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July  2013, 9(3): 643-656. doi: 10.3934/jimo.2013.9.643

## Multi-period mean-variance portfolio selection with fixed and proportional transaction costs

 1 Department of Mathematics, Xidian University, Xi'an, 710071, China, China

Received  May 2012 Revised  November 2012 Published  April 2013

Portfolio selection problem is one of the core research fields in modern financial economics. Considering the transaction costs in multi-period investments makes portfolio selection problems hard to solve. In this paper, the multi-period mean-variance portfolio selection problems with fixed and proportional transaction costs are investigated. By introducing the Lagrange multiplier and using the dynamic programming approach, the indirect utility function is defined for solving the portfolio selection problem constructed in this paper. The optimal strategies and the boundaries of the no-transaction region are obtained in the explicit form. And the efficient frontier for the original portfolio selection problems is also given. Numerical result shows that the method provided in this paper works well.
Citation: Zhen Wang, Sanyang Liu. Multi-period mean-variance portfolio selection with fixed and proportional transaction costs. Journal of Industrial & Management Optimization, 2013, 9 (3) : 643-656. doi: 10.3934/jimo.2013.9.643
##### References:
 [1] M. Akian, J. Menaldi and A. Sulem, On an investment-consumption model with transaction costs,, Journal of Control and Optimization, 34 (1996), 329.  doi: 10.1137/S0363012993247159.  Google Scholar [2] A. Balbas and S. Mayral, Nonconvex optimization for pricing and hedging in imperfect markets,, Computers and Mathematics with Applications, 52 (2006), 121.  doi: 10.1016/j.camwa.2006.08.009.  Google Scholar [3] D. Bertsimas and D. Pachamanova, Robust multiperiod portfolio management in the presence of transaction costs,, Computers and Operations Research, 35 (2008), 3.  doi: 10.1016/j.cor.2006.02.011.  Google Scholar [4] M. Best and J. Hlouskova, Quadratic programming with transaction costs,, Computers $&$ Operations Research, 35 (2008), 18.  doi: 10.1016/j.cor.2006.02.013.  Google Scholar [5] P. Boyle and X. Lin, Portfolio selection with transaction costs,, North American Actuarial Journal, 1 (1997), 27.  doi: 10.1080/10920277.1997.10595602.  Google Scholar [6] T. Chellathurai and T. Draviam, Dynamic portfolio selection with fixed and/or proportional transaction costs using non-singular stochastic optimal control theory,, Journal of Economic Dynamics $&$ Control, 31 (2007), 2168.  doi: 10.1016/j.jedc.2006.06.006.  Google Scholar [7] U. Çlikyurt and S. Ökici, Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach,, European Journal of Operational Research, 179 (2007), 186.   Google Scholar [8] G. Constantinides, Optimal portfolio revision with proportional transaction costs: Extension to hara utility function and exogenous deterministic income,, Management Science, 22 (1976), 921.  doi: 10.1287/mnsc.22.8.921.  Google Scholar [9] M. Davis and A. Norman, Portfolio selection with transaction costs,, Mathematics of Operations Research, 15 (1990), 676.  doi: 10.1287/moor.15.4.676.  Google Scholar [10] N. Framstad, B. Øksendal and A. Sulem, Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs,, Journal of Mathematical Economics, 35 (2001), 233.  doi: 10.1016/S0304-4068(00)00067-7.  Google Scholar [11] G. Gennotte and A. Jung, Investment strategies under transaction costs: The finite horizon case,, Management Science, 40 (1994), 385.  doi: 10.1287/mnsc.40.3.385.  Google Scholar [12] B. Jang, Optimal portfolio selection with transaction costs when an illiquid asset pays cash dividends,, Journal of the Korean Mathematical Society, 44 (2007), 139.  doi: 10.4134/JKMS.2007.44.1.139.  Google Scholar [13] J. Kamin, Optimal portfolio revision with a proportional transaction cost,, Management Science, 21 (1975), 1263.  doi: 10.1287/mnsc.21.11.1263.  Google Scholar [14] H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to tokyo stock market,, Management Science, 37 (1991), 519.  doi: 10.1287/mnsc.37.5.519.  Google Scholar [15] D. G. Luenberger, "Opitimization by Vector Space Methods,", Wiley, (1968).   Google Scholar [16] H. Markowitz, "Mean-Variance Analysis in Portfolio Choice and Capital Markets,", Blackwell, (1992).   Google Scholar [17] R. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case,, Review of Economics and Statistics, 51 (1969), 247.  doi: 10.2307/1926560.  Google Scholar [18] K. Muthuraman, A computational scheme for optimal investment-consumption with proportional transaction costs,, Journal of Economic Dynamics $&$ Control, 31 (2007), 1132.  doi: 10.1016/j.jedc.2006.04.005.  Google Scholar [19] P. Samuelson, Lifetime portfolio selection by dynamic stochastic programming,, Review of Economics and Statistics, 51 (1969), 239.  doi: 10.2307/1926559.  Google Scholar [20] M. Woodside-Oriakhi, C. Lucas and J. E. Beasley, Portfolio rebalancing with an investment horizon and transaction costs,, Omega, 41 (2013), 406.  doi: 10.1016/j.omega.2012.03.003.  Google Scholar [21] H. Yao, A simple method for solving multiperiod mean-variance asset-liability management problem,, Procedia Engineering, 23 (2011), 387.  doi: 10.1016/j.proeng.2011.11.2518.  Google Scholar [22] L. Yi, Z. Li and D. Li, Multi-period portfolio selection for asset-liability management with uncertain investment horizon,, Journal of Industrial and Management Optimization, 4 (2008), 535.  doi: 10.3934/jimo.2008.4.535.  Google Scholar [23] M. Yu, S. Takahashib, H. Inoueb and S. Wang, Dynamic portfolio optimization with risk control for absolute deviation model,, European Journal of Operational Research, 201 (2010), 349.  doi: 10.1016/j.ejor.2009.03.009.  Google Scholar

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##### References:
 [1] M. Akian, J. Menaldi and A. Sulem, On an investment-consumption model with transaction costs,, Journal of Control and Optimization, 34 (1996), 329.  doi: 10.1137/S0363012993247159.  Google Scholar [2] A. Balbas and S. Mayral, Nonconvex optimization for pricing and hedging in imperfect markets,, Computers and Mathematics with Applications, 52 (2006), 121.  doi: 10.1016/j.camwa.2006.08.009.  Google Scholar [3] D. Bertsimas and D. Pachamanova, Robust multiperiod portfolio management in the presence of transaction costs,, Computers and Operations Research, 35 (2008), 3.  doi: 10.1016/j.cor.2006.02.011.  Google Scholar [4] M. Best and J. Hlouskova, Quadratic programming with transaction costs,, Computers $&$ Operations Research, 35 (2008), 18.  doi: 10.1016/j.cor.2006.02.013.  Google Scholar [5] P. Boyle and X. Lin, Portfolio selection with transaction costs,, North American Actuarial Journal, 1 (1997), 27.  doi: 10.1080/10920277.1997.10595602.  Google Scholar [6] T. Chellathurai and T. Draviam, Dynamic portfolio selection with fixed and/or proportional transaction costs using non-singular stochastic optimal control theory,, Journal of Economic Dynamics $&$ Control, 31 (2007), 2168.  doi: 10.1016/j.jedc.2006.06.006.  Google Scholar [7] U. Çlikyurt and S. Ökici, Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach,, European Journal of Operational Research, 179 (2007), 186.   Google Scholar [8] G. Constantinides, Optimal portfolio revision with proportional transaction costs: Extension to hara utility function and exogenous deterministic income,, Management Science, 22 (1976), 921.  doi: 10.1287/mnsc.22.8.921.  Google Scholar [9] M. Davis and A. Norman, Portfolio selection with transaction costs,, Mathematics of Operations Research, 15 (1990), 676.  doi: 10.1287/moor.15.4.676.  Google Scholar [10] N. Framstad, B. Øksendal and A. Sulem, Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs,, Journal of Mathematical Economics, 35 (2001), 233.  doi: 10.1016/S0304-4068(00)00067-7.  Google Scholar [11] G. Gennotte and A. Jung, Investment strategies under transaction costs: The finite horizon case,, Management Science, 40 (1994), 385.  doi: 10.1287/mnsc.40.3.385.  Google Scholar [12] B. Jang, Optimal portfolio selection with transaction costs when an illiquid asset pays cash dividends,, Journal of the Korean Mathematical Society, 44 (2007), 139.  doi: 10.4134/JKMS.2007.44.1.139.  Google Scholar [13] J. Kamin, Optimal portfolio revision with a proportional transaction cost,, Management Science, 21 (1975), 1263.  doi: 10.1287/mnsc.21.11.1263.  Google Scholar [14] H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to tokyo stock market,, Management Science, 37 (1991), 519.  doi: 10.1287/mnsc.37.5.519.  Google Scholar [15] D. G. Luenberger, "Opitimization by Vector Space Methods,", Wiley, (1968).   Google Scholar [16] H. Markowitz, "Mean-Variance Analysis in Portfolio Choice and Capital Markets,", Blackwell, (1992).   Google Scholar [17] R. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case,, Review of Economics and Statistics, 51 (1969), 247.  doi: 10.2307/1926560.  Google Scholar [18] K. Muthuraman, A computational scheme for optimal investment-consumption with proportional transaction costs,, Journal of Economic Dynamics $&$ Control, 31 (2007), 1132.  doi: 10.1016/j.jedc.2006.04.005.  Google Scholar [19] P. Samuelson, Lifetime portfolio selection by dynamic stochastic programming,, Review of Economics and Statistics, 51 (1969), 239.  doi: 10.2307/1926559.  Google Scholar [20] M. Woodside-Oriakhi, C. Lucas and J. E. Beasley, Portfolio rebalancing with an investment horizon and transaction costs,, Omega, 41 (2013), 406.  doi: 10.1016/j.omega.2012.03.003.  Google Scholar [21] H. Yao, A simple method for solving multiperiod mean-variance asset-liability management problem,, Procedia Engineering, 23 (2011), 387.  doi: 10.1016/j.proeng.2011.11.2518.  Google Scholar [22] L. Yi, Z. Li and D. Li, Multi-period portfolio selection for asset-liability management with uncertain investment horizon,, Journal of Industrial and Management Optimization, 4 (2008), 535.  doi: 10.3934/jimo.2008.4.535.  Google Scholar [23] M. Yu, S. Takahashib, H. Inoueb and S. Wang, Dynamic portfolio optimization with risk control for absolute deviation model,, European Journal of Operational Research, 201 (2010), 349.  doi: 10.1016/j.ejor.2009.03.009.  Google Scholar
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