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 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 & 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.  Google Scholar

[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.  Google Scholar

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S. BoydM. T. MuellerB. O'Donoghue and Y. Wang, Performance bounds and suboptimal policies for multi-period investment, Foundations and Trends in Optimization, 1 (2014), 1-72.   Google Scholar

[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.  Google Scholar

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P. ChenH. 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.  Google Scholar

[7]

Z. P. ChenG. 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.  Google Scholar

[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.  Google Scholar

[9]

V. DeMiguelA. 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.  Google Scholar

[10]

Y. H. FuK. M. NgB. 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.  Google Scholar

[11]

N. Gârleanu and L. H. Pedersen, Dynamic trading with predictable returns and transaction costs, The Journal of Finance, 68 (2013), 2309-2340.   Google Scholar

[12]

N. GülpinarD. 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.  Google Scholar

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S. GuoL. YuX. 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.  Google Scholar

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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.  Google Scholar

[15]

M. LeippoldF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

X. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

L. M. Viceira, Bond risk, bond return volatility, and the term structure of interest rates, International Journal of Forecasting, 28 (2012), 97-117.   Google Scholar

[26]

J. WeiK. C. WongS. 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.  Google Scholar

[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.  Google Scholar

[28]

S. XieZ. 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.  Google Scholar

[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.  Google Scholar

[30]

H. YaoZ. 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.  Google Scholar

[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.  Google Scholar

[32]

L. YuS. Y. WangF. 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.  Google Scholar

[33]

L. YuS. 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.  Google Scholar

[34]

L. YuS. 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.  Google Scholar

[35]

J. ZhangZ. 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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

T. Bjork and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, SSRN: 1694759. doi: 10.2139/ssrn.1694759.  Google Scholar

[4]

S. BoydM. T. MuellerB. O'Donoghue and Y. Wang, Performance bounds and suboptimal policies for multi-period investment, Foundations and Trends in Optimization, 1 (2014), 1-72.   Google Scholar

[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.  Google Scholar

[6]

P. ChenH. 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.  Google Scholar

[7]

Z. P. ChenG. 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.  Google Scholar

[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.  Google Scholar

[9]

V. DeMiguelA. 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.  Google Scholar

[10]

Y. H. FuK. M. NgB. 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.  Google Scholar

[11]

N. Gârleanu and L. H. Pedersen, Dynamic trading with predictable returns and transaction costs, The Journal of Finance, 68 (2013), 2309-2340.   Google Scholar

[12]

N. GülpinarD. 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.  Google Scholar

[13]

S. GuoL. YuX. 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.  Google Scholar

[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.  Google Scholar

[15]

M. LeippoldF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

X. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

L. M. Viceira, Bond risk, bond return volatility, and the term structure of interest rates, International Journal of Forecasting, 28 (2012), 97-117.   Google Scholar

[26]

J. WeiK. C. WongS. 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.  Google Scholar

[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.  Google Scholar

[28]

S. XieZ. 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.  Google Scholar

[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.  Google Scholar

[30]

H. YaoZ. 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.  Google Scholar

[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.  Google Scholar

[32]

L. YuS. Y. WangF. 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.  Google Scholar

[33]

L. YuS. 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.  Google Scholar

[34]

L. YuS. 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.  Google Scholar

[35]

J. ZhangZ. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  The M-V frontiers under different strategies with/without cost
Figure 2.  The M-V frontiers under different strategies
Figure 3.  The efficient frontiers of strategies under different costaversion coefficient
Figure 4.  The efficient frontiers of strategies under different parameter
Table 1.  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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