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 authors survey the development of continuous-time methods in finance during the last 30 years and assess the use of continuous time models in finance, including derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices.
Abstract: I survey and assess the development of continuous-time methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuous-time models. Capital market frictions and bargaining issues are being increasingly incorporated in continuous-time theory. THE ROOTS OF MODERN CONTINUOUS-TIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuous-time modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting. Merton ~1973b! also showed how such a framework can be used to develop equilibrium asset pricing implications, thereby significantly extending the asset pricing theory to richer dynamic settings and expanding the scope of applications of continuous-time methods to study problems in financial economics. 1 Within a span of about 30 years from the publication of Merton’s inf luential papers, continuous-time methods have become an integral part of financial economics. Indeed, in certain core areas in finance ~such as, e.g., asset pricing, derivatives valuation, term structure theory, and portfolio selection! continuoustime methods have proved to be the most attractive way to conduct research and gain economic intuition. The continuous-time approach in these areas has produced models with a rich variety of testable implications. The econometric theory for testing continuous-time models has made rapid strides in the last decade and has thus kept pace with the impressive progress on the theoretical front. One hopes that the actual empirical investigations and estimation using the new procedures will follow suit soon.
232 citations
TL;DR: This paper proposes a practical scheme to obtain a portfolio with a large third moment under the constraints on the first and second moment and solves the problem is a linear programming problem, so that a large scale model can be optimized without difficulty.
Abstract: It is assumed in the standard portfolio analysis that an investor is risk averse and that his utility is a function of the mean and variance of the rate of the return of the portfolio or can be approximated as such. It turns out, however, that the third moment (skewness) plays an important role if the distribution of the rate of return of assets is asymmetric around the mean. In particular, an investor would prefer a portfolio with larger third moment if the mean and variance are the same. In this paper, we propose a practical scheme to obtain a portfolio with a large third moment under the constraints on the first and second moment. The problem we need to solve is a linear programming problem, so that a large scale model can be optimized without difficulty. It is demonstrated that this model generates a portfolio with a large third moment very quickly.
232 citations
Cites background from "Optimum consumption and portfolio r..."
...Also, Merton [ 14 ] argues that if the utility function is locally quadratic and the return distribution is locally normal, then the MV- model is valid for a broad range of utility functions and a broad range of probability distributions if we allow continuous trading....
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TL;DR: In this paper, the authors analyzed an equilibrium in which a call option (derivative asset) is traded and the equilibrium stock price (primary asset) increases when the options market is opened.
Abstract: The traditional pricing methodology in finance values derivative securities as redundant assets that have no impact on equilibrium prices and allocations. This paper demonstrates that, when the market is incomplete, primary and derivative asset markets, generically, interact: the valuation of derivative and primary security prices depend on the contractual characteristics of the derivative assets available. In a version of the Mossin mean-variance model, the authors analyze an equilibrium in which a call option (derivative asset) is traded and the equilibrium stock price (primary asset) increases when the options market is opened. Copyright 1991 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.
231 citations
TL;DR: In this paper, the authors derive the optimal investment policy for a risk-averse investor in a market where there are arbitrage opportunities and show that it is often optimal to underinvest in the arbitrage by taking a smaller position than margin constraints allow.
Abstract: In theory, an investor can make infinite profits by taking unlimited positions in an arbitrage. In reality, however, investors must satisfy margin requirements which completely change the economics of arbitrage. We derive the optimal investment policy for a risk-averse investor in a market where there are arbitrage opportunities. We show that it is often optimal to underinvest in the arbitrage by taking a smaller position than margin constraints allow. In some cases, it is actually optimal for an investor to walk away from a pure arbitrage opportunity. Even when the optimal policy is followed, the arbitrage strategy may underperform the riskless asset or have an unimpressive Sharpe ratio. Furthermore, the arbitrage portfolio typically experiences losses at some point before the final convergence date. These results have important implications for the role of arbitrageurs in financial markets.
229 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