Dynamic Optimal Lifetime Portfolio Selection in Discrete Framework

Ashri Putri Rahadi, Nora Amelda Rizal, Budhi Arta Surya


By this study, we want to provide tools for an economic agent who wants to maintain her maximized intertemporal consumption rate over time period regularly by trading activity in capital market. She would use the diversification principle to compose portfolio that is consisted of risky and risk free assets, with initial wealth by using principle of dynamic programming in stochastic control problem. While our target function is to maximize the discounted utility function over consumption rate, the agent may use this portion of wealth for hedonistic lifestyle without thinking the bequest aim. Motivated by observation that seminal work of Samuelson (1969)’s optimal investment on risky asset was independent of time and wealth process, we were replacing the source of uncertainty in the risky asset by binomial process in order to improve Samuelson's solution. The risky asset price evolution is determined to move in two possible directions of up and down in the next time instantly with certain probability, following binomial trees scheme. Working within this setting, we manage to obtain trading strategies as explicit solution for the optimal investment and consumption decisions which were being expressed as time-dependent functions as well as wealth process feedback form. This solution goes along with inter-temporal concept that our consumption and investment strategies tomorrow will be affected with our wealth today. We exemplify the results by numerical examples and perform Monte Carlo simulation.


Dynamic programming, Hamilton-Jacobi-Bellman equation, intertemporal consumption, stochastic optimal control.

Full Text:



D.P. Bertsekas, and S.E. Shreve, Stochastic Optimal Control: The Discrete- Time Case, Academic Press, 1978.

D.P. Bertsekas, (1987). Dynamic Programming: Deterministic and Stochastic Models, Englewood Clis, N.J.: Prentice-Hall, 1987.

T. Bjork, Arbitrage Theory in Continuous Time, Oxford University Press Inc., New York,2009.

M. J. Brennan, M.J. and Y. Xia. Dynamic asset allocation under inflation, Journal of Finance, 57, 1201-1238, 2002.

J. Y. Campbell, and L. M. Viceira, (2001). Strategic Asset Allocation: Portfolio Choice for Long-Term Investors. New York: Oxford University Press.

J.M. Cox, S.A. Ross, and M. Rubinstein. Option pricing: A simplied approach, Journal of Financial Economics, 7, 229-263, 1979.

J. C. Cox, and C. F. Huang. Optimal consumption and portfolio policies when asset prices follow a diusion process, Journal of Economic Theory, 49, 33-88, 1989.

I. Friend, and and M. Blum. The Demand for risky assets, The American Economic Review, 65, 900-922, 1975.

R. R. Grauer, and N. H. Hakansson. Higher return, lower risk: historical returns on long run actively managed portfolios of stocks bonds and bills:

-1978, Financial Analysts Journal, 38, 39-53, 1982.

R. R. Grauer, and N. H. Hakansson. Returns on levered, actively managed long run portfolios of stocks, bonds and bills: 1934-1983, Financial Analysts Journal, 41, 24-43, 1985.

H. He, and N. Pearson. Consumption and portfolio policies with incomplete markets and short-sale constrains: the innite dimensional case, Journal of Economic Theory, 54, 259-304, 1991.

J. C. Hull. Options, Futures, and Other Derivatives , 4th Edition, Prentice Hall, 2000.

I. Karatzas, J. P. Lehoczky, J.P. and S. E. Shreve. Optimal portfolio and consumption decisions for a "small investor" on a nite horizon, SIAM Journal of Control and Optimization, 25, 1557-1586, 1987.

I. Karatzas, J. P. Lehoczky, S. E. Shreve and G. L. Xu. Martingale and duality methods for utility maximization in an incomplete market, SIAM Journal of Control and Optimization, 29, 702-730, 1991.

I. Karatzas, and S. E. Shreve. Methods of Mathematical Finance, Springer Verlag, New York, 1998.

D. G. Luenberger, D.G. (1998). Investment Science, Oxford University Press.

H. M. Markowitz. Portfolio Selection. Journal of Finance 7, 7791, 1952.

R. C. Merton. Lifetime portfolio selection under uncertainty. the continuous time case. Review of Economic Statistics., 51, 247257, 1969.

R. C. Merton. Optimum consumption and portfolio rules in continuous time model, Journal of Economic Theory, 3, 373-413, 1971.

Nelson, D.B. and Ramaswamy, K. (1990). Simple binomial processes as diusion approximations in nancial models, Review of Financial Studies, 3, 393- 430.

P.A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 51, 239 - 246, 1969.

Dynamic Optimal Lifetime Portfolio Selection in Discrete Framework 15

T. Zariphopoulou, A solution approach to valuation with unhedgeable risks, Finance and Stochastics, 5, 61 - 82, 2001.

DOI: http://dx.doi.org/10.21535%2Fjust.v3i1.187


  • There are currently no refbacks.

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.