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Journal ArticleDOI

Optimum consumption and portfolio rules in a continuous-time model☆

01 Dec 1971-Journal of Economic Theory (Academic Press)-Vol. 3, Iss: 4, pp 373-413
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
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Journal ArticleDOI
TL;DR: In this article, the authors show that Von Neumann-Morgenstern preferences over terminal consumption can be inferred from wealth on a single sample path when markets are complete and returns follow a known law in a neoclassical investment problem in either a discrete-time i.i.d. binomial model or a continuous-time diffusion model with a Gaussian state variable.
Abstract: Von Neumann-Morgenstern preferences over terminal consumption can be inferred from wealth on a single sample path when markets are complete and returns follow a known law in a neoclassical investment problem in either a discrete-time i.i.d. binomial model or a continuous-time diffusion model with a Gaussian state variable. Numerical results suggest that useful information about preferences can be obtained from even a single noisy sample of monthly observations of a portfolio over 5 years. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

42 citations

Journal ArticleDOI
TL;DR: In this article, the portfolio problem of an investor maximizing the expected exponential utility of his terminal real wealth is analyzed and an exact solution is proposed. But the solution is not suitable for the case of a stochastic inflation risk.
Abstract: This paper analyses the portfolio problem of an investor maximizing the expected exponential utility of his terminal real wealth. The investor must cope with both a set of stochastic investment opportunities and a set of background risks. If the market is complete we are able to find an exact solution. If the market is incomplete, we suggest an approximated general solution. Contrary to other exact solutions obtained in the literature, all our results are obtained considering a stochastic inflation risk and without specifying any particular functional form for the stochastic variables involved in the problem. (C) 2002 Elsevier Science B.V. All rights reserved.

42 citations


Cites methods from "Optimum consumption and portfolio r..."

  • ...In this paper, we follow the traditional route to use the stochastic dynamic programming technique(Merton, 1969, 1971)leading to the Hamilton–Jacobi–Bellman (HJB) equation.1 For the method called “martingale approach” the reader is referred toCox and Huang (1989, 1991)andLioui and Poncet (2001)....

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Journal ArticleDOI
TL;DR: In this article, the entropy of the factor exposure distribution is used as the number of uncorrelated bets (also known as the effective number of bets, or ENB in short), implicitly embedded within a given asset allocation decision.
Abstract: While it is often argued that allocation decisions can be best expressed in terms of exposure to rewarded risk factors, as opposed to somewhat arbitrary asset class decompositions, the practical implications of this paradigm shift for the optimal design of the policy portfolio still remain largely unexplored. This paper aims at analyzing whether the use of uncorrelated underlying risk factors, as opposed to correlated asset returns, can lead to a more efficient framework for measuring and managing portfolio diversification. Following Meucci (2009), we use the entropy of the factor exposure distribution as the number of uncorrelated bets (also known as the effective number of bets, or ENB in short), implicitly embedded within a given asset allocation decision. We present a set of formal results regarding the existence and unicity of portfolios designed to achieve the maximum effective number of bets. We also provide empirical evidence that incorporating constraints, or target levels, on a portfolio effective number of bets generates an improvement in out-of-sample risk-adjusted performance with respect to standard mean-variance analysis.

42 citations


Additional excerpts

  • ...(3.5) If 1′NΣ −1µ > 0, this problem has a well-known solution10 (Merton, 1971): wMSR = Σ−1µ 1′NΣ −1µ ....

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Journal ArticleDOI
TL;DR: In this paper, the authors studied a stochastic optimization problem under constraints in a general framework including financial models with constrained portfolios, labor income and large investor models and reinsurance models and also imposed American-type constraint on the state space process.

42 citations

Journal ArticleDOI
TL;DR: The main result of the paper shows that the value function of the Markov decision problem (MDP) corresponding to the approximating controlled Markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits.
Abstract: We consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs. Numerical approximations for the value function using the Markov chain approximation method of Kushner and Dupuis are studied. The main result of the paper shows that the value function of the Markov decision problem (MDP) corresponding to the approximating controlled Markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits. All our convergence arguments are probabilistic; the main assumption that we make is that the value function be finite and continuous. In particular, uniqueness of the solutions of the associated HJB equations is neither needed nor available (in the generality under which the problem is considered). Specific features of the problem that make the convergence analysis nontrivial include unboundedness of the state and control space and the cost function; degeneracies in the dynamics; mixed boundary (Dirichlet-Neumann) conditions; and presence of both singular and absolutely continuous controls in the dynamics. Finally, schemes for computing the value function and optimal control policies for the MDP are presented and illustrated with a numerical study.

42 citations

References
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Journal ArticleDOI
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

Book ChapterDOI
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