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

*(English)*Zbl 1317.90248Summary: 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.

##### MSC:

90C26 | Nonconvex programming, global optimization |

91G10 | Portfolio theory |

91G80 | Financial applications of other theories |

49N15 | Duality theory (optimization) |

##### Keywords:

asset allocation; inflation; stochastic interest rate; dynamic mean-variance; Hamilton-Jacobi-Bellman equation
PDF
BibTeX
XML
Cite

\textit{H. Yao} et al., J. Ind. Manag. Optim. 12, No. 1, 187--209 (2016; Zbl 1317.90248)

Full Text:
DOI

**OpenURL**

##### References:

[1] | I. Bajeux-Besnainou, Dynamic asset allocation in a mean-variance framework,, Management Science, 44, (1998) · Zbl 0999.91037 |

[2] | T. R. Bielecki, Continuous-time mean-variance portfolio selection with bankruptcy prohibition,, Mathematical Finance, 15, 213, (2005) · Zbl 1153.91466 |

[3] | M. J. Brennan, Dynamic asset allocation under inflation,, Journal of Finance, 57, 1201, (2002) |

[4] | T. Chellathurai, Dynamic portfolio selection with fixed and/or proportional transaction costs using non-singular stochastic optimal control theory,, Journal of Economic Dynamics and Control, 31, 2168, (2007) · Zbl 1163.91385 |

[5] | P. Chen, Markowitz’s mean-variance asset-liability management with regime switching: A continuous-time model,, Insurance: Mathematics and Economics, 43, 456, (2008) · Zbl 1152.91496 |

[6] | Y. Y. Chou, Optimal portfolio-consumption choice under stochastic inflation with nominal and indexed bonds,, Applied Stochastic Models in Business and Industry, 27, 691, (2011) · Zbl 1274.91376 |

[7] | O. L. V. Costa, Optimal mean-variance control for discrete-time linear systems with Markovian jumps and multiplicative noises,, Automatica, 48, 304, (2012) · Zbl 1260.93173 |

[8] | R. Ferland, Mean-variance efficiency with extended CIR interest rates,, Applied Stochastic Models in Business and Industry, 26, 71, (2010) · Zbl 1224.91136 |

[9] | W. H. Fleming, <em>Controlled Markov Processes and Viscosity Solutions</em>,, 2ed. Springer, (2006) · Zbl 1105.60005 |

[10] | C. P. Fu, Dynamic mean-variance portfolio selection with borrowing constraint,, European Journal of Operational Research, 200, 313, (2010) · Zbl 1183.91192 |

[11] | J. W. Gao, Stochastic optimal control of DC pension funds,, Insurance: Mathematics and Economics, 42, 1159, (2008) · Zbl 1141.91439 |

[12] | D. Hainaut, Dynamic asset allocation under VaR constraint with stochastic interest rates,, Annals Of Operations Research, 172, 97, (2009) · Zbl 1183.91165 |

[13] | N. W. Han, Optimal asset allocation for DC pension plans under inflation,, Insurance: Mathematics and Economics, 51, 172, (2012) · Zbl 1284.91520 |

[14] | R. Josa-Fombellida, Optimal asset allocation for aggregated defined benefit pension funds with stochastic interest rates,, European Journal of Operational Research, 201, 211, (2010) · Zbl 1177.91125 |

[15] | R. Korn, A Stochastic control approach to portfolio problems with stochastic interest rates,, SIAM Journal on Control and Optimization, 40, 1250, (2002) · Zbl 1020.93029 |

[16] | P. Lakner, Portfolio optimization with downside constraints,, Mathematical Finance, 16, 283, (2006) · Zbl 1145.91350 |

[17] | M. Leippold, Multiperiod mean-variance efficient portfolios with endogenous liabilities,, Quantitative Finance, 11, 1535, (2011) · Zbl 1258.91199 |

[18] | D. Li, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation,, Mathematical Finance, 10, 387, (2000) · Zbl 0997.91027 |

[19] | X. Li, Continuous-time mean-variance efficiency: The 80, The Annals of Applied Probability, 16, 1751, (2006) · Zbl 1132.91472 |

[20] | X. Li, Dynamic mean-variance portfolio selection with no-shorting constraints,, SIAM Journal on Control and Optimization, 40, 1540, (2002) · Zbl 1027.91040 |

[21] | A. E. B. Lim, Mean-variance portfolio selection with random parameters in a complete market,, Mathematics of Operations Research, 27, 101, (2002) · Zbl 1082.91521 |

[22] | D. G. Luenberger, <em>Optimization by Vector Space Methods</em>,, Wiley, (1969) · Zbl 0176.12701 |

[23] | H. Markowitz, Portfolio selection,, Journal of Finance, 7, 77, (1952) |

[24] | R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time model,, Review of Economic and Statistics, 51, 247, (1969) |

[25] | C. Munk, Optimal consumption and investment strategies with stochastic interest rates,, Journal of Banking & Finance, 28, 1987, (2004) |

[26] | C. Munk, Dynamic asset allocation with stochastic income and interest rates,, Journal of Financial Economics, 96, 433, (2010) |

[27] | C. Munk, 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, 141, (2004) |

[28] | P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming,, Review of Economics and Statistics, 51, 239, (1969) |

[29] | Z. Wang, Multi-period mean-variance portfolio selection with fixed and proportional transaction costs,, Journal of Industrial and Management Optimization, 9, 643, (2013) · Zbl 1281.90080 |

[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, 918, (2013) · Zbl 1280.91160 |

[31] | H. X. Yao, Continuous-time mean-variance asset-liability management with endogenous liabilities,, Insurance: Mathematics and Economics, 52, 6, (2013) · Zbl 1291.91199 |

[32] | H. X. Yao, Uncertain exit time multi-period mean-variance portfolio selection with endogenous liabilities and Markov jumps,, Automatica, 49, 3258, (2013) · Zbl 1358.93166 |

[33] | L. Yi, Mutli-period portfolio selection for asset-liability management with uncertain investment horizon,, Journal of Industrial and Management Optimization, 4, 535, (2008) · Zbl 1160.90544 |

[34] | H. L. Yuan, Optimal consumption and portfolio policies with the consumption habit constraints and the terminal wealth downside constraints,, Insurance: Mathematics and Economics, 45, 405, (2009) · Zbl 1231.91418 |

[35] | A. Zhang, Optimal investment for a pension fund under inflation risk,, Mathematical Methods of Operations Research, 71, 353, (2010) · Zbl 1189.93147 |

[36] | Y. Zeng, Optimal Strategies Of Benchmark And Mean-Variance Portfolio Selection Problems For Insurers,, Journal of Industrial and Management Optimization, 6, 483, (2010) · Zbl 1269.90085 |

[37] | X. Y. Zhou, Continuous-time mean-variance portfolio selection: A stochastic LQ framework,, Applied Mathematics and Optimization, 42, 19, (2000) · Zbl 0998.91023 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.