Article Contents
Article Contents

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

• * Corresponding author: zhangck@gdut.edu.cn

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.

Mathematics Subject Classification: Primary: 90B50, 90C39; Secondary: 93E20.

 Citation:

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

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

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

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

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

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

•  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. M. J. Brennan  and  Y. Xia , Dynamic asset allocation under inflation, Journal of Finance, 57 (2002) , 1201-1238. 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. H. Chang , Dynamic mean-variance portfolio selection with liability and stochastic interest rate, Economic Modelling, 51 (2015) , 172-182. 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. 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. 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. 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. 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. W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer Science & Business Media, New York, 2006. D. Giamouridis  and  A. Sakkas , Dynamic asset allocation with liabilities, European Financial Management, 23 (2017) , 254-291. 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. J. Hull  and  A. White , Pricing interest-rate-derivative securities, Review of Financial Studies, 3 (1990) , 573-592. H. K. Koo , Consumption and portfolio selection with labor income: A continuous time approach, Mathematical Finance, 81 (1998) , 49-56. 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. 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. 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. H. Markowitz , Portfolio selection, Journal of Finance, 7 (1952) , 77-91. 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. 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. 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. 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. T. K. Siu , Long-term strategic asset allocation with infation risk and regime switching, Quantitative Finance, 11 (2011) , 1565-1580.  doi: 10.1080/14697680903055588. 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. 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. 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. 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. 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. J. Yu , Optimal asset-liability management for an insurer under markov regime switching jump-diffusion market, Asia-Pacific Financial Markets, 21 (2014) , 317-330. A. Zhang, Stochastic Optimization in Finance and Life Insurance: Applications of the Martingale Method, Ph.D thesis, University of Kaiserslautern in Kaiserslautern, 2008. 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. 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.

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