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.
Citations
More filters
TL;DR: In this paper, optimal portfolio management policies for an investor who must pay a transaction cost equal to a fixed Traction of his portfolio value each time he trades are studied. But the authors focus on the infinite horizon objective function of maximizing the asymptotic growth rate, so the optimal policies they derive approximate those of an investor with logarithmic utility at a distant horizon.
Abstract: We study optimal portfolio management policies for an investor who must pay a transaction cost equal to a fixed Traction of his portfolio value each time he trades. We focus on the infinite horizon objective function of maximizing the asymptotic growth rate, so me optimal policies we derive approximate those of an investor with logarithmic utility at a distant horizon. When investment opportunities are modeled as m correlated geometric Brownian motion stocks and a riskless bond, we show that the optimal policy reduces to solving a single stopping time problem. When there is a single risky stock, we give a system of equations whose solution determines the optima! rule. We use numerical methods to solve for the optima! policy when there are two risky stocks. We study several specific examples and observe the general qualitative result that, even with very low transaction cost levels, the optimal policy entails very infrequent trading.
258 citations
TL;DR: The authors survey the recent literature on learning in financial markets and show that many financial market phenomena that appear puzzling at first sight are easier to understand once we recognize that parameters in financial models are uncertain and subject to learning.
258 citations
TL;DR: In this article, the value function of the stochastic control problem is a smooth solution of the associated Hamilton-Jacobi-Bellman (HJB) equation and the optimal policy is shown to exist and given in a feedback form from the optimality conditions in the HJB equation.
256 citations
Cites background from "Optimum consumption and portfolio r..."
...A detailed analysis of u near the origin is given in Section 7 and the asymptotic behavior of u at infinity is analyzed in Section 8. Here, we characterize the behavior of the optimal policy as the ratio of wealth to income becomes large, showing it to converge to the optimal behavior in the original Merton (1971) problem with no stochastic income....
[...]
...For part (ii), the case of y = 0 is handled by the fact that, if Y,, = y = 0, then Y, = 0 for all t almost surely, reducing the problem to that of Merton (1971) ....
[...]
...Here, we characterize the behavior of the optimal policy as the ratio of wealth to income becomes large, showing it to converge to the optimal behavior in the original Merton (1971) problem with no stochastic income....
[...]
...For part (ii), the case of y = 0 is handled by the fact that, if Y,, = y = 0, then Y, = 0 for all t almost surely, reducing the problem to that of Merton (1971)....
[...]
Posted Content•
TL;DR: In this article, a continuous time equilibrium model is proposed to analyze the effects of a capacity constraint in the learning process of a representative investor, where the investor optimally allocates her information capacity across the various sources.
Abstract: Motivated by the fact that investors have limited time and attention in processing information, this paper provides a continuous time equilibrium model to analyze the effects of a capacity constraint in the learning process of a representative investor. Facing multiple sources of uncertainty, the investor optimally allocates her information capacity across the various sources. Consequently, the cross-sectional structure of information and the resulting asset price dynamics are determined endogenously. The model provides implications on consumption behavior and on the cross-sectional differences in price informativeness in terms of supply of information, speed of price adjustments to fundamental shocks, and price reactions to firm disclosures.
255 citations
Cites methods from "Optimum consumption and portfolio r..."
...Following Merton (1971), I solve the investor’s problem using the Bellman equation, 0 C X u C J + JW rW C + X T + 1 2 JWWX T TX The following first-order condition can be derived as u C JW X Jw JWW T 1 The value function is solved as J W e r W r 1 r r 1 2 T T 1 (A-7) X 1 r T 1 and(A-8) C rW 1 r r 1…...
[...]
TL;DR: In this paper, the authors investigate the performance of intertemporal optimization models that relax the restriction imposed by expected utility that risk aversion and inter-temporal substitution are negatively related.
Abstract: This paper investigates the empirical performance of intertemporal optimization models that relax the restriction imposed by expected utility that risk aversion and intertemporal substitution are negatively related. The authors estimate a system of rates of return and consumption growth equations, and interpret their results in the light of the expected utility, the ordinal certainty equivalence, and the Kreps and Porteus (1978) models. Their results are based on average cohort data for consumption, and thus should not reflect births or deaths. They suggest that non-expected-utility models afford major efficiency gains in the estimation of the elasticity of intertemporal substitution by providing simple cross-equation restrictions.
254 citations
References
More filters
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