# American Institute of Mathematical Sciences

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. Bielecki, H. Q. Jin, S. 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. [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. [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. [4] X. Y. Cui, J. J. Gao, X. 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. [5] X. Y. Cui, X. 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. [6] E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Thoery and Investment Analysis, John Wiley & Sons, 2007. [7] C. P. Fu, A. 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. [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. [9] 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. [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. [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. [12] X. Li, X. 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. [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. [14] H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. doi: 10.1111/j.1540-6261.1952.tb01525.x. [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. [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. [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. [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. [19] S. Xie, Z. 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. [20] L. Yi, Z. 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. [21] L. Yi, X. P. Wu, X. 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. [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. [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. [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. [25] S. S. Zhu, D. 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.

show all references

##### References:
 [1] T. R. Bielecki, H. Q. Jin, S. 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. [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. [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. [4] X. Y. Cui, J. J. Gao, X. 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. [5] X. Y. Cui, X. 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. [6] E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Thoery and Investment Analysis, John Wiley & Sons, 2007. [7] C. P. Fu, A. 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. [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. [9] 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. [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. [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. [12] X. Li, X. 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. [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. [14] H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. doi: 10.1111/j.1540-6261.1952.tb01525.x. [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. [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. [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. [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. [19] S. Xie, Z. 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. [20] L. Yi, Z. 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. [21] L. Yi, X. P. Wu, X. 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. [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. [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. [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. [25] S. S. Zhu, D. 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.
Efficient frontiers with different correlation coefficients
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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