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Time-consistent strategy for a multi-period mean-variance asset-liability management problem with stochastic interest rate
1. | School of Finance and Economics, Qinghai University, Xining 810016, PR China |
2. | School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, PR China |
3. | Sun Yat-sen Business School, Sun Yat-sen University, Guangzhou 510275, China |
4. | School of Management, Xinhua College of Sun Yat-sen University, Guangzhou 510520, China |
5. | School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China |
In this paper, we investigate a multi-period mean-variance asset-liability management problem with stochastic interest rate and seek its time-consistent strategy. The financial market is assumed to be composed of one risk-free asset and multiple risky assets, and the stochastic interest rate is characterized by the discrete-time Vasicek model proposed by Yao et al. (2016a)[
References:
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L. H. Bian, Z. F. Li and H. X. Yao,
Pre-commitment and equilibrium investment strategies for the DC pension plan with regime switching and a return of premiums clause, Insurance: Mathematics and Economics, 81 (2018), 78-94.
doi: 10.1016/j.insmatheco.2018.05.005. |
[2] |
T. Björk, M. Khapko and A. Murgoci,
On time-inconsistent stochastic control in continuous time, Finance and Stochastics, 21 (2017), 331-360.
doi: 10.1007/s00780-017-0327-5. |
[3] |
T. Björk and A. Murgoci,
A theory of Markovian time-inconsitent stochastic control in discrete time, Finance and Stochastics, 18 (2014), 545-592.
doi: 10.1007/s00780-014-0234-y. |
[4] |
C. 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. |
[5] |
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. |
[6] |
P. Chen, H. L. 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] |
M. C. Chiu and H. Y. Wong,
Mean-variance asset-liability management with asset correlation risk and insurance liabilities, Insurance: Mathematics and Economics, 59 (2014), 300-310.
doi: 10.1016/j.insmatheco.2014.10.003. |
[8] |
R. Ferland and F. Watier,
Mean-variance efficiency with extended CIR interest rates, Applied Stochastic Models in Business and Industry, 26 (2010), 71-84.
doi: 10.1002/asmb.767. |
[9] |
D. Hainaut,
Dynamic asset allocation under VaR constraint with stochastic interest rates, Annals of Operations Research, 172 (2009), 97-117.
doi: 10.1007/s10479-008-0509-9. |
[10] |
L. He and Z. X. Liang,
Optimal investment strategy for the DC plan with the return of premiums clauses in a mean-variance framework, Insurance: Mathematics and Economics, 53 (2013), 643-649.
doi: 10.1016/j.insmatheco.2013.09.002. |
[11] |
R. Korn and H. Kraft,
A stochastic control approach to portfolio problems with stochastic interest rates, SIAM Journal on Control and Optimization, 40 (2002), 1250-1269.
doi: 10.1137/S0363012900377791. |
[12] |
M. Leippold, F. Trojani and P. Vanini,
A geomeric 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. |
[13] |
M. Leippold, F. Trojani and P. Vanini,
Multiperiod mean-variance efficient portfolios with endogenous liabilities, Quantitative Finance, 11 (2011), 1535-1546.
doi: 10.1080/14697680902950813. |
[14] |
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. |
[15] |
C. J. Li, Z. F. Li, K. Fu and H. Q. Song,
Time-consistent optimal portfolio strategy for asset-liability management under mean-variance criterion, Accounting and Finance Research, 2 (2013), 89-104.
doi: 10.5430/afr.v2n2p89. |
[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] |
Z. X. Liang and M. Ma,
Optimal dynamic asset allocation of pension fund in mortality and salary risks framework, Insurance: Mathematics and Economics, 64 (2015), 151-161.
doi: 10.1016/j.insmatheco.2015.05.008. |
[18] |
A. Lioui and P. Poncet,
On optimal portfolio choice under stochastic interest rates, Journal of Economic Dynamics and Control, 25 (2001), 1841-1865.
doi: 10.1016/S0165-1889(00)00005-1. |
[19] |
H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. Google Scholar |
[20] |
F. Menoncin and E. Vigna,
Mean-variance target-based optimization for defined contribution pension schemes in a stochastic framework, Insurance: Mathematics and Economics, 76 (2017), 172-184.
doi: 10.1016/j.insmatheco.2017.08.002. |
[21] |
R. J. Muirhead, Aspects of Multivariate Statistical Theory, John Wiley, New York, 1982. Google Scholar |
[22] |
C. Munk and C. Sørensen, Dynamic asset allocation with stochastic income and interest rates, Journal of Financial Economics, 96 (2010), 433-462. Google Scholar |
[23] |
S. Mushtaq and D. A. Siddiqui,
Effect of interest rate on economic performance: Evidence from Islamic and non-Islamic economies, Financial Innovation, 2 (2016), 2-9.
doi: 10.1186/s40854-016-0028-7. |
[24] |
J. Pan and Q. X. Xiao,
Optimal asset-liability management with liquidity constraints and stochastic interest rates in the expected utility framework, Journal of Computational and Applied Mathematics, 317 (2017a), 371-387.
doi: 10.1016/j.cam.2016.11.037. |
[25] |
J. Pan and Q. X. Xiao,
Optimal mean-variance asset-liability management with stochastic interest rates and inflation risks, Mathematical Methods of Operations Research, 85 (2017b), 491-519.
doi: 10.1007/s00186-017-0580-6. |
[26] |
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. |
[27] |
Y. Shen and T. K. Siu,
Asset allocation under stochastic interest rate with regime switching, Economic Modelling, 29 (2012), 1126-1136.
doi: 10.1016/j.econmod.2012.03.024. |
[28] |
R. Strotz, Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, (1973), 128–143.
doi: 10.1007/978-1-349-15492-0_10. |
[29] |
J. Y. Sun, Z. F. Li and Y. Zeng,
Precommitment and equilibrium investment strategies for defined contribution pension plans under a jump-diffusion model, Insurance: Mathematics and Economics, 67 (2016), 158-172.
doi: 10.1016/j.insmatheco.2016.01.005. |
[30] |
J. Tobin,
Liquidity preference as behavior toward risk, Review of Economic Studies, 67 (1958), 65-86.
doi: 10.2307/2296205. |
[31] |
J. Wei and T. X. Wang,
Time-consistent mean-variance asset-liability management with random coefficients, Insurance: Mathematics and Economics, 77 (2017), 84-96.
doi: 10.1016/j.insmatheco.2017.08.011. |
[32] |
J. Wei, K. C. Wong, S. C. P. Yam and S. P. Yung, Liquidity preference as behavior toward risk, Review of Economic Studies, 67 (1958), 65-86. Google Scholar |
[33] |
H. L. Wu, Time-consistent strategies for a multiperiod mean-variance portfolio selection problem, Journal of Applied Mathematics, 2013 (2013), Art. ID 841627, 13 pp.
doi: 10.1155/2013/841627. |
[34] |
H. L. Wu and H. Chen,
Nash equilibrium strategy for a multi-period mean-variance portfolio selection problem with regime switching, Economic Modelling, 46 (2015), 79-90.
doi: 10.1016/j.econmod.2014.12.024. |
[35] |
S. X. Xie,
Continuous-time mean-variance portfolio selection with liability and regime switching, Insurance: Mathematics and Economics, 45 (2009), 148-155.
doi: 10.1016/j.insmatheco.2009.05.005. |
[36] |
H. X. Yao, X. Li, Z. F. Hao and Y. Li,
Dynamic asset-liability management in a Markov market with stochastic cash flows, Quantitative Finance, 16 (2016b), 1575-1597.
doi: 10.1080/14697688.2016.1151070. |
[37] |
H. X. Yao, Z. F. Li and Y. Z. Lai,
Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate, Journal of Industrial and Management Optimization, 12 (2016c), 187-209.
doi: 10.3934/jimo.2016.12.187. |
[38] |
H. X. Yao, Z. F. Li and D. Li,
Multi-period mean-variance portfolio selection with stochastic interest rate and uncontrollable liability, European Journal of Operational Research, 252 (2016a), 837-851.
doi: 10.1016/j.ejor.2016.01.049. |
[39] |
H. X. Yao, Y. Zeng and S. M. Chen,
Multi-period mean-variance asset-liability management with uncontrolled cash flow and uncertain time-horizon, Economic Modelling, 30 (2013), 492-500.
doi: 10.1016/j.econmod.2012.10.004. |
[40] |
F. Z. Zhang, Matrix Theory: Basic Results and Techniques, 2$^{nd}$ edition, Springer-Verlag, New York, 2011. Google Scholar |
[41] |
L. Zhang, H. Zhang and H. X. Yao,
Optimal investment management for a defined contribution pension fund under imperfect information, Insurance: Mathematics and Economics, 79 (2018), 210-224.
doi: 10.1016/j.insmatheco.2018.01.007. |
[42] |
X. Y. Zhou and D. Li,
Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics Optimization, 42 (2000), 19-33.
doi: 10.1007/s002450010003. |
show all references
References:
[1] |
L. H. Bian, Z. F. Li and H. X. Yao,
Pre-commitment and equilibrium investment strategies for the DC pension plan with regime switching and a return of premiums clause, Insurance: Mathematics and Economics, 81 (2018), 78-94.
doi: 10.1016/j.insmatheco.2018.05.005. |
[2] |
T. Björk, M. Khapko and A. Murgoci,
On time-inconsistent stochastic control in continuous time, Finance and Stochastics, 21 (2017), 331-360.
doi: 10.1007/s00780-017-0327-5. |
[3] |
T. Björk and A. Murgoci,
A theory of Markovian time-inconsitent stochastic control in discrete time, Finance and Stochastics, 18 (2014), 545-592.
doi: 10.1007/s00780-014-0234-y. |
[4] |
C. 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. |
[5] |
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. |
[6] |
P. Chen, H. L. 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] |
M. C. Chiu and H. Y. Wong,
Mean-variance asset-liability management with asset correlation risk and insurance liabilities, Insurance: Mathematics and Economics, 59 (2014), 300-310.
doi: 10.1016/j.insmatheco.2014.10.003. |
[8] |
R. Ferland and F. Watier,
Mean-variance efficiency with extended CIR interest rates, Applied Stochastic Models in Business and Industry, 26 (2010), 71-84.
doi: 10.1002/asmb.767. |
[9] |
D. Hainaut,
Dynamic asset allocation under VaR constraint with stochastic interest rates, Annals of Operations Research, 172 (2009), 97-117.
doi: 10.1007/s10479-008-0509-9. |
[10] |
L. He and Z. X. Liang,
Optimal investment strategy for the DC plan with the return of premiums clauses in a mean-variance framework, Insurance: Mathematics and Economics, 53 (2013), 643-649.
doi: 10.1016/j.insmatheco.2013.09.002. |
[11] |
R. Korn and H. Kraft,
A stochastic control approach to portfolio problems with stochastic interest rates, SIAM Journal on Control and Optimization, 40 (2002), 1250-1269.
doi: 10.1137/S0363012900377791. |
[12] |
M. Leippold, F. Trojani and P. Vanini,
A geomeric 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. |
[13] |
M. Leippold, F. Trojani and P. Vanini,
Multiperiod mean-variance efficient portfolios with endogenous liabilities, Quantitative Finance, 11 (2011), 1535-1546.
doi: 10.1080/14697680902950813. |
[14] |
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. |
[15] |
C. J. Li, Z. F. Li, K. Fu and H. Q. Song,
Time-consistent optimal portfolio strategy for asset-liability management under mean-variance criterion, Accounting and Finance Research, 2 (2013), 89-104.
doi: 10.5430/afr.v2n2p89. |
[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] |
Z. X. Liang and M. Ma,
Optimal dynamic asset allocation of pension fund in mortality and salary risks framework, Insurance: Mathematics and Economics, 64 (2015), 151-161.
doi: 10.1016/j.insmatheco.2015.05.008. |
[18] |
A. Lioui and P. Poncet,
On optimal portfolio choice under stochastic interest rates, Journal of Economic Dynamics and Control, 25 (2001), 1841-1865.
doi: 10.1016/S0165-1889(00)00005-1. |
[19] |
H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. Google Scholar |
[20] |
F. Menoncin and E. Vigna,
Mean-variance target-based optimization for defined contribution pension schemes in a stochastic framework, Insurance: Mathematics and Economics, 76 (2017), 172-184.
doi: 10.1016/j.insmatheco.2017.08.002. |
[21] |
R. J. Muirhead, Aspects of Multivariate Statistical Theory, John Wiley, New York, 1982. Google Scholar |
[22] |
C. Munk and C. Sørensen, Dynamic asset allocation with stochastic income and interest rates, Journal of Financial Economics, 96 (2010), 433-462. Google Scholar |
[23] |
S. Mushtaq and D. A. Siddiqui,
Effect of interest rate on economic performance: Evidence from Islamic and non-Islamic economies, Financial Innovation, 2 (2016), 2-9.
doi: 10.1186/s40854-016-0028-7. |
[24] |
J. Pan and Q. X. Xiao,
Optimal asset-liability management with liquidity constraints and stochastic interest rates in the expected utility framework, Journal of Computational and Applied Mathematics, 317 (2017a), 371-387.
doi: 10.1016/j.cam.2016.11.037. |
[25] |
J. Pan and Q. X. Xiao,
Optimal mean-variance asset-liability management with stochastic interest rates and inflation risks, Mathematical Methods of Operations Research, 85 (2017b), 491-519.
doi: 10.1007/s00186-017-0580-6. |
[26] |
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. |
[27] |
Y. Shen and T. K. Siu,
Asset allocation under stochastic interest rate with regime switching, Economic Modelling, 29 (2012), 1126-1136.
doi: 10.1016/j.econmod.2012.03.024. |
[28] |
R. Strotz, Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, (1973), 128–143.
doi: 10.1007/978-1-349-15492-0_10. |
[29] |
J. Y. Sun, Z. F. Li and Y. Zeng,
Precommitment and equilibrium investment strategies for defined contribution pension plans under a jump-diffusion model, Insurance: Mathematics and Economics, 67 (2016), 158-172.
doi: 10.1016/j.insmatheco.2016.01.005. |
[30] |
J. Tobin,
Liquidity preference as behavior toward risk, Review of Economic Studies, 67 (1958), 65-86.
doi: 10.2307/2296205. |
[31] |
J. Wei and T. X. Wang,
Time-consistent mean-variance asset-liability management with random coefficients, Insurance: Mathematics and Economics, 77 (2017), 84-96.
doi: 10.1016/j.insmatheco.2017.08.011. |
[32] |
J. Wei, K. C. Wong, S. C. P. Yam and S. P. Yung, Liquidity preference as behavior toward risk, Review of Economic Studies, 67 (1958), 65-86. Google Scholar |
[33] |
H. L. Wu, Time-consistent strategies for a multiperiod mean-variance portfolio selection problem, Journal of Applied Mathematics, 2013 (2013), Art. ID 841627, 13 pp.
doi: 10.1155/2013/841627. |
[34] |
H. L. Wu and H. Chen,
Nash equilibrium strategy for a multi-period mean-variance portfolio selection problem with regime switching, Economic Modelling, 46 (2015), 79-90.
doi: 10.1016/j.econmod.2014.12.024. |
[35] |
S. X. Xie,
Continuous-time mean-variance portfolio selection with liability and regime switching, Insurance: Mathematics and Economics, 45 (2009), 148-155.
doi: 10.1016/j.insmatheco.2009.05.005. |
[36] |
H. X. Yao, X. Li, Z. F. Hao and Y. Li,
Dynamic asset-liability management in a Markov market with stochastic cash flows, Quantitative Finance, 16 (2016b), 1575-1597.
doi: 10.1080/14697688.2016.1151070. |
[37] |
H. X. Yao, Z. F. Li and Y. Z. Lai,
Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate, Journal of Industrial and Management Optimization, 12 (2016c), 187-209.
doi: 10.3934/jimo.2016.12.187. |
[38] |
H. X. Yao, Z. F. Li and D. Li,
Multi-period mean-variance portfolio selection with stochastic interest rate and uncontrollable liability, European Journal of Operational Research, 252 (2016a), 837-851.
doi: 10.1016/j.ejor.2016.01.049. |
[39] |
H. X. Yao, Y. Zeng and S. M. Chen,
Multi-period mean-variance asset-liability management with uncontrolled cash flow and uncertain time-horizon, Economic Modelling, 30 (2013), 492-500.
doi: 10.1016/j.econmod.2012.10.004. |
[40] |
F. Z. Zhang, Matrix Theory: Basic Results and Techniques, 2$^{nd}$ edition, Springer-Verlag, New York, 2011. Google Scholar |
[41] |
L. Zhang, H. Zhang and H. X. Yao,
Optimal investment management for a defined contribution pension fund under imperfect information, Insurance: Mathematics and Economics, 79 (2018), 210-224.
doi: 10.1016/j.insmatheco.2018.01.007. |
[42] |
X. Y. Zhou and D. Li,
Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics Optimization, 42 (2000), 19-33.
doi: 10.1007/s002450010003. |








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