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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01 May 1994
TL;DR: In this paper, the problem of portfolio optimization under the drawdown constraint was studied, where the wealth process never falls below a fraction of its maximum-to-date, and one strives to maximize the long-term growth rate of its expected utility.
Abstract: We study the problem of portfolio optimization under the \drawdown constraint" that the wealth process never falls below a xed fraction of its maximum-to-date, and one strives to maximize the long-term growth rate of its expected utility. This problem was introduced and solved explicitly by Grossman and Zhou; we present an approach which simpli es and extends their results.
169 citations
TL;DR: In this paper, the authors consider the portfolio problem in continuous-time where the objective of the investor or money manager is to exceed the performance of a given stochastic benchmark, as is often the case in institutional money management.
Abstract: We consider the portfolio problem in continuous-time where the objective of the investor or money manager is to exceed the performance of a given stochastic benchmark, as is often the case in institutional money management. The benchmark is driven by a stochastic process that need not be perfectly correlated with the investment opportunities, and so the market is in a sense incomplete. We first solve a variety of investment problems related to the achievement of goals: for example, we find the portfolio strategy that maximizes the probability that the return of the investor's portfolio beats the return of the benchmark by a given percentage without ever going below it by another predetermined percentage. We also consider objectives related to the minimization of the expected time until the investor beats the benchmark. We show that there are two cases to consider, depending upon the relative favorability of the benchmark to the investment opportunity the investor faces. The problem of maximizing the expected discounted reward of outperforming the benchmark, as well as minimizing the discounted penalty paid upon being outperformed by the benchmark is also discussed. We then solve a more standard expected utility maximization problem which allows new connections to be made between some specific utility functions and the nonstandard goal problems treated here.
169 citations
168 citations
TL;DR: In this article, a more intuitive derivation of the factor pricing equation is presented which describes the deviation on an asset by asset basis, and the deviation is shown to be small for assets in a realistic finite economy and is arbitrarily close to zero for those assets with arbitrarily small size relative to aggregate wealth.
168 citations
Posted Content•
TL;DR: The portfolio problem in continuous-time where the objective of the investor or money manager is to exceed the performance of a given stochastic benchmark, as is often the case in institutional money management, is considered.
Abstract: We consider the portfolio problem in continuous-time where the objective of the investor or money manager is to exceed the performance of a given stochastic benchmark, as is often the case in institutional money management. The benchmark is driven by a stochastic process that need not be perfectly correlated with the investment opportunities, and so the market is in a sense incomplete. We first solve a variety of investment problems related to the achievement of goals: for example, we find the portfolio strategy that maximizes the probability that the return of the investor's portfolio beats the return of the benchmark by a given percentage without ever going below it by another predetermined percentage. We also consider objectives related to the minimization of the expected time until the investor beats the benchmark. We show that there are two cases to consider, depending upon the relative favorability of the benchmark to the investment opportunity the investor faces. The problem of maximizing the expected discounted reward of outperforming the benchmark, as well as minimizing the discounted penalty paid upon being outperformed by the benchmark is also discussed. We then solve a more standard expected utility maximization problem which allows new connections to be made between some specific utility functions and the nonstandard goal problems treated here.
167 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