Optimum consumption and portfolio rules in a continuous-time model☆
TL;DR: In this paper, the authors considered the continuous-time consumption-portfolio problem for an individual whose income is generated by capital gains on investments in assets with prices assumed to satisfy the geometric Brownian motion hypothesis, which implies that asset prices are stationary and lognormally distributed.
About: This article is published in Journal of Economic Theory.The article was published on 1971-12-01 and is currently open access. It has received 4952 citations till now. The article focuses on the topics: Geometric Brownian motion & Intertemporal portfolio choice.
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TL;DR: In this paper, a characterization of the value function as the maximal subsolution of a backward stochastic differential equation (BSDE) and an optimality criterium is provided.
Abstract: In this paper, we study the exponential utility maximization problem in an incomplete market with a default time inducing a discontinuity in the price of stock. We consider the case of strategies valued in a closed set. Using dynamic programming and BSDEs techniques, we provide a characterization of the value function as the maximal subsolution of a backward stochastic differential equation (BSDE) and an optimality criterium. Moreover, in the case of bounded coefficients, the value function is shown to be the maximal solution of a BSDE. Moreover, the value function can be written as the limit of a sequence of processes which can be characterized as the solutions of Lipschitz BSDEs in the case of bounded coefficients. In the case of convex constraints and under some exponential integrability assumptions on the coefficients, some complementary properties are provided. These results can be generalized to the case of several default times or a Poisson process.
53 citations
TL;DR: In this article, an optimal portfolio and consumption choice problem of a family that combines life insurance for parents who receive deterministic labor income until the fixed time T was studied, where the goal was to maximize the weighted average of utility of parents and that of children.
Abstract: We study an optimal portfolio and consumption choice problem of a family that combines life insurance for parents who receive deterministic labor income until the fixed time T . We consider utility functions of parents and children separately and assume that parents have an uncertain lifetime. If parents die before time T , children have no labor income and they choose the optimal consumption and portfolio with remaining wealth and life insurance benefit. The object of the family is to maximize the weighted average of utility of parents and that of children. We obtain analytic solutions for the value function and the optimal policies, and then analyze how the changes of the weight of the parents’ utility function and other factors affect the optimal policies.
53 citations
TL;DR: In this paper, the conditional variance of changes in income increases with its level, and the authors show that a larger realization of income not only implies a higher level of human wealth, but also signals a riskier stream of future labor income, inducing a higher precautionary saving, and thus giving rise to Friedman's conjecture.
53 citations
Cites background or methods from "Optimum consumption and portfolio r..."
...7 Merton (1971) derived an explicitly solved consumption rule using Poisson processes to model the income...
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...For tractability reasons, I assume that the agent is endowed with CARA utility, following Merton (1971) , Kimball and Mankiw (1989), and Caballero (1991)....
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...Following Merton (1971), Kimball and Mankiw (1989), and Caballero (1991), I assume CARA utility for technical convenience.7 This is not surprising....
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...For tractability reasons, I assume that the agent is endowed with CARA utility, following Merton (1971), Kimball and Mankiw (1989), and Caballero (1991)....
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TL;DR: In this paper, the authors give an overview of stochastic optimal control theory and its applications to operational research, and their actual and potential impact on OR is discussed from a methodological point of view.
53 citations
TL;DR: In this paper, a dynamic asset allocation problem in the presence of liabilities is considered and the optimal asset allocation rule is derived and its sensitivity with respect to the parameters of the model is analyzed.
Abstract: A dynamic asset allocation problem in the presence of liabilities is considered. The fund manager has von Neumann–Morgenstern preferences with terminal utility function defined over the excess of liquid wealth over a minimum liability coverage tolerated and intermediate utility function defined over dividends, the excess of expenditures over liability cash flows. Preferences incorporate a parameter controlling the tolerance for a shortfall in the funding ratio at the terminal date. The optimal asset allocation rule is derived and its sensitivity with respect to the parameters of the model is analyzed.
53 citations
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TL;DR: In this paper, the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous-time model was examined, where his income is generated by returns on assets and these returns or instantaneous "growth rates" are stochastic.
Abstract: OST models of portfolio selection have M been one-period models. I examine the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous-time model whzere his income is generated by returns on assets and these returns or instantaneous "growth rates" are stochastic. P. A. Samuelson has developed a similar model in discrete-time for more general probability distributions in a companion paper [8]. I derive the optimality equations for a multiasset problem when the rate of returns are generated by a Wiener Brownian-motion process. A particular case examined in detail is the two-asset model with constant relative riskaversion or iso-elastic marginal utility. An explicit solution is also found for the case of constant absolute risk-aversion. The general technique employed can be used to examine a wide class of intertemporal economic problems under uncertainty. In addition to the Samuelson paper [8], there is the multi-period analysis of Tobin [9]. Phelps [6] has a model used to determine the optimal consumption rule for a multi-period example where income is partly generated by an asset with an uncertain return. Mirrless [5] has developed a continuous-time optimal consumption model of the neoclassical type with technical progress a random variable.
4,908 citations
Book•
01 Jan 1965
TL;DR: This book should be of interest to undergraduate and postgraduate students of probability theory.
Abstract: This book should be of interest to undergraduate and postgraduate students of probability theory.
3,597 citations
TL;DR: In this paper, the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions), is discussed.
Abstract: Publisher Summary This chapter reviews the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions). It presents a generalization of Phelps' model to include portfolio choice and consumption. The explicit form of the optimal solution is derived for the special case of utility functions having constant relative risk aversion. The optimal portfolio decision is independent of time, wealth, and the consumption decision at each stage. Most analyses of portfolio selection, whether they are of the Markowitz–Tobin mean-variance or of more general type, maximize over one period. The chapter only discusses special and easy cases that suffice to illustrate the general principles involved and presents the lifetime model that reveals that investing for many periods does not itself introduce extra tolerance for riskiness at early or any stages of life.
2,369 citations
Book•
17 Jan 2012
TL;DR: In this article, a book on stochastic stability and control dealing with Liapunov function approach to study of Markov processes is presented, which is based on the work of this article.
Abstract: Book on stochastic stability and control dealing with Liapunov function approach to study of Markov processes
1,293 citations
987 citations