American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Generalized ADMM with optimal indefinite proximal term for linearly constrained convex optimization
  • JIMO Home
  • This Issue
  • Next Article
    An executive model for network-level pavement maintenance and rehabilitation planning based on linear integer programming
March  2020, 16(2): 813-834. doi: 10.3934/jimo.2018180

Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks

Huai-Nian Zhu 1, , Cheng-Ke Zhang 1,, and  Zhuo Jin 2,

1. 

School of Economics and Commence, Guangdong University of Technology, Guangzhou 510520, China
2. 

Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia

* Corresponding author: zhangck@gdut.edu.cn

Received  December 2017 Revised  July 2018 Published  March 2020 Early access  December 2018

Fund Project: This research is supported by National Natural Science Foundation of China (Nos.71571053, 71673061), Natural Science Foundation of Guangdong Province (Nos.2015A030310218, 2016A030313701, 2018A030313687) and Distinguished Innovation Program of Education Commission of Guangdong Province (N0.2015WTSCX014)

This paper investigates a continuous-time Markowitz mean-variance asset-liability management (ALM) problem under stochastic interest rates and inflation risks. We assume that the company can invest in $n + 1$ assets: one risk-free bond and $n$ risky stocks. The risky stock's price is governed by a geometric Brownian motion (GBM), and the uncontrollable liability follows a Brownian motion with drift, respectively. The correlation between the risky assets and the liability is considered. The objective is to minimize the risk (measured by variance) of the terminal wealth subject to a given expected terminal wealth level. By applying the Lagrange multiplier method and stochastic control approach, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation, which can be converted into six partial differential equations (PDEs). The closed-form solutions for these six PDEs are derived by using the homogenization approach and the variable transformation technique. Then the closed-form expressions for the efficient strategy and efficient frontier are obtained. In addition, a numerical example is presented to illustrate the results.

Keywords: Asset-liability management, mean-variance, stochastic interest rate, inflation, Hamilton-Jacobi-Bellman equation.
Mathematics Subject Classification: Primary: 90B50, 90C39; Secondary: 93E20.
Citation: Huai-Nian Zhu, Cheng-Ke Zhang, Zhuo Jin. Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks. Journal of Industrial & Management Optimization, 2020, 16 (2) : 813-834. doi: 10.3934/jimo.2018180
References:
[1]

A. Bensoussan, J. Keppo and S. P. Sethi, Optimal consumption and portfolio decisions with partially observed real prices, Mathematical Finance, 19 (2009), 215-236.  doi: 10.1111/j.1467-9965.2009.00362.x.  Google Scholar

[2]

M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, Journal of Finance, 57 (2002), 1201-1238.   Google Scholar

[3]

U. Celikyurt and S. Özekici, Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach, European Journal of Operational Research, 179 (2007), 186-202.   Google Scholar

[4]

H. Chang, Dynamic mean-variance portfolio selection with liability and stochastic interest rate, Economic Modelling, 51 (2015), 172-182.   Google Scholar

[5]

P. Chen, H. 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

[6]

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

[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]

O. L. Costa and M. V. Araujo, A generalized multi-period mean-variance portfolio optimization with markov switching parameters, Automatica, 44 (2008), 2487-2497.  doi: 10.1016/j.automatica.2008.02.014.  Google Scholar

[9]

X. Cui, J. Gao, X. Li and D. Li, Optimal multi-period 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

[10]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer Science & Business Media, New York, 2006.  Google Scholar

[11]

D. Giamouridis and A. Sakkas, Dynamic asset allocation with liabilities, European Financial Management, 23 (2017), 254-291.   Google Scholar

[12]

R. P. Hoevenaars, R. D. Molenaar, P. C. Schotman and T. B. Steenkamp, Strategic asset allocation with liabilities: Beyond stocks and bonds, Journal of Economic Dynamics and Control, 32 (2008), 2939-2970.  doi: 10.1016/j.jedc.2007.11.003.  Google Scholar

[13]

J. Hull and A. White, Pricing interest-rate-derivative securities, Review of Financial Studies, 3 (1990), 573-592.   Google Scholar

[14]

H. K. Koo, Consumption and portfolio selection with labor income: A continuous time approach, Mathematical Finance, 81 (1998), 49-56.   Google Scholar

[15]

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

[16]

A. E. Lim and X. Y. Zhou, Mean-variance portfolio selection with random parameters in a complete market, Mathematics of Operations Research, 27 (2002), 101-120.  doi: 10.1287/moor.27.1.101.337.  Google Scholar

[17]

S. Lv, Z. Wu and Z. Yu, Continuous-time mean-variance portfolio selection with random horizon in an incomplete market, Automatica, 69 (2016), 176-180.  doi: 10.1016/j.automatica.2016.02.017.  Google Scholar

[18]

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

[19]

C. Munk, C. Sorensen and T. N. Vinther, Dynamic asset allocation under mean-reverting returns, stochastic interest rates and inflation uncertainty, International Review of Economics and Finance, 13 (2004), 141-166.   Google Scholar

[20]

J. Pan and Q. Xiao, Optimal asset-liability management with liquidity constraints and stochastic interest rates in the expected utility framework, Journal of Computational and Applied Mathematics, 317 (2017), 371-387.  doi: 10.1016/j.cam.2016.11.037.  Google Scholar

[21]

J. Pan and Q. Xiao, Optimal mean-variance asset-liability management with stochastic interest rates and inflation risks, Mathematical Methods of Operations Research, 85 (2017), 491-519.  doi: 10.1007/s00186-017-0580-6.  Google Scholar

[22]

Y. Shen, Mean-variance portfolio selection in a complete market with unbounded random coefficients, Automatica, 55 (2015), 165-175.  doi: 10.1016/j.automatica.2015.03.009.  Google Scholar

[23]

T. K. Siu, Long-term strategic asset allocation with infation risk and regime switching, Quantitative Finance, 11 (2011), 1565-1580.  doi: 10.1080/14697680903055588.  Google Scholar

[24]

J. Wang and P. A. Forsyth, Continuous time mean variance asset allocation: A time-consistent strategy, European Journal of Operational Research, 209 (2011), 184-201.  doi: 10.1016/j.ejor.2010.09.038.  Google Scholar

[25]

S. Xie, Z. 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

[26]

S. 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

[27]

H. Yao, Z. Yang and P. Chen, Markowitz's mean-variance defined contribution pension fund management under inflation: A continuous-time model, Insurance: Mathematics and Economics, 53 (2013), 851-863.  doi: 10.1016/j.insmatheco.2013.10.002.  Google Scholar

[28]

H. Yao, Z. Li and Y. Lai, Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate, Journal of Industrial and Management Optimization, 12 (2016), 187-209.  doi: 10.3934/jimo.2016.12.187.  Google Scholar

[29]

J. Yu, Optimal asset-liability management for an insurer under markov regime switching jump-diffusion market, Asia-Pacific Financial Markets, 21 (2014), 317-330.   Google Scholar

[30]

A. Zhang, Stochastic Optimization in Finance and Life Insurance: Applications of the Martingale Method, Ph.D thesis, University of Kaiserslautern in Kaiserslautern, 2008. Google Scholar

[31]

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

[32]

X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.  Google Scholar

show all references

References:
[1]

A. Bensoussan, J. Keppo and S. P. Sethi, Optimal consumption and portfolio decisions with partially observed real prices, Mathematical Finance, 19 (2009), 215-236.  doi: 10.1111/j.1467-9965.2009.00362.x.  Google Scholar

[2]

M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, Journal of Finance, 57 (2002), 1201-1238.   Google Scholar

[3]

U. Celikyurt and S. Özekici, Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach, European Journal of Operational Research, 179 (2007), 186-202.   Google Scholar

[4]

H. Chang, Dynamic mean-variance portfolio selection with liability and stochastic interest rate, Economic Modelling, 51 (2015), 172-182.   Google Scholar

[5]

P. Chen, H. 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

[6]

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

[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]

O. L. Costa and M. V. Araujo, A generalized multi-period mean-variance portfolio optimization with markov switching parameters, Automatica, 44 (2008), 2487-2497.  doi: 10.1016/j.automatica.2008.02.014.  Google Scholar

[9]

X. Cui, J. Gao, X. Li and D. Li, Optimal multi-period 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

[10]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer Science & Business Media, New York, 2006.  Google Scholar

[11]

D. Giamouridis and A. Sakkas, Dynamic asset allocation with liabilities, European Financial Management, 23 (2017), 254-291.   Google Scholar

[12]

R. P. Hoevenaars, R. D. Molenaar, P. C. Schotman and T. B. Steenkamp, Strategic asset allocation with liabilities: Beyond stocks and bonds, Journal of Economic Dynamics and Control, 32 (2008), 2939-2970.  doi: 10.1016/j.jedc.2007.11.003.  Google Scholar

[13]

J. Hull and A. White, Pricing interest-rate-derivative securities, Review of Financial Studies, 3 (1990), 573-592.   Google Scholar

[14]

H. K. Koo, Consumption and portfolio selection with labor income: A continuous time approach, Mathematical Finance, 81 (1998), 49-56.   Google Scholar

[15]

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

[16]

A. E. Lim and X. Y. Zhou, Mean-variance portfolio selection with random parameters in a complete market, Mathematics of Operations Research, 27 (2002), 101-120.  doi: 10.1287/moor.27.1.101.337.  Google Scholar

[17]

S. Lv, Z. Wu and Z. Yu, Continuous-time mean-variance portfolio selection with random horizon in an incomplete market, Automatica, 69 (2016), 176-180.  doi: 10.1016/j.automatica.2016.02.017.  Google Scholar

[18]

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

[19]

C. Munk, C. Sorensen and T. N. Vinther, Dynamic asset allocation under mean-reverting returns, stochastic interest rates and inflation uncertainty, International Review of Economics and Finance, 13 (2004), 141-166.   Google Scholar

[20]

J. Pan and Q. Xiao, Optimal asset-liability management with liquidity constraints and stochastic interest rates in the expected utility framework, Journal of Computational and Applied Mathematics, 317 (2017), 371-387.  doi: 10.1016/j.cam.2016.11.037.  Google Scholar

[21]

J. Pan and Q. Xiao, Optimal mean-variance asset-liability management with stochastic interest rates and inflation risks, Mathematical Methods of Operations Research, 85 (2017), 491-519.  doi: 10.1007/s00186-017-0580-6.  Google Scholar

[22]

Y. Shen, Mean-variance portfolio selection in a complete market with unbounded random coefficients, Automatica, 55 (2015), 165-175.  doi: 10.1016/j.automatica.2015.03.009.  Google Scholar

[23]

T. K. Siu, Long-term strategic asset allocation with infation risk and regime switching, Quantitative Finance, 11 (2011), 1565-1580.  doi: 10.1080/14697680903055588.  Google Scholar

[24]

J. Wang and P. A. Forsyth, Continuous time mean variance asset allocation: A time-consistent strategy, European Journal of Operational Research, 209 (2011), 184-201.  doi: 10.1016/j.ejor.2010.09.038.  Google Scholar

[25]

S. Xie, Z. 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

[26]

S. 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

[27]

H. Yao, Z. Yang and P. Chen, Markowitz's mean-variance defined contribution pension fund management under inflation: A continuous-time model, Insurance: Mathematics and Economics, 53 (2013), 851-863.  doi: 10.1016/j.insmatheco.2013.10.002.  Google Scholar

[28]

H. Yao, Z. Li and Y. Lai, Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate, Journal of Industrial and Management Optimization, 12 (2016), 187-209.  doi: 10.3934/jimo.2016.12.187.  Google Scholar

[29]

J. Yu, Optimal asset-liability management for an insurer under markov regime switching jump-diffusion market, Asia-Pacific Financial Markets, 21 (2014), 317-330.   Google Scholar

[30]

A. Zhang, Stochastic Optimization in Finance and Life Insurance: Applications of the Martingale Method, Ph.D thesis, University of Kaiserslautern in Kaiserslautern, 2008. Google Scholar

[31]

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

[32]

X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.  Google Scholar

Figure 1.  Impact of $r_0$ on the efficient frontier
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Impact of $\sigma_{r}(t)$ on the efficient frontier
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Impact of $\rho(t)$ on the efficient frontier
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Impact of $I_0$ on the efficient frontier
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Impact of $v(t)$ on the efficient frontier
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Impact of $\eta(t)$ on the efficient frontier
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

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, 2021, 17 (3) : 1383-1410. doi: 10.3934/jimo.2020026
[2]

Haixiang Yao, Zhongfei Li, Yongzeng Lai. Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate. Journal of Industrial & Management Optimization, 2016, 12 (1) : 187-209. doi: 10.3934/jimo.2016.12.187
[3]

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
[4]

Jean-Claude Zambrini. On the geometry of the Hamilton-Jacobi-Bellman equation. Journal of Geometric Mechanics, 2009, 1 (3) : 369-387. doi: 10.3934/jgm.2009.1.369
[5]

Bian-Xia Yang, Shanshan Gu, Guowei Dai. Existence and multiplicity for Hamilton-Jacobi-Bellman equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021130
[6]

Yan Zhang, Yonghong Wu, Benchawan Wiwatanapataphee, Francisca Angkola. Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework. Journal of Industrial & Management Optimization, 2020, 16 (1) : 71-101. doi: 10.3934/jimo.2018141
[7]

Daniele Castorina, Annalisa Cesaroni, Luca Rossi. On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1251-1263. doi: 10.3934/cpaa.2016.15.1251
[8]

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
[9]

Yu Yuan, Hui Mi. Robust optimal asset-liability management with penalization on ambiguity. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021121
[10]

Yingxu Tian, Junyi Guo, Zhongyang Sun. Optimal mean-variance reinsurance in a financial market with stochastic rate of return. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1887-1912. doi: 10.3934/jimo.2020051
[11]

Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial & Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161
[12]

Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, , () : -. doi: 10.3934/era.2021046
[13]

Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521
[14]

Lan Yi, Zhongfei Li, Duan Li. Multi-period portfolio selection for asset-liability management with uncertain investment horizon. Journal of Industrial & Management Optimization, 2008, 4 (3) : 535-552. doi: 10.3934/jimo.2008.4.535
[15]

Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933
[16]

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295
[17]

Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. The optimal mean variance problem with inflation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 185-203. doi: 10.3934/dcdsb.2016.21.185
[18]

Hao Chang, Jiaao Li, Hui Zhao. Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021025
[19]

Qian Zhao, Yang Shen, Jiaqin Wei. Mean-variance investment and contribution decisions for defined benefit pension plans in a stochastic framework. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1147-1171. doi: 10.3934/jimo.2020015
[20]

Zhiping Chen, Jia Liu, Gang Li. Time consistent policy of multi-period mean-variance problem in stochastic markets. Journal of Industrial & Management Optimization, 2016, 12 (1) : 229-249. doi: 10.3934/jimo.2016.12.229

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (348)
  • HTML views (1064)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences