dinh, T. P. (2009), Portfolio selection under downside risk measures and cardinality constraints based on DC programming and DCA, Computational Management Science, 6, 459-475. https://doi.org/10.1007/s10287-009-0098-3
  19. Le Thi, H. A. and Moeini, M. (2014), Long-short portfolio optimization under cardinality constraints by difference of convex functions algorithm, Journal of Optimization Theory and Applications, 161, 199-224. https://doi.org/10.1007/s10957-012-0197-0
  20. Li, C. J. and Li, Z. F. (2012), Multi-period portfolio optimization for asset-liability management with bankrupt control, Applied Mathematics and Computation, 218, 11196-11208. https://doi.org/10.1016/j.amc.2012.05.010
  21. Li, D. and Ng, W. L. (2000), Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10(3), 387-406. https://doi.org/10.1111/1467-9965.00100
  22. Li, D., Sun, X., and Wang, J. (2006), Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection, Mathematical Finance, 16, 83-101. https://doi.org/10.1111/j.1467-9965.2006.00262.x
  23. Li, X., Qin, Z., and Kar, S. (2010), Mean-variance-skewness model for portfolio selection with fuzzy returns, European Journal of operational Research, 202, 239-247. https://doi.org/10.1016/j.ejor.2009.05.003
  24. Liu, Y. J., Zhang, W. G., and Xu, W. J. (2012), Fuzzy multi-period portfolio selection optimization models using multiple criteria, Automatica, 48, 3042-3053. https://doi.org/10.1016/j.automatica.2012.08.036
  25. Liu, Y. J., Zhang, W. G. and Zhang, P. (2013), A multiperiod portfolio selection optimization model by using interval analysis, Economic Modelling, 33, 113-119. https://doi.org/10.1016/j.econmod.2013.03.006
  26. Mansini, R., Ogryczak, W., and Speranza, M. G. (2007), Conditional value at risk and related linear programming models for portfolio optimization, Annals of Operations Research, 152, 227-256. https://doi.org/10.1007/s10479-006-0142-4
  27. Markowitz, H. M. (1952), Portfolio selection, Journal of Finance, 7, 77-91.
  28. Murray, W. and Shek, H. (2012), A local relaxation method for the cardinality constrained portfolio optimization problem, Computational Optimization and Applications, 53, 681-709. https://doi.org/10.1007/s10589-012-9471-1
  29. Ruiz-Torrubiano, R. and Suarez, A. (2010), Hybrid approaches and dimensionality reduction for portfolio selection with cardinality constrains, IEEE Computational Intelligence Magazine, 5, 92-107. https://doi.org/10.1109/MCI.2010.936308
  30. Shaw, D. X., Liu, M. S., and Kopman, L. (2008), Lagrangian relaxation procedure for cardinality-constrained portfolio optimization, Optimization Methods & Software, 23, 411-420. https://doi.org/10.1080/10556780701722542
  31. Sun, X. L., Zheng, X. J., and Li, D. (2013), Recent advances in mathematical programming with semicontinuous variables and cardinality constraint, Journal of the Operations Research Society of China, 1, 55-77. https://doi.org/10.1007/s40305-013-0004-0
  32. Tanaka, H., Guo, P., and Turksen, I. B. (2000), Portfolio selection based on fuzzy probabilities and possibility distributions, Fuzzy Sets and Systems, 111, 387-397. https://doi.org/10.1016/S0165-0114(98)00041-4
  33. van Binsbergen, J. H. and Brandt, M. (2007), Solving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function?, Computational Economics, 29, 355-367. https://doi.org/10.1007/s10614-006-9073-z
  34. Woodside-Oriakhi, M., Lucas, C., and Beasley, J. E. (2011), Heuristic algorithms for the cardinality constrained efficient frontier, European Journal of Operational Research, 213, 538-550. https://doi.org/10.1016/j.ejor.2011.03.030
  35. Wu, H. L. and Li, Z. F. (2012), Multi-period meanvariance portfolio selection with regime switching and a stochastic cash flow, Insurance: Mathematics and Economics, 50, 371-384. https://doi.org/10.1016/j.insmatheco.2012.01.003
  36. Yan, W. and Li, S. R. (2009), A class of multi-period semi-variance portfolio selection with a four-factor futures price model, Journal of Applied Mathematics and Computing, 29, 19-34. https://doi.org/10.1007/s12190-008-0086-8
  37. Yan, W., Miao, R., and Li, S. R. (2007), Multi-period semi-variance portfolio selection: Model and numerical solution, Applied Mathematics and Computation, 194, 128-134. https://doi.org/10.1016/j.amc.2007.04.036
  38. Yu, M., Takahashi, S., Inoue, H., and Wang, S. Y. (2010), Dynamic portfolio optimization with risk control for absolute deviation model, European Journal of Operational Research, 201(2), 349-364. https://doi.org/10.1016/j.ejor.2009.03.009
  39. Yu, M. and Wang, S. Y. (2012), Dynamic optimal portfolio with maximum absolute deviation model, Journal of Global Optimization, 53, 363-380. https://doi.org/10.1007/s10898-012-9887-2
  40. Zhang, W. G. and Nie, Z. K. (2004), On admissible efficient portfolio selection problem, Applied Mathematics and Computation, 159, 357-371. https://doi.org/10.1016/j.amc.2003.10.019
  41. Zhang, W. G., Liu, W. A., and Wang, Y. L. (2006), On admissible efficient portfolio selection problem: Models and algorithms, Applied Mathematics and Computation, 176, 208-218. https://doi.org/10.1016/j.amc.2005.09.085
  42. Zhang, W. G. and Wang, Y. L. (2008), An analytic derivation of admissible efficient frontier with borrowing, European Journal of Operational Research, 184, 229-243. https://doi.org/10.1016/j.ejor.2006.09.058
  43. Zhang, W. G., Liu, Y. J., and Xu, W. J. (2012), A possibilistic mean-semivariance-entropy model for multiperiod portfolio selection with transaction costs, European Journal of Operational Research, 222, 41-349.
  44. Zhang, W. G., Liu, Y. J., and Xu, W. J. (2014), A new fuzzy programming approach for multi-period portfolio Optimization with return demand and risk control, Fuzzy Sets and Systems, 246, 107-126. https://doi.org/10.1016/j.fss.2013.09.002
  45. Zhang, P. and Zhang, W. G. (2014), Multiperiod mean absolute deviation fuzzy portfolio selection model with risk control and cardinality constraints, Fuzzy Sets and Systems, 255, 74-91. https://doi.org/10.1016/j.fss.2014.07.018
  46. Zhu, S. S., Li, D., and Wang, S. Y. (2004), Risk control over bankruptcy in dynamic portfolio selection: a generalized mean-variance formulation, IEEE Transactions on Automatic Control, 49(3), 447-457. https://doi.org/10.1109/TAC.2004.824474