January  2018, 14(1): 249-265. doi: 10.3934/jimo.2017045

A mean-field formulation for multi-period asset-liability mean-variance portfolio selection with probability constraints

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou 51061, China

2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

3. 

Department of Finance and Investment, Sun Yat-Sen Business School, Sun Yat-Sen University, Guangzhou 510275, China

Received  March 2016 Revised  February 2017 Published  April 2017

Fund Project: This work was partially supported by Research Grants Council of Hong Kong under grants 519913, 15209614 and 15224215, grants of National Natural Science Foundation of China(No. 11571124 and No. 11671158), and China Postdoctoral Science Foundation (No. 2016M592505).

This paper is concerned with studying an optimal multi-period asset-liability mean-variance portfolio selection with probability constraints using mean-field formulation without embedding technique. We strictly derive its analytical optimal strategy and efficient frontier. Numerical examples shed light on efficiency and accuracy of our method when dealing with this class of multi-period non-separable mean-variance portfolio selection problems.

Citation: Xianping Wu, Xun Li, Zhongfei Li. A mean-field formulation for multi-period asset-liability mean-variance portfolio selection with probability constraints. Journal of Industrial & Management Optimization, 2018, 14 (1) : 249-265. doi: 10.3934/jimo.2017045
References:
[1]

T. R. BieleckiH. Q. JinS. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244.  doi: 10.1111/j.0960-1627.2005.00218.x.  Google Scholar

[2]

P. Chen and H. L. Yang, Markowitz's mean-variance asset{liability management with regime switching: A multi-period model, Applied Mathematical Finance, 18 (2011), 29-50.  doi: 10.1080/13504861003703633.  Google Scholar

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M. C. Chiu and D. Li, Asset and liability management under a continuous-time mean-variance optimization framework, Insurance: Mathematics and Economics, 39 (2006), 330-355.  doi: 10.1016/j.insmatheco.2006.03.006.  Google Scholar

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X. Y. CuiJ. J. GaoX. Li and D. Li, Optimal multiperiod mean-variance policy under no-shorting constraint, European Journal of Operational Research, 234 (2014), 459-468.  doi: 10.1016/j.ejor.2013.02.040.  Google Scholar

[5]

X. Y. CuiX. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Transactions on Automatic Control, 59 (2014), 1833-1844.  doi: 10.1109/TAC.2014.2311875.  Google Scholar

[6]

E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Thoery and Investment Analysis, John Wiley & Sons, 2007. Google Scholar

[7]

C. P. FuA. Lari-Lavassani and X. Li, Dynamic mean variance portfolio selection with borrowing constraint, European Journal of Operational Research, 200 (2010), 312-319.  doi: 10.1016/j.ejor.2009.01.005.  Google Scholar

[8]

C. G. Krouse, Portfolio Balancing Corporate Assets and Liabilities with Special Application to InsuranceManagement, The Journal of Financial and Quantitative Analysis, 5 (1970), 77-104.   Google Scholar

[9]

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

[10]

C. J. Li and Z. F. Li, Multi-period 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

[11]

D. Li and W.L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.  Google Scholar

[12]

X. LiX. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with noshorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504.  Google Scholar

[13]

Z. F. Li and S. X. Xie, Mean-variance portfolio optimization under stochastic income and uncertain exit time, Dynamics of Continuous, Discrete and Impulsive Systems, 17 (2010), 131-147.   Google Scholar

[14]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[15]

W. F. Sharpe and L. G. Tint, Liabilities-a new approach, Journal of Portfolio Management, 16 (1990), 5-10.  doi: 10.3905/jpm.1990.409248.  Google Scholar

[16]

S. Z. Wei and Z. X. Ye, Multi-period optimization portfolio with bankruptcy control in stochastic market, Applied Mathematics and Computation, 186 (2007), 414-425.  doi: 10.1016/j.amc.2006.07.108.  Google Scholar

[17]

H. L. Wu and Z. F. Li, Multi-period mean-variance portfolio selection with regime switching and a stochastic cash flow, Insurance: Mathematics and Economics, 50 (2012), 371-384.  doi: 10.1016/j.insmatheco.2012.01.003.  Google Scholar

[18]

H. L. Wu and Y. Zeng, Multi-period mean-variance portfolio selection in a regime-switching market with a bankruptcy state, Optimal Control Applications and Methods, 34 (2013), 415-432.  doi: 10.1002/oca.2027.  Google Scholar

[19]

S. XieZ. F. Li and S. Y. 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

[20]

L. YiZ. F. Li and D. Li, Multi-period portfolio selection for asset-liability management with uncertain investment horizon, Journal of Industrial and Management Optimization, 4 (2008), 535-552.   Google Scholar

[21]

L. YiX. P. WuX. Li and X. Y. Cui, Mean-field formulation for optimal multi-period meanvariance portfolio selection with uncertain exit time, Operations Research Letters, 42 (2014), 489-494.  doi: 10.1016/j.orl.2014.08.007.  Google Scholar

[22]

Y. Zeng and Z. F. Li, Asset-liability management under benchmark and mean-variance criteria in a jump diffusion market, Journal of Systems Science and Complexity, 24 (2011), 317-327.  doi: 10.1007/s11424-011-9105-1.  Google Scholar

[23]

L. Zhang and Z. F. Li, Multi-period mean-varinace portfolio selection with uncertain time horizon when returns are serially correlated, Mathematical Probelems in Engineering, 2012 (2012), Art. ID 216891, 17 pp.  Google Scholar

[24]

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

[25]

S. S. ZhuD. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection:A generalized mean-variance formulation, Automatic Control, IEEE Transactions on, 49 (2004), 447-457.  doi: 10.1109/TAC.2004.824474.  Google Scholar

show all references

References:
[1]

T. R. BieleckiH. Q. JinS. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244.  doi: 10.1111/j.0960-1627.2005.00218.x.  Google Scholar

[2]

P. Chen and H. L. Yang, Markowitz's mean-variance asset{liability management with regime switching: A multi-period model, Applied Mathematical Finance, 18 (2011), 29-50.  doi: 10.1080/13504861003703633.  Google Scholar

[3]

M. C. Chiu and D. Li, Asset and liability management under a continuous-time mean-variance optimization framework, Insurance: Mathematics and Economics, 39 (2006), 330-355.  doi: 10.1016/j.insmatheco.2006.03.006.  Google Scholar

[4]

X. Y. CuiJ. J. GaoX. Li and D. Li, Optimal multiperiod mean-variance policy under no-shorting constraint, European Journal of Operational Research, 234 (2014), 459-468.  doi: 10.1016/j.ejor.2013.02.040.  Google Scholar

[5]

X. Y. CuiX. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Transactions on Automatic Control, 59 (2014), 1833-1844.  doi: 10.1109/TAC.2014.2311875.  Google Scholar

[6]

E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Thoery and Investment Analysis, John Wiley & Sons, 2007. Google Scholar

[7]

C. P. FuA. Lari-Lavassani and X. Li, Dynamic mean variance portfolio selection with borrowing constraint, European Journal of Operational Research, 200 (2010), 312-319.  doi: 10.1016/j.ejor.2009.01.005.  Google Scholar

[8]

C. G. Krouse, Portfolio Balancing Corporate Assets and Liabilities with Special Application to InsuranceManagement, The Journal of Financial and Quantitative Analysis, 5 (1970), 77-104.   Google Scholar

[9]

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

[10]

C. J. Li and Z. F. Li, Multi-period 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

[11]

D. Li and W.L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.  Google Scholar

[12]

X. LiX. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with noshorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504.  Google Scholar

[13]

Z. F. Li and S. X. Xie, Mean-variance portfolio optimization under stochastic income and uncertain exit time, Dynamics of Continuous, Discrete and Impulsive Systems, 17 (2010), 131-147.   Google Scholar

[14]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[15]

W. F. Sharpe and L. G. Tint, Liabilities-a new approach, Journal of Portfolio Management, 16 (1990), 5-10.  doi: 10.3905/jpm.1990.409248.  Google Scholar

[16]

S. Z. Wei and Z. X. Ye, Multi-period optimization portfolio with bankruptcy control in stochastic market, Applied Mathematics and Computation, 186 (2007), 414-425.  doi: 10.1016/j.amc.2006.07.108.  Google Scholar

[17]

H. L. Wu and Z. F. Li, Multi-period mean-variance portfolio selection with regime switching and a stochastic cash flow, Insurance: Mathematics and Economics, 50 (2012), 371-384.  doi: 10.1016/j.insmatheco.2012.01.003.  Google Scholar

[18]

H. L. Wu and Y. Zeng, Multi-period mean-variance portfolio selection in a regime-switching market with a bankruptcy state, Optimal Control Applications and Methods, 34 (2013), 415-432.  doi: 10.1002/oca.2027.  Google Scholar

[19]

S. XieZ. F. Li and S. Y. 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

[20]

L. YiZ. F. Li and D. Li, Multi-period portfolio selection for asset-liability management with uncertain investment horizon, Journal of Industrial and Management Optimization, 4 (2008), 535-552.   Google Scholar

[21]

L. YiX. P. WuX. Li and X. Y. Cui, Mean-field formulation for optimal multi-period meanvariance portfolio selection with uncertain exit time, Operations Research Letters, 42 (2014), 489-494.  doi: 10.1016/j.orl.2014.08.007.  Google Scholar

[22]

Y. Zeng and Z. F. Li, Asset-liability management under benchmark and mean-variance criteria in a jump diffusion market, Journal of Systems Science and Complexity, 24 (2011), 317-327.  doi: 10.1007/s11424-011-9105-1.  Google Scholar

[23]

L. Zhang and Z. F. Li, Multi-period mean-varinace portfolio selection with uncertain time horizon when returns are serially correlated, Mathematical Probelems in Engineering, 2012 (2012), Art. ID 216891, 17 pp.  Google Scholar

[24]

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

[25]

S. S. ZhuD. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection:A generalized mean-variance formulation, Automatic Control, IEEE Transactions on, 49 (2004), 447-457.  doi: 10.1109/TAC.2004.824474.  Google Scholar

Figure 1.  Efficient frontiers with different correlation coefficients
Table 1.  Data for the asset allocation example
SP EM MS liability
Expected return 14% 16% 17% 10%
Standard deviation 18.5% 30% 24% 20%
Correlation coefficient
SP 1 0.64 0.79 $\rho_1$
EM 0.64 1 0.75 $\rho_2$
MS 0.79 0.75 1 $\rho_3$
liability $\rho_1$ $\rho_2$ $\rho_3$ 1
SP EM MS liability
Expected return 14% 16% 17% 10%
Standard deviation 18.5% 30% 24% 20%
Correlation coefficient
SP 1 0.64 0.79 $\rho_1$
EM 0.64 1 0.75 $\rho_2$
MS 0.79 0.75 1 $\rho_3$
liability $\rho_1$ $\rho_2$ $\rho_3$ 1
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