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: A backward stochastic differential equation, (BSDE), approach to a risk-based, optimal investment problem of an insurer, and leads to a simple and natural approach for the existence and uniqueness of an optimal strategy of the game problem without Markov assumptions.
38 citations
TL;DR: This paper decompose the optimal portfolio weight into components that correspond to a collection of fictitious economies, one of which is a standard diffusion economy, and the others of which are pure-jump economies.
Abstract: This article solves the portfolio choice problem in a multi-asset jump-diffusion model. We decompose the optimal portfolio weight into components that correspond to a collection of fictitious economies, one of which is a standard diffusion economy, and the others of which are pure-jump economies. We derive a semi-closed-form solution for the optimal portfolio weight, and investigate its properties with or without ambiguity aversion. We find that an investor may not reduce her investment in risky assets when facing more frequent jumps, as suggested by a single-asset jump-diffusion model. Moreover, an investor who is extremely cautious about her estimates of higher moments of asset returns may still hold risky assets, contrary to the prediction of a pure-diffusion model with ambiguity aversion to the first moment.
38 citations
TL;DR: In this article, a representation explicite of the strategy of cout le plus bas (strategie de cout optimal) which reproduit a distribution donnee is presented, which permits d'etendre la theorie en presence de contraintes sur la dependance avec le marche.
Abstract: Dans cet article, nous donnons une representation explicite de la strategie de cout le plus bas (strategie de cout optimal) qui reproduit une distribution donnee. Pour toute strategie non optimale, nous proposons des instruments financiers derives qui dominent dans le sens de la dominance stochastique de premier ou de second ordre. Nous mettons en evidence les liens entre l’optimalite du cout d’une strategie et sa dependance avec le marche. Cela nous permet d’etendre la theorie en presence de contraintes sur la dependance avec le marche. Nous montrons en particulier que les strategies dependant de la trajectoire du sous-jacent ne sont pas optimales dans le cadre de Black et Scholes mais peuvent le devenir en presence de contraintes sur la dependance avec le marche financier.
38 citations
TL;DR: In this paper, the authors extend Rietz and Barro's framework and subject consumption and dividends to rare disasters in the growth persistence by means of two hidden types of economic slowdowns: recessions and lost decades.
Abstract: Rietz (1988) and Barro (2006) subject consumption and dividends to rare disasters in the growth rate. We extend their framework and subject consumption and dividends to rare disasters in the growth persistence. We model growth persistence by means of two hidden types of economic slowdowns: recessions and lost decades. We estimate the model based on the post-war U.S. data using maximum likelihood and find that it can simultaneously match a wide array of dynamic pricing phenomena in the equity and bond markets. The key intuition for our results stems from the inability to discriminate between the short and the long recessions ex ante.
38 citations
Cites background from "Optimum consumption and portfolio r..."
...The dynamic budget constraint takes the standard form as in Merton (1971),...
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...The dynamic budget constraint takes the standard form as in Merton (1971), dWt = ( ∑ a∈A ωat (dR a t − rtdt) + rtdt ) Wt − ctdt, (40) where we still need to specify the law of motion for the asset return dRat ....
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TL;DR: In this article, the authors provide new results on the structure of optimal portfolios, economic insights on the behavior of the hedging components and simulation-based methods for numerical implementation of allocation rules.
Abstract: This paper provides (i) new results on the structure of optimal portfolios, (ii) economic insights on the behavior of the hedging components and (iii) simulation-based methods for numerical implementation of allocation rules. The core of our approach relies on closed-form solutions for functionals of diffusion processes which simplify their numerical simulation and facilitate the computation and simulation of the hedging components of optimal portfolios. One of our procedures relies on a variance-stabilizing transformation of the underlying process which eliminates stochastic integrals from the representation of random variables in hedging terms and ensures the existence of an exact weak approximation scheme. This improves the performance of Monte-Carlo methods in the numerical implementation of portfolio rules derived on the basis of probabilistic arguments. Our approach is flexible and can be used even when the dimensionality of the set of underlying state variables is large. We implement the procedure for a class of bivariate and trivariate models in which the uncertainty is described by diffusion processes for the market price of risk (MPR), the interest rate (IR) and other relevant factors. After calibrating the models to the data we document the behavior of the portfolio demand and the hedging components relative to the parameters of the model such as risk aversion, investment horizon, speeds of mean-reversion, IR and MPR levels and volatilities. We show that the hedging terms are important and cannot be ignored for asset allocation purposes. Risk aversion and investment horizon emerge as the most relevant factors: they have a substantial impact on the size of the optimal portfolio and on its economic properties for realistic values of the models' parameters.
38 citations
References
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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