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January  2016, 12(1): 187-209. doi: 10.3934/jimo.2016.12.187

Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate

 1 School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China 2 Lingnan (University) College/Sun Yat-sen Business School, Sun Yat-sen University, Guangzhou 510275 3 Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada

Received  May 2014 Revised  December 2014 Published  April 2015

This paper studies dynamic asset allocation with stochastic interest rates and inflation rates under the continuous-time mean-variance model in a more general market that may be incomplete. First, by the Lagrange method and the dynamic programming approach, we derive the associated Hamilton-Jacobi-Bellman equation and solve it explicitly. Then, closed form expressions for the efficient strategy and the efficient frontier are derived by applying the Lagrange dual theory. In addition, we state a necessary and sufficient condition under which the efficient frontier is a straight line in the standard deviation-mean plane, and some degenerate cases are discussed. Finally, empirical analysis based on real data from the Chinese market is presented to illustrate applications of the results obtained in this paper.
Citation: Haixiang Yao, Zhongfei Li, Yongzeng Lai. Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate. Journal of Industrial and Management Optimization, 2016, 12 (1) : 187-209. doi: 10.3934/jimo.2016.12.187
References:
 [1] I. Bajeux-Besnainou and R. Portait, Dynamic asset allocation in a mean-variance framework, Management Science, 44 (1998), S79-S95. [2] T. R. Bielecki, H. Q. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244. doi: 10.1111/j.0960-1627.2005.00218.x. [3] M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, Journal of Finance, 57 (2002), 1201-1238. doi: 10.1111/1540-6261.00459. [4] T. Chellathurai and T. Draviam, Dynamic portfolio selection with fixed and/or proportional transaction costs using non-singular stochastic optimal control theory, Journal of Economic Dynamics and Control, 31 (2007), 2168-2195. doi: 10.1016/j.jedc.2006.06.006. [5] 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. [6] Y. Y. Chou, N. W. Han and M. W. Hung, Optimal portfolio-consumption choice under stochastic inflation with nominal and indexed bonds, Applied Stochastic Models in Business and Industry, 27 (2011), 691-706. doi: 10.1002/asmb.886. [7] O. L. V. Costa and A. D. Oliveira, Optimal mean-variance control for discrete-time linear systems with Markovian jumps and multiplicative noises, Automatica, 48 (2012), 304-315. doi: 10.1016/j.automatica.2011.11.009. [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. [9] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2ed. Springer, New York, 2006. [10] C. P. Fu, A. Lari-lavassani and X. Li, Dynamic mean-variance portfolio selection with borrowing constraint, European Journal of Operational Research, 200 (2010), 313-319. doi: 10.1016/j.ejor.2009.01.005. [11] J. W. Gao, Stochastic optimal control of DC pension funds, Insurance: Mathematics and Economics, 42 (2008), 1159-1164. doi: 10.1016/j.insmatheco.2008.03.004. [12] 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. [13] N. W. Han and M. W. Hung, Optimal asset allocation for DC pension plans under inflation, Insurance: Mathematics and Economics, 51 (2012), 172-181. doi: 10.1016/j.insmatheco.2012.03.003. [14] R. Josa-Fombellida and J. P. Rincón-Zapatero, Optimal asset allocation for aggregated defined benefit pension funds with stochastic interest rates, European Journal of Operational Research, 201 (2010), 211-221. [15] 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. [16] P. Lakner and L. M. Nygren, Portfolio optimization with downside constraints, Mathematical Finance, 16 (2006), 283-299. doi: 10.1111/j.1467-9965.2006.00272.x. [17] 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. [18] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. [19] X. Li and X. Y. Zhou, Continuous-time mean-variance efficiency: The 80 The Annals of Applied Probability, 16 (2006), 1751-1763. doi: 10.1214/105051606000000349. [20] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555. doi: 10.1137/S0363012900378504. [21] A. E. B. Lim and X. Y. Zhou, Mean-variance portfolio selection with random parameters in a complete market, Mathematics of Operations Research, 27 (2002), 101-120. [22] D. G. Luenberger, Optimization by Vector Space Methods, Wiley, New York, 1969. [23] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [24] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time model, Review of Economic and Statistics, 51 (1969), 247-256. [25] C. Munk and C. Sørensen, Optimal consumption and investment strategies with stochastic interest rates, Journal of Banking & Finance, 28 (2004), 1987-2013. doi: 10.1016/j.jbankfin.2003.07.002. [26] C. Munk and C. Sørensen, Dynamic asset allocation with stochastic income and interest rates, Journal of Financial Economics, 96 (2010), 433-462. doi: 10.1016/j.jfineco.2010.01.004. [27] C. Munk and C. Sørensen and T. N. Vinther, Dynamic asset allocation under mean-reverting returns, stochastic interest rates, and inflation uncertainty: Are popular recommendations consistent with rational behavior?, International Review of Economics and Finance, 13 (2004), 141-166. [28] P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 51 (1969), 239-246. doi: 10.2307/1926559. [29] Z. Wang and S. Y. Liu, Multi-period mean-variance portfolio selection with fixed and proportional transaction costs, Journal of Industrial and Management Optimization, 9 (2013), 643-657. doi: 10.3934/jimo.2013.9.643. [30] H. L. Wu, Mean-variance portfolio selection with a stochastic cash flow in a markov-switching jump-diffusion market, Journal of Optimization Theory and Applications, 158 (2013), 918-934. doi: 10.1007/s10957-013-0292-x. [31] H. X. Yao, Y. Z. Lai and Y. Li, Continuous-time mean-variance asset-liability management with endogenous liabilities, Insurance: Mathematics and Economics, 52 (2013), 6-17. doi: 10.1016/j.insmatheco.2012.10.001. [32] H. X. Yao, Y. Z. Lai and Z. F. Hao, Uncertain exit time multi-period mean-variance portfolio selection with endogenous liabilities and Markov jumps, Automatica, 49 (2013), 3258-3269. doi: 10.1016/j.automatica.2013.08.023. [33] L. Yi, Z. F. Li and D. Li, Mutli-period portfolio selection for asset-liability management with uncertain investment horizon, Journal of Industrial and Management Optimization, 4 (2008), 535-552. doi: 10.3934/jimo.2008.4.535. [34] H. L. Yuan and Y. J. Hu, Optimal consumption and portfolio policies with the consumption habit constraints and the terminal wealth downside constraints, Insurance: Mathematics and Economics, 45 (2009), 405-409. doi: 10.1016/j.insmatheco.2009.08.012. [35] A. Zhang and C. O. Ewald, Optimal investment for a pension fund under inflation risk, Mathematical Methods of Operations Research, 71 (2010), 353-369. doi: 10.1007/s00186-009-0294-5. [36] Y. Zeng, Z. F. Li and J. J. Liu, Optimal Strategies Of Benchmark And Mean-Variance Portfolio Selection Problems For Insurers, Journal of Industrial and Management Optimization, 6 (2010), 483-496. doi: 10.3934/jimo.2010.6.483. [37] 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.

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References:
 [1] I. Bajeux-Besnainou and R. Portait, Dynamic asset allocation in a mean-variance framework, Management Science, 44 (1998), S79-S95. [2] T. R. Bielecki, H. Q. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244. doi: 10.1111/j.0960-1627.2005.00218.x. [3] M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, Journal of Finance, 57 (2002), 1201-1238. doi: 10.1111/1540-6261.00459. [4] T. Chellathurai and T. Draviam, Dynamic portfolio selection with fixed and/or proportional transaction costs using non-singular stochastic optimal control theory, Journal of Economic Dynamics and Control, 31 (2007), 2168-2195. doi: 10.1016/j.jedc.2006.06.006. [5] 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. [6] Y. Y. Chou, N. W. Han and M. W. Hung, Optimal portfolio-consumption choice under stochastic inflation with nominal and indexed bonds, Applied Stochastic Models in Business and Industry, 27 (2011), 691-706. doi: 10.1002/asmb.886. [7] O. L. V. Costa and A. D. Oliveira, Optimal mean-variance control for discrete-time linear systems with Markovian jumps and multiplicative noises, Automatica, 48 (2012), 304-315. doi: 10.1016/j.automatica.2011.11.009. [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. [9] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2ed. Springer, New York, 2006. [10] C. P. Fu, A. Lari-lavassani and X. Li, Dynamic mean-variance portfolio selection with borrowing constraint, European Journal of Operational Research, 200 (2010), 313-319. doi: 10.1016/j.ejor.2009.01.005. [11] J. W. Gao, Stochastic optimal control of DC pension funds, Insurance: Mathematics and Economics, 42 (2008), 1159-1164. doi: 10.1016/j.insmatheco.2008.03.004. [12] 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. [13] N. W. Han and M. W. Hung, Optimal asset allocation for DC pension plans under inflation, Insurance: Mathematics and Economics, 51 (2012), 172-181. doi: 10.1016/j.insmatheco.2012.03.003. [14] R. Josa-Fombellida and J. P. Rincón-Zapatero, Optimal asset allocation for aggregated defined benefit pension funds with stochastic interest rates, European Journal of Operational Research, 201 (2010), 211-221. [15] 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. [16] P. Lakner and L. M. Nygren, Portfolio optimization with downside constraints, Mathematical Finance, 16 (2006), 283-299. doi: 10.1111/j.1467-9965.2006.00272.x. [17] 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. [18] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. [19] X. Li and X. Y. Zhou, Continuous-time mean-variance efficiency: The 80 The Annals of Applied Probability, 16 (2006), 1751-1763. doi: 10.1214/105051606000000349. [20] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555. doi: 10.1137/S0363012900378504. [21] A. E. B. Lim and X. Y. Zhou, Mean-variance portfolio selection with random parameters in a complete market, Mathematics of Operations Research, 27 (2002), 101-120. [22] D. G. Luenberger, Optimization by Vector Space Methods, Wiley, New York, 1969. [23] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [24] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time model, Review of Economic and Statistics, 51 (1969), 247-256. [25] C. Munk and C. Sørensen, Optimal consumption and investment strategies with stochastic interest rates, Journal of Banking & Finance, 28 (2004), 1987-2013. doi: 10.1016/j.jbankfin.2003.07.002. [26] C. Munk and C. Sørensen, Dynamic asset allocation with stochastic income and interest rates, Journal of Financial Economics, 96 (2010), 433-462. doi: 10.1016/j.jfineco.2010.01.004. [27] C. Munk and C. Sørensen and T. N. Vinther, Dynamic asset allocation under mean-reverting returns, stochastic interest rates, and inflation uncertainty: Are popular recommendations consistent with rational behavior?, International Review of Economics and Finance, 13 (2004), 141-166. [28] P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 51 (1969), 239-246. doi: 10.2307/1926559. [29] Z. Wang and S. Y. Liu, Multi-period mean-variance portfolio selection with fixed and proportional transaction costs, Journal of Industrial and Management Optimization, 9 (2013), 643-657. doi: 10.3934/jimo.2013.9.643. [30] H. L. Wu, Mean-variance portfolio selection with a stochastic cash flow in a markov-switching jump-diffusion market, Journal of Optimization Theory and Applications, 158 (2013), 918-934. doi: 10.1007/s10957-013-0292-x. [31] H. X. Yao, Y. Z. Lai and Y. Li, Continuous-time mean-variance asset-liability management with endogenous liabilities, Insurance: Mathematics and Economics, 52 (2013), 6-17. doi: 10.1016/j.insmatheco.2012.10.001. [32] H. X. Yao, Y. Z. Lai and Z. F. Hao, Uncertain exit time multi-period mean-variance portfolio selection with endogenous liabilities and Markov jumps, Automatica, 49 (2013), 3258-3269. doi: 10.1016/j.automatica.2013.08.023. [33] L. Yi, Z. F. Li and D. Li, Mutli-period portfolio selection for asset-liability management with uncertain investment horizon, Journal of Industrial and Management Optimization, 4 (2008), 535-552. doi: 10.3934/jimo.2008.4.535. [34] H. L. Yuan and Y. J. Hu, Optimal consumption and portfolio policies with the consumption habit constraints and the terminal wealth downside constraints, Insurance: Mathematics and Economics, 45 (2009), 405-409. doi: 10.1016/j.insmatheco.2009.08.012. [35] A. Zhang and C. O. Ewald, Optimal investment for a pension fund under inflation risk, Mathematical Methods of Operations Research, 71 (2010), 353-369. doi: 10.1007/s00186-009-0294-5. [36] Y. Zeng, Z. F. Li and J. J. Liu, Optimal Strategies Of Benchmark And Mean-Variance Portfolio Selection Problems For Insurers, Journal of Industrial and Management Optimization, 6 (2010), 483-496. doi: 10.3934/jimo.2010.6.483. [37] 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.
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