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 article, the mean-variance model was used to determine whether the fluctuations of conditional first and second moments observed for many assets are consistent with the SharpeLintner-Mossin capital asset pricing model, and the results indicated that estimated conditional variances cannot explain the observed time variation of risk premia.
Abstract: This paper attempts to determine whether the fluctuations of conditional first and second moments-which are observed for many assets-are consistent with the SharpeLintner-Mossin capital asset pricing model. We test the mean-variance model under several different assumptions about the time variation of conditional second moments of returns, using weekly data from July 1974 to December 1986, that include returns on a portfolio composed of dollar, Deutsche mark, sterling, and Swiss franc assets, together with the U.S. stock market. The results indicate that estimated conditional variances cannot explain the observed time variation of risk premia. RATES OF RETURN ON international financial assets are characterized by statistical properties that are quite common to all financial markets: they are highly volatile and largely unpredictable. These properties make it very difficult to extract statistically reliable estimates of systematic exchange-rate and assetprice movements and are at the root of the generally poor empirical performance of international asset pricing models. Nevertheless, two important results have been uncovered by empirical researchers and can be considered a fair characterization of the data: (a) expected returns on foreign assets vary over time (Cumby and Obstfeld (1981) and the numerous articles that followed, recently surveyed by Frankel and Meese (1987)); (b) the volatility of returns of foreign assets also changes over time (Cumby and Obstfeld (1984), Hodrick and Srivastava (1984), and Hsieh (1988), among others). As Giovannini and Jorion (1987) suggest, the
207 citations
TL;DR: In this paper, the authors examine the implications of such ecological uncertainty for competitive equilibrium in a market with property rights and show that stochastic fluctuations add a risk premium to the rate of return required to keep a unit of stock in situ, and examine the effects of fluctuations on resource rent.
Abstract: The natural growth rate of most renewable resource stocks is in part stochastic. This paper examines the implications of such ecological uncertainty for competitive equilibrium in a market with property rights. We show that stochastic fluctuations add a risk premium to the rate of return required to keep a unit of stock in situ, and we examine the effects of fluctuations on resource rent. Examples are used to show that extraction can increase, decrease, or be left unchanged as the variance of the fluctuations increases, depending on the extent of market "self-correction". Regulatory implications are also discussed. Renewable resource economics has traditionally been concerned with the study of dynamically optimal harvesting policies given a deterministic function for the natural growth of the resource stock. Issues have included the existence and characteristics of steady-state equilibria for the optimally managed resource, the need for and design of regulatory policies to prevent over-exploitation, and conditions under which (as a social optimum or otherwise) the resource will be exploited to extinction.1 Much of this work has been based on the assumption of a fixed and exogenous price for the harvested resource (typically resulting in "bang-bang" solutions for the harvesting policy). However some recent papers make price endogenous, and thereby describe how the extraction rate, and the rate of return and asset value of the resource behave in a competitive market with property rights.2
207 citations
Cites methods from "Optimum consumption and portfolio r..."
...For an introduction to the techniques used in this paper, see Chow (1979) and Merton (1971, 1975)....
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TL;DR: In this article, the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and $n$ risky assets whose prices are log-normal diffusions is considered.
Abstract: This paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and $n$ risky assets whose prices are log-normal diffusions. We suppose that transactions between the assets incur a cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption. Dynamic programming leads to a variational inequality for the value function. Existence and uniqueness of a viscosity solution are proved. The variational inequality is solved by using a numerical algorithm based on policies, iterations, and multigrid methods. Numerical results are displayed for $n=1$ and $n=2$.
207 citations
TL;DR: In this article, the authors present some asset pricing results for the general case in which asset prices can jump, where asset gains (price plus cumulative dividends) processes are assumed to be special semimartingales.
207 citations
TL;DR: This chapter develops Markov Chain Monte Carlo methods for Bayesian inference in continuous-time asset pricing models with detailed examples for equity price models, option pricing models, term structure models, and regime-switching models.
Abstract: Publisher Summary This chapter describes various Markov Chain Monte Carlo (MCMC) methods for exploring the posterior distributions generated by continuous-time asset pricing models. The MCMC methods are particularly well suited for continuous-time finance applications for several reasons. MCMC is a unified estimation procedure, which simultaneously estimates both parameters and latent variables. MCMC directly computes the distribution of the latent variables and parameters given the observed data and allows the researcher to quantify estimation and model risk. Estimation risk is the inherent uncertainty present in estimating parameters or state variables, while model risk is the uncertainty over model specification. The simplest MCMC algorithm is called the Gibbs sampler, which requires one to conveniently draw from the complete set of conditional distributions. In many cases, implementing the Gibbs sampler requires drawing random variables from standard continuous distributions such as normal, t, beta, or gamma or discrete distributions such as binomial, multinomial, or Dirichlet. The Griddy Gibbs sampler is an approximation that can be applied to approximate the conditional distribution by a discrete set of points. The Metropolis–Hastings algorithm allows the functional form of the density to be nonanalytic, where one only has to evaluate the true density at two given points. Random-walk Metropolis is the original algorithm considered by Metropolis et al. in 1953, and it is the mirror image of the independence Metropolis–Hastings algorithm.
207 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