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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

 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  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.

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:
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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
Impact of $r_0$ on the efficient frontier
Impact of $\sigma_{r}(t)$ on the efficient frontier
Impact of $\rho(t)$ on the efficient frontier
Impact of $I_0$ on the efficient frontier
Impact of $v(t)$ on the efficient frontier
Impact of $\eta(t)$ on the efficient frontier
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