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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 paper, the authors studied the impact of yield tax, lump-sum tax and tax on interest rate earnings on the privately optimal rotation period when forest stand value is stochastic and forest owners are risk averse or risk neutral.

69 citations

Journal ArticleDOI
TL;DR: This article reviewed and discussed the contribution of Modigliani's work to macroeconomic model building for economist forecasting and policy making, focusing on the relationship between the short and the long term.
Abstract: This paper reviews and discusses the contribution by Franco Modigliani to macroeconomic model building for economist forecasting and policy making. As Paul Samuelson observed, Modigliani's theoretical work was fundamental in the development of the basic framework within which the "post-Keynesian eclecticism" of the later twentieth century developed. In the move from theory to practice two aspects of Modigliani's work are considered: 1) the systemic approach that focuses on the relationship between the short and the long term; 2) the special reference to the mechanism of monetary policy transmission in the use of the macroeconomic model for economic stabilisation policy. What is left today of this contribution is finally briefly discussed, also with reference to some recent proposals on macroeconomic model building. JEL Codes: B21, B22, B31, D91, E21

69 citations

Journal ArticleDOI
TL;DR: In this article, the optimal demand for futures contracts by an investor with a logarithmic utility function who attempts to hedge a nontraded cash position is analyzed in the "wealth-commodity-price" space.
Abstract: In this paper, we focus on the optimal demand for futures contracts by an investor with a logarithmic utility function who attempts to hedge a nontraded cash position. When the analysis is conducted in the "cash-commodity-price" space, we show that the value function associated with the Bernoulli investor program is not additively separable, thus suggesting that this investor hedges against shifts in the opportunity set as represented by the commodity price. By establishing the equivalence between the cash formulation of the problem and the wealth formulation, we are able to analyze the problem in the "wealth-commodity-price" space. In this space, we show the additive separability of the value function when the futures settlement price process is perfectly locally correlated with the commodity price process. The demand for futures in this instance is composed of (a) a mean-variance term and (b) a minimum-variance component that is a classic feature of models with nontraded assets. Since the first-best (nonmyopic) optimum is attained, however, the deviation from a mean-variance demand should not be interpreted as the expression of a nonmyopic behavior but rather as an attempt to restore a first-best optimum. On the other hand, when the correlation between the futures price and the underlying commodity price is imperfect, in general, the value function does not separate additively, the first-best solution cannot be attained, and the optimal futures trading strategy involves a hedging term against shifts in the opportunity set. IN THE CLASSIC DYNAMIC portfolio problem, the investor selects a trading strategy in risky and riskless assets so as to maximize his or her expected lifetime utility. In this environment, where all assets can be freely traded, the logarithmic investor behaves myopically to the extent that he or she ignores shifts in the opportunity set when making current choices (Samuelson [271; Mossin [24]; Hakansson [16]; Merton [22]). In particular, in models with continuous trading, it turns out that the value function corresponding to the investor's program is additively separable in wealth and state variables. The implication is that his or her behavior will be myopic, i.e., that his or her demand functions for assets will be of the pure meanvariance efficiency type and in particular will not include any dynamic hedging components that are characteristic of financial markets with shifting opportunities. In this paper, we investigate the structure of the Bernoulli investor's demand functions in the situation where one asset cannot be freely traded in the market.

69 citations

Journal ArticleDOI
TL;DR: In this article, the optimal portfolio, consumption-leisure and retirement choice of an infinitely-lived economic agent whose instantaneous preference is characterized by a constant elasticity of substitution (CES) function of consumption and leisure is studied.
Abstract: We study optimal portfolio, consumption-leisure and retirement choice of an infinitely lived economic agent whose instantaneous preference is characterized by a constant elasticity of substitution (CES) function of consumption and leisure. We integrate in one model the optimal consumption-leisure-work choice, the optimal portfolio selection, and the optimal stopping problem in which the agent chooses her retirement time. The economic agent derives utility from both consumption and leisure, and is able to adjust her supply of labor flexibly above a certain minimum work-hour, and also has a retirement option. We solve the problem analytically by considering a variational inequality arising from the dual functions of the optimal stopping problem. The optimal retirement time is characterized as the first time when her wealth exceeds a certain critical level. We provide the critical wealth level for retirement and characterize the optimal consumption-leisure and portfolio policies before and after retirement in closed forms. We also derive properties of the optimal policies. In particular, we show that consumption in general jumps around retirement.

69 citations

Journal ArticleDOI
TL;DR: In this paper, the authors used martingale representation and projection to obtain an intertemporal CAPM; the history of a scalar Brownian motion plays the role of a sufficient statistic.
Abstract: The intertemporal models developed in this paper have been stimulated by the capital asset pricing model (CAPM) of Sharpe and Lintner, and the role of factor structure in Ross' arbitrage pricing theory (APT). Suppose that some (one-dimensional) stochastic process S provides a sufficient statistic for aggregate consumption. Then a (heuristic) dynamic programming argument shows that S, market wealth, and the wealth derivative of the value function (for any agent) are all locally perfectly correlated. It follows from Merton's work that there is a linear relationship between the local mean return on a security and the local covariance of that return with the return on the market portfolio. The formal development uses martingale representation and martingale projection to obtain an intertemporal CAPM; the history of a scalar Brownian motion plays the role of a sufficient statistic. We motivate the assumption of a sufficient statistic for aggregate consumption by considering a countable set of securities whose payoffs have an approximate factor structure, where the factor components are in the information set generated by an N-dimensional Brownian motion and the idiosyncratic components are weakly correlated. The approximate factor structure on the security payoffs implies that the rates of return have (locally) an approximate factor structure. The role of the market portfolio can now be played by a set of N well-diversified portfolios.

69 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