# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020026

## 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

* Corresponding author: Zhongfei Li

Received  December 2018 Revised  October 2019 Published  February 2020

Fund Project: This research is supported by the National Natural Science Foundation of China (Nos. 71721001, 71871071, 71471045), the Natural Science Foundation of Guangdong Province of China (Nos. 2014A030312003, 2017A030313399, 2018B030311004), the Innovation Team Project of Guangdong Colleges and Universities (No. 2016WCXTD012), and the Innovative School Project in Higher Education of Guangdong Province of China (No. GWTP-GC-2017-03)

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)[38]. We regard this problem as a non-cooperative game whose equilibrium strategy is the desired time-consistent strategy. We derive the analytical expressions of the equilibrium strategy, the equilibrium value function and the equilibrium efficient frontier by the extended Bellman equation. Some special cases of our model are discussed, and some properties of our equilibrium strategy, including a two-fund separation theorem, are proposed. Finally, a numerical example with real data is given to illustrate our theoretical results.

Citation: Lihua Bian, Zhongfei Li, Haixiang Yao. Time-consistent strategy for a multi-period mean-variance asset-liability management problem with stochastic interest rate. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020026
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] J. Tobin, Liquidity preference as behavior toward risk, Review of Economic Studies, 67 (1958), 65-86.  doi: 10.2307/2296205.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] J. Tobin, Liquidity preference as behavior toward risk, Review of Economic Studies, 67 (1958), 65-86.  doi: 10.2307/2296205.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
The equilibrium strategies with and without liability
The surplus process of the equilibrium strategies with and without liability
The effect of liability on the equilibrium strategy
The effect of stochastic interest rate on the equilibrium strategy
A comparison of the equilibrium strategy and precommitment strategy
The effect of liability on the equilibrium efficient frontier
The effect of stochastic interest rate on the equilibrium efficient frontier
A comparison of the equilibrium efficient frontier and pre-commitment efficient frontier
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