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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 |
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, doi: 10.3934/jimo.2018180
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References:
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Optimal consumption and portfolio decisions with partially observed real prices, Mathematical Finance, 19 (2009), 215-236.
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M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, Journal of Finance, 57 (2002), 1201-1238. |
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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. |
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H. Chang, Dynamic mean-variance portfolio selection with liability and stochastic interest rate, Economic Modelling, 51 (2015), 172-182. |
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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. |
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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. |
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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. |
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O. L. Costa and M. V. Araujo,
A generalized multi-period mean-variance portfolio optimization with markov switching parameters, Automatica, 44 (2008), 2487-2497.
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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. |
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W. H. Fleming and H. M. Soner,
Controlled Markov Processes and Viscosity Solutions, Springer Science & Business Media, New York, 2006. |
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D. Giamouridis and A. Sakkas, Dynamic asset allocation with liabilities, European Financial Management, 23 (2017), 254-291. |
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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. |
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J. Hull and A. White, Pricing interest-rate-derivative securities, Review of Financial Studies, 3 (1990), 573-592. |
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H. K. Koo, Consumption and portfolio selection with labor income: A continuous time approach, Mathematical Finance, 81 (1998), 49-56. |
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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. |
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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. |
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S. Lv, Z. Wu and Z. Yu,
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doi: 10.1016/j.automatica.2016.02.017. |
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H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [30] |
A. Zhang, Stochastic Optimization in Finance and Life Insurance: Applications of the Martingale Method, Ph.D thesis, University of Kaiserslautern in Kaiserslautern, 2008. |
| [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. |
| [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. |
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. |
| [2] |
M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, Journal of Finance, 57 (2002), 1201-1238. |
| [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. |
| [4] |
H. Chang, Dynamic mean-variance portfolio selection with liability and stochastic interest rate, Economic Modelling, 51 (2015), 172-182. |
| [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. |
| [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. |
| [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] |
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. |
| [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. |
| [10] |
W. H. Fleming and H. M. Soner,
Controlled Markov Processes and Viscosity Solutions, Springer Science & Business Media, New York, 2006. |
| [11] |
D. Giamouridis and A. Sakkas, Dynamic asset allocation with liabilities, European Financial Management, 23 (2017), 254-291. |
| [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. |
| [13] |
J. Hull and A. White, Pricing interest-rate-derivative securities, Review of Financial Studies, 3 (1990), 573-592. |
| [14] |
H. K. Koo, Consumption and portfolio selection with labor income: A continuous time approach, Mathematical Finance, 81 (1998), 49-56. |
| [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. |
| [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. |
| [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. |
| [18] |
H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [30] |
A. Zhang, Stochastic Optimization in Finance and Life Insurance: Applications of the Martingale Method, Ph.D thesis, University of Kaiserslautern in Kaiserslautern, 2008. |
| [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. |
| [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. |
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
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