# American Institute of Mathematical Sciences

July  2019, 15(3): 1493-1515. doi: 10.3934/jimo.2018106

## Multiperiod portfolio optimization for asset-liability management with quadratic transaction costs

 1 School of Business Administration, Hunan University, Changsha 410082, China 2 Business School, Hunan Normal University, Changsha 410081, China 3 Business School, University of Kent, Kent, CT2 7PE, UK

* Corresponding author: Zhongbao Zhou

Received  January 2018 Revised  March 2018 Published  July 2019 Early access  July 2018

This paper investigates the multiperiod asset-liability management problem with quadratic transaction costs. Under the mean-variance criteria, we construct tractability models with/without the riskless asset and obtain the pre-commitment and time-consistent investment strategies through the application of embedding scheme and backward induction approach, respectively. In addition, some conclusions in the existing literatures can be regarded as the degenerated cases under our setting. Finally, the numerical simulations are given to show the difference of frontiers derived by different strategies. Also, some interesting findings on the impact of quadratic transaction cost parameters on efficient frontiers are discussed.

Citation: Zhongbao Zhou, Ximei Zeng, Helu Xiao, Tiantian Ren, Wenbin Liu. Multiperiod portfolio optimization for asset-liability management with quadratic transaction costs. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1493-1515. doi: 10.3934/jimo.2018106
##### References:
 [1] R. D. Arnott and W. H. Wagner, The measurement and control of trading costs, Financial Analysts Journal, 46 (1990), 73-80.  doi: 10.2469/faj.v46.n6.73. [2] A. Bensoussan, K. C. Wong and S. C. P. Yam, Mean-variance pre-commitment policies revisited via a mean-field technique, in 2012 Recent Advances in Financial Engineering: Proceedings of the International Workshop on Finance, (2014), 177-198. doi: 10.1142/9789814571647_0008. [3] T. Bjork and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, SSRN: 1694759. doi: 10.2139/ssrn.1694759. [4] S. Boyd, M. T. Mueller, B. O'Donoghue and Y. Wang, Performance bounds and suboptimal policies for multi-period investment, Foundations and Trends in Optimization, 1 (2014), 1-72. [5] H. Chang, Dynamic mean-variance portfolio selection with liability and stochastic interest rate, Economic Modelling, 51 (2015), 172-182.  doi: 10.1016/j.econmod.2015.07.017. [6] P. Chen, H. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance: Mathematics and Economics, 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001. [7] Z. P. Chen, G. Li and J. E. Guo, Optimal investment policy in the time consistent mean-variance formulation, Insurance: Mathematics and Economics, 52 (2013), 145-156.  doi: 10.1016/j.insmatheco.2012.11.007. [8] M. C. Chiu, Asset-liability management in continuous-time: Cointegration and exponential utility, Optimization and Control for Systems in the Big-Data Era, (2017), 85-100.  doi: 10.1007/978-3-319-53518-0_6. [9] V. DeMiguel, A. Martín-Utrera and F. J. Nogales, Parameter uncertainty in multiperiod portfolio optimization with transaction costs, Journal of Financial and Quantitative Analysis, 50 (2015), 1443-1471.  doi: 10.1017/S002210901500054X. [10] Y. H. Fu, K. M. Ng, B. Huang and H. C. Huang, Portfolio optimization with transaction costs: A two-period mean-variance model, Annals of Operations Research, 233 (2015), 135-156.  doi: 10.1007/s10479-014-1574-x. [11] N. Gârleanu and L. H. Pedersen, Dynamic trading with predictable returns and transaction costs, The Journal of Finance, 68 (2013), 2309-2340. [12] N. Gülpinar, D. Pachamanova and E. Çanakoğlu, A robust asset-liability management framework for investment products with guarantees, OR Spectrum, 38 (2016), 1007-1041.  doi: 10.1007/s00291-016-0437-z. [13] S. Guo, L. Yu, X. Li and S. Kar, Fuzzy multi-period portfolio selection with different investment horizons, European Journal of Operational Research, 254 (2016), 1026-1035.  doi: 10.1016/j.ejor.2016.04.055. [14] A. Keel and H. H. Müller, Efficient portfolios in the asset liability context, ASTIN Bulletin: The Journal of the IAA, 25 (1995), 33-48.  doi: 10.2143/AST.25.1.563252. [15] M. Leippold, F. Trojani and P. Vanini, A geometric approach to multiperiod mean variance optimization of assets and liabilities, Journal of Economic Dynamics and Control, 28 (2004), 1079-1113.  doi: 10.1016/S0165-1889(03)00067-8. [16] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [17] C. Li and Z. Li, Multiperiod portfolio optimization for asset-liability management with bankrupt control, Applied Mathematics and Computation, 218 (2012), 11196-11208.  doi: 10.1016/j.amc.2012.05.010. [18] X. Li, S. Guo and L. Yu, Skewness of fuzzy numbers and its applications in portfolio selection, IEEE Transactions on Fuzzy Systems, 23 (2015), 2135-2143.  doi: 10.1109/TFUZZ.2015.2404340. [19] J. Long and S. Zeng, Equilibrium time-consistent strategy for corporate international investment problem with mean-variance criterion, Mathematical Problems in Engineering, 2016 (2016), Art. ID 3295041, 20 pp. doi: 10.1155/2016/3295041. [20] H. Q. Ma, M. Wu and N. J. Huang, Time consistent strategies for mean-variance asset-liability management problems, Mathematical Problems in Engineering, 2013 (2013), Art. ID 709129, 16 pp. [21] H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. [22] M. Papi and S. Sbaraglia, Optimal asset-liability management with constraints: A dynamic programming approach, Applied Mathematics and Computation, 173 (2006), 306-349.  doi: 10.1016/j.amc.2005.04.078. [23] R. W. Reid and S. J. Citron, On noninferior performance index vectors, Journal of Optimization Theory and Applications, 7 (1971), 11-28.  doi: 10.1007/BF00933589. [24] W. F. Sharpe and L. G. Tint, Liabilities-A new approach, The Journal of Portfolio Management, 16 (1990), 5-10.  doi: 10.3905/jpm.1990.409248. [25] L. M. Viceira, Bond risk, bond return volatility, and the term structure of interest rates, International Journal of Forecasting, 28 (2012), 97-117. [26] J. Wei, K. C. Wong, S. C. P. Yam and S. P. Yung, Markowitz's mean-variance asset-liability management with regime switching: A time-consistent approach, Insurance: Mathematics and Economics, 53 (2013), 281-291.  doi: 10.1016/j.insmatheco.2013.05.008. [27] H. Wu, Time-consistent strategies for a multiperiod mean-variance portfolio selection problem, Journal of Applied Mathematics, 2013 (2013), Art. ID 841627, 13 pp. [28] S. Xie, Z. Li and S. Wang, Continuous-time portfolio selection with liability: Mean-variance model and stochastic LQ approach, Insurance: Mathematics and Economics, 42 (2008), 943-953.  doi: 10.1016/j.insmatheco.2007.10.014. [29] H. Yang and P. Chen, Markowitz's mean-variance asset-liability management with regime switching: A multiperiod model, Applied Mathematical Finance, 18 (2011), 29-50.  doi: 10.1080/13504861003703633. [30] H. Yao, Z. Li and S. Chen, Continuous-time mean-variance portfolio selection with only risky assets, Economic Modelling, 36 (2014), 244-251.  doi: 10.1016/j.econmod.2013.09.041. [31] A. Yoshimoto, The mean-variance approach to portfolio optimization subject to transaction costs, Journal of the Operations Research Society of Japan, 39 (1996), 99-117.  doi: 10.15807/jorsj.39.99. [32] L. Yu, S. Y. Wang, F. H. Wen and K. K. Lai, Genetic algorithm-based multi-criteria project portfolio selection, Annals of Operations Research, 197 (2012), 71-86.  doi: 10.1007/s10479-010-0819-6. [33] L. Yu, S. Y. Wang and K. K. Lai, Multi-attribute portfolio selection with genetic optimization algorithms, INFOR: Information Systems and Operational Research, 47 (2009), 23-30.  doi: 10.3138/infor.47.1.23. [34] L. Yu, S. Y. Wang and K. K. Lai, Neural network-based mean-variance -skewness model for portfolio selection, Computers and Operations Research, 35 (2008), 34-46.  doi: 10.1016/j.cor.2006.02.012. [35] J. Zhang, Z. Jin and Y. An, Dynamic portfolio optimization with ambiguity aversion, Journal of Banking and Finance, 79 (2017), 95-109.  doi: 10.1016/j.jbankfin.2017.03.007. [36] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.

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##### References:
 [1] R. D. Arnott and W. H. Wagner, The measurement and control of trading costs, Financial Analysts Journal, 46 (1990), 73-80.  doi: 10.2469/faj.v46.n6.73. [2] A. Bensoussan, K. C. Wong and S. C. P. Yam, Mean-variance pre-commitment policies revisited via a mean-field technique, in 2012 Recent Advances in Financial Engineering: Proceedings of the International Workshop on Finance, (2014), 177-198. doi: 10.1142/9789814571647_0008. [3] T. Bjork and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, SSRN: 1694759. doi: 10.2139/ssrn.1694759. [4] S. Boyd, M. T. Mueller, B. O'Donoghue and Y. Wang, Performance bounds and suboptimal policies for multi-period investment, Foundations and Trends in Optimization, 1 (2014), 1-72. [5] H. Chang, Dynamic mean-variance portfolio selection with liability and stochastic interest rate, Economic Modelling, 51 (2015), 172-182.  doi: 10.1016/j.econmod.2015.07.017. [6] P. Chen, H. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance: Mathematics and Economics, 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001. [7] Z. P. Chen, G. Li and J. E. Guo, Optimal investment policy in the time consistent mean-variance formulation, Insurance: Mathematics and Economics, 52 (2013), 145-156.  doi: 10.1016/j.insmatheco.2012.11.007. [8] M. C. Chiu, Asset-liability management in continuous-time: Cointegration and exponential utility, Optimization and Control for Systems in the Big-Data Era, (2017), 85-100.  doi: 10.1007/978-3-319-53518-0_6. [9] V. DeMiguel, A. Martín-Utrera and F. J. Nogales, Parameter uncertainty in multiperiod portfolio optimization with transaction costs, Journal of Financial and Quantitative Analysis, 50 (2015), 1443-1471.  doi: 10.1017/S002210901500054X. [10] Y. H. Fu, K. M. Ng, B. Huang and H. C. Huang, Portfolio optimization with transaction costs: A two-period mean-variance model, Annals of Operations Research, 233 (2015), 135-156.  doi: 10.1007/s10479-014-1574-x. [11] N. Gârleanu and L. H. Pedersen, Dynamic trading with predictable returns and transaction costs, The Journal of Finance, 68 (2013), 2309-2340. [12] N. Gülpinar, D. Pachamanova and E. Çanakoğlu, A robust asset-liability management framework for investment products with guarantees, OR Spectrum, 38 (2016), 1007-1041.  doi: 10.1007/s00291-016-0437-z. [13] S. Guo, L. Yu, X. Li and S. Kar, Fuzzy multi-period portfolio selection with different investment horizons, European Journal of Operational Research, 254 (2016), 1026-1035.  doi: 10.1016/j.ejor.2016.04.055. [14] A. Keel and H. H. Müller, Efficient portfolios in the asset liability context, ASTIN Bulletin: The Journal of the IAA, 25 (1995), 33-48.  doi: 10.2143/AST.25.1.563252. [15] M. Leippold, F. Trojani and P. Vanini, A geometric approach to multiperiod mean variance optimization of assets and liabilities, Journal of Economic Dynamics and Control, 28 (2004), 1079-1113.  doi: 10.1016/S0165-1889(03)00067-8. [16] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [17] C. Li and Z. Li, Multiperiod portfolio optimization for asset-liability management with bankrupt control, Applied Mathematics and Computation, 218 (2012), 11196-11208.  doi: 10.1016/j.amc.2012.05.010. [18] X. Li, S. Guo and L. Yu, Skewness of fuzzy numbers and its applications in portfolio selection, IEEE Transactions on Fuzzy Systems, 23 (2015), 2135-2143.  doi: 10.1109/TFUZZ.2015.2404340. [19] J. Long and S. Zeng, Equilibrium time-consistent strategy for corporate international investment problem with mean-variance criterion, Mathematical Problems in Engineering, 2016 (2016), Art. ID 3295041, 20 pp. doi: 10.1155/2016/3295041. [20] H. Q. Ma, M. Wu and N. J. Huang, Time consistent strategies for mean-variance asset-liability management problems, Mathematical Problems in Engineering, 2013 (2013), Art. ID 709129, 16 pp. [21] H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. [22] M. Papi and S. Sbaraglia, Optimal asset-liability management with constraints: A dynamic programming approach, Applied Mathematics and Computation, 173 (2006), 306-349.  doi: 10.1016/j.amc.2005.04.078. [23] R. W. Reid and S. J. Citron, On noninferior performance index vectors, Journal of Optimization Theory and Applications, 7 (1971), 11-28.  doi: 10.1007/BF00933589. [24] W. F. Sharpe and L. G. Tint, Liabilities-A new approach, The Journal of Portfolio Management, 16 (1990), 5-10.  doi: 10.3905/jpm.1990.409248. [25] L. M. Viceira, Bond risk, bond return volatility, and the term structure of interest rates, International Journal of Forecasting, 28 (2012), 97-117. [26] J. Wei, K. C. Wong, S. C. P. Yam and S. P. Yung, Markowitz's mean-variance asset-liability management with regime switching: A time-consistent approach, Insurance: Mathematics and Economics, 53 (2013), 281-291.  doi: 10.1016/j.insmatheco.2013.05.008. [27] H. Wu, Time-consistent strategies for a multiperiod mean-variance portfolio selection problem, Journal of Applied Mathematics, 2013 (2013), Art. ID 841627, 13 pp. [28] S. Xie, Z. Li and S. Wang, Continuous-time portfolio selection with liability: Mean-variance model and stochastic LQ approach, Insurance: Mathematics and Economics, 42 (2008), 943-953.  doi: 10.1016/j.insmatheco.2007.10.014. [29] H. Yang and P. Chen, Markowitz's mean-variance asset-liability management with regime switching: A multiperiod model, Applied Mathematical Finance, 18 (2011), 29-50.  doi: 10.1080/13504861003703633. [30] H. Yao, Z. Li and S. Chen, Continuous-time mean-variance portfolio selection with only risky assets, Economic Modelling, 36 (2014), 244-251.  doi: 10.1016/j.econmod.2013.09.041. [31] A. Yoshimoto, The mean-variance approach to portfolio optimization subject to transaction costs, Journal of the Operations Research Society of Japan, 39 (1996), 99-117.  doi: 10.15807/jorsj.39.99. [32] L. Yu, S. Y. Wang, F. H. Wen and K. K. Lai, Genetic algorithm-based multi-criteria project portfolio selection, Annals of Operations Research, 197 (2012), 71-86.  doi: 10.1007/s10479-010-0819-6. [33] L. Yu, S. Y. Wang and K. K. Lai, Multi-attribute portfolio selection with genetic optimization algorithms, INFOR: Information Systems and Operational Research, 47 (2009), 23-30.  doi: 10.3138/infor.47.1.23. [34] L. Yu, S. Y. Wang and K. K. Lai, Neural network-based mean-variance -skewness model for portfolio selection, Computers and Operations Research, 35 (2008), 34-46.  doi: 10.1016/j.cor.2006.02.012. [35] J. Zhang, Z. Jin and Y. An, Dynamic portfolio optimization with ambiguity aversion, Journal of Banking and Finance, 79 (2017), 95-109.  doi: 10.1016/j.jbankfin.2017.03.007. [36] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.
The M-V frontiers under different strategies with/without cost
The M-V frontiers under different strategies
The efficient frontiers of strategies under different costaversion coefficient
The efficient frontiers of strategies under different parameter
The parameter set
 $\Lambda^1$ $\Lambda^2$ $\Lambda^3$ $\Lambda^4$ $0.001 *\begin{bmatrix} 1& 0 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 1 \end{bmatrix}$ $0.001*\begin{bmatrix} 3& 0 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 1 \end{bmatrix}$ $0.001*\begin{bmatrix} 1& 0 & 0 \\ 0 & 3& 0 \\ 0 & 0 & 1 \end{bmatrix}$ $0.001*\begin{bmatrix} 1& 0 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 3 \end{bmatrix}$
 $\Lambda^1$ $\Lambda^2$ $\Lambda^3$ $\Lambda^4$ $0.001 *\begin{bmatrix} 1& 0 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 1 \end{bmatrix}$ $0.001*\begin{bmatrix} 3& 0 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 1 \end{bmatrix}$ $0.001*\begin{bmatrix} 1& 0 & 0 \\ 0 & 3& 0 \\ 0 & 0 & 1 \end{bmatrix}$ $0.001*\begin{bmatrix} 1& 0 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 3 \end{bmatrix}$
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