An optimal consumption and investment problem with quadratic utility and subsistence consumption constraints: a dynamic programming approach
Abstract
In this paper, we analyze the optimal consumption and investment problem of an agent who has a quadratic-type utility function and faces a subsistence consumption constraint. We use the dynamic programming method to solve the optimization problem in continuous-time. We further provide the sufficient conditions for the optimization problem to be well-defined.
Keyword : portfolio selection, quadratic utility, subsistence consumption constraints, dynamic programming method
How to Cite
Shin, Y. H., Koo, J. L., & Roh, K. H. (2018). An optimal consumption and investment problem with quadratic utility and subsistence consumption constraints: a dynamic programming approach. Mathematical Modelling and Analysis, 23(4), 627-638. https://doi.org/10.3846/mma.2018.038
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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N. Gong and T. Li. Role of index bonds in an optimal dynamic asset allocation model with real subsistence consumption. Applied Mathematics and Computation, 174(1):710–731, 2006.https://doi.org/10.1016/j.amc.2005.04.089
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J.L. Koo, S.R. Ahn, B.L. Koo, H.K. Koo and Y.H. Shin. Optimal consumption and portfolio selection with quadratic utility and a subsistence consumption constraint. Stochastic Analysis and Applications, 34(1):165–177, 2016.https://doi.org/10.1080/07362994.2015.1112748
P. Lakner and L.M. Nygren. Portfolio optimization with downside constraints. Mathematical Finance, 16(2):283–299, 2006. https://doi.org/10.1111/j.1467-9965.2006.00272.x
H. S. Lee and Y.H. Shin. A dynamic programming approach to subsistence consumption constraints on optimal consumption and portfolio.Journal of Computational Analysis & Applications, 22(1):79–99, 2017.
B.H. Lim, Y.H. Shin and U.J. Choi. Optimal investment, consumption and retirement choice problem with disutility and subsistence consumption constraints. Journal of Mathematical Analysis and Applications, 345(1):109–122, 2008.https://doi.org/10.1016/j.jmaa.2008.04.011
H. M. Markowitz. Foundations of portfolio theory. Journal of Finance,46(2):469–477, 1991.https://doi.org/10.1111/j.1540-6261.1991.tb02669.x
R.C. Merton. Lifetime portfolio selection under uncertainty: The continuous-time case. Review of Economics and Statistics, 51(3):247–257, 1969.https://doi.org/10.2307/1926560
R.C. Merton. Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3(4):373–413, 1971.https://doi.org/10.1016/0022-0531(71)90038-X
G. Shim and Y.H. Shin. Portfolio selection with subsistence consumption constraints and CARA utility. Mathematical Problems in Engineering, 2014.https://doi.org/10.1155/2014/153793
Y.H. Shin and B.H. Lim. Comparison of optimal portfolios with and without subsistence consumption constraints. Nonlinear Analysis: Theory, Methods &Applications, 74(1):50–58, 2011. https://doi.org/10.1016/j.na.2010.08.014
Y.H. Shin, B.H. Lim and U.J. Choi. Optimal consumption and portfolio selection problem with downside consumption constraints. Applied Mathematics and Computation, 188(2):1801–1811, 2007.https://doi.org/10.1016/j.amc.2006.11.053
H. Yuan and Y. Hu. Optimal consumption and portfolio policies with the consumption habit constraints and the terminal wealth downside constraints. Insurance: Mathematics and Economics, 45(3):405–409, 2009.https://doi.org/10.1016/j.insmatheco.2009.08.012
J.C. Cox and C.-F. Huang. Optimal consumption and portfolio policies when asset prices follow a diffusion process. Journal of Economic Theory, 49(1):33–83, 1989. https://doi.org/10.1016/0022-0531(89)90067-7
P.H. Dybvig. Dusenberry’s ratcheting of consumption: Optimal dynamic consumption and investment given intolerance for any decline in standard of living.The Review of Economic Studies, 62(2):287–313, 1995.https://doi.org/10.2307/2297806
N. Gong and T. Li. Role of index bonds in an optimal dynamic asset allocation model with real subsistence consumption. Applied Mathematics and Computation, 174(1):710–731, 2006.https://doi.org/10.1016/j.amc.2005.04.089
G. Hanoch and H. Levy. Efficient portfolio selection with quadratic and cubicutility. The Journal of Business, 43:181–189, 1970.
C.F. Huang and H. Pages. Optimal consumption and portfolio policies with an infinite horizon: Existence and convergence. The Annals of Applied Probability, 2(1):36–64, 1992. https://doi.org/10.1214/aoap/1177005770
I. Karatzas, J.P. Lehoczky, S.P. Sethi and S.E. Shreve. Explicit solution of a general consumption/investment problem. Mathematics of Operations Research, 11(2):261–294, 1986.https://doi.org/10.1287/moor.11.2.261
J.L. Koo, S.R. Ahn, B.L. Koo, H.K. Koo and Y.H. Shin. Optimal consumption and portfolio selection with quadratic utility and a subsistence consumption constraint. Stochastic Analysis and Applications, 34(1):165–177, 2016.https://doi.org/10.1080/07362994.2015.1112748
P. Lakner and L.M. Nygren. Portfolio optimization with downside constraints. Mathematical Finance, 16(2):283–299, 2006. https://doi.org/10.1111/j.1467-9965.2006.00272.x
H. S. Lee and Y.H. Shin. A dynamic programming approach to subsistence consumption constraints on optimal consumption and portfolio.Journal of Computational Analysis & Applications, 22(1):79–99, 2017.
B.H. Lim, Y.H. Shin and U.J. Choi. Optimal investment, consumption and retirement choice problem with disutility and subsistence consumption constraints. Journal of Mathematical Analysis and Applications, 345(1):109–122, 2008.https://doi.org/10.1016/j.jmaa.2008.04.011
H. M. Markowitz. Foundations of portfolio theory. Journal of Finance,46(2):469–477, 1991.https://doi.org/10.1111/j.1540-6261.1991.tb02669.x
R.C. Merton. Lifetime portfolio selection under uncertainty: The continuous-time case. Review of Economics and Statistics, 51(3):247–257, 1969.https://doi.org/10.2307/1926560
R.C. Merton. Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3(4):373–413, 1971.https://doi.org/10.1016/0022-0531(71)90038-X
G. Shim and Y.H. Shin. Portfolio selection with subsistence consumption constraints and CARA utility. Mathematical Problems in Engineering, 2014.https://doi.org/10.1155/2014/153793
Y.H. Shin and B.H. Lim. Comparison of optimal portfolios with and without subsistence consumption constraints. Nonlinear Analysis: Theory, Methods &Applications, 74(1):50–58, 2011. https://doi.org/10.1016/j.na.2010.08.014
Y.H. Shin, B.H. Lim and U.J. Choi. Optimal consumption and portfolio selection problem with downside consumption constraints. Applied Mathematics and Computation, 188(2):1801–1811, 2007.https://doi.org/10.1016/j.amc.2006.11.053
H. Yuan and Y. Hu. Optimal consumption and portfolio policies with the consumption habit constraints and the terminal wealth downside constraints. Insurance: Mathematics and Economics, 45(3):405–409, 2009.https://doi.org/10.1016/j.insmatheco.2009.08.012