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: The authors analyzes the market for financial assets in a production and exchange economy with several realized outputs and a single unobservable source of non-consistency risk and shows that, for a large class of diffusion outputs and preferences, optimizing consumers first estimate the realizations of the unobservability factor and then use these estimates to determine portfolio and consumption rules.
Abstract: This paper analyzes the market for financial assets in a production and exchange economy with several realized outputs and a single unobservable source of nondiversifiable risk. The paper demonstrates that, for a large class of diffusion outputs and preferences, optimizing consumers first estimate the realizations of the unobservable factor and then use these estimates to determine portfolio and consumption rules. Moreover, the explicit consideration of this unobservable productivity factor affects equilibrium demands and prices. The equilibrium spot rate of interest emerges as the "best estimate" of the unobservable factor, and multiperiod default-free bonds arise as the optimal hedge for the unobservable changes of the stochastic investment opportunity set. THIS PAPER ANALYZES THE market for financial assets in a production and exchange economy with several realized outputs and a single unobservable source of nondiversifiable risk. The paper poses and answers three questions: 1. How do consumers determine equilibrium demands and prices when their investment opportunities are unobservable? 2. In equilibrium, what financial assets provide an optimal hedge for the
280 citations
Cites background or methods from "Optimum consumption and portfolio r..."
...2 Some additional works that contributed to the analysis of equilibrium in a partially observable economy are Duffie [14], Duffie and Huang [15, 16], Kunita and Watanabe [28], and Merton [31, 32]....
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...Williams [40] examined a "Merton [31]-type economy" which assumed individuals did not know constant parameters of the distributions of exogenously determined asset prices....
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...Procedures developed by Merton [31], Breeden [2], and Cox, Ingersoll, and Ross [8, 9] identify the necessary and sufficient conditions which determine equilibrium interest rates and optimal portfolio rules as,...
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Book•
29 Sep 2011
TL;DR: In this paper, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest-rate, and credit markets, and present and analyze multiscale stochastically volatility models and asymptotic approximations.
Abstract: Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest-rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approximations. These can be used in equity markets, for instance, to link the prices of path-dependent exotic instruments to market implied volatilities. The methods are also used for interest rate and credit derivatives. Other applications considered include variance-reduction techniques, portfolio optimization, forward-looking estimation of CAPM 'beta', and the Heston model and generalizations of it. 'Off-the-shelf' formulas and calibration tools are provided to ease the transition for practitioners who adopt this new method. The attention to detail and explicit presentation make this also an excellent text for a graduate course in financial and applied mathematics.
279 citations
TL;DR: In this paper, a new type of risk aversion found only in muitivariate utility functions is defined and certain behavioral assumptions, which are necessary and sufficient for one of three forms of separable utility functions including the well-known additive form, are given.
Abstract: This paper concerns utility functions for more than one attribute. A new type of risk aversion found only in muitivariate utility functions is defined. Certain behavioral assumptions, which are necessary and sufficient for one of three forms of separable utility functions including the well-known additive form, are given. It is shown that only one of these separable forms, the “negative multiplicative form,” possesses this new type of risk aversion and in particular that the additive form does not.
278 citations
Cites background from "Optimum consumption and portfolio r..."
...For example in the investment literature such as Samuelson [11], Merton [6] and Hakansson [3], an individual makes decisions to maximize his expected utility for a lifetime consumption stream....
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TL;DR: The authors study a consumption-based asset pricing model in which some investors form beliefs about future price changes in the stock market by extrapolating past price changes, while other investors hold fully rational beliefs.
274 citations
TL;DR: In this article, Detemple et al. proposed a new simulation-based approach for optimal portfolio allocation in realistic environments with complex dynamics for the state variables and large numbers of factors and assets.
Abstract: This paper proposes a new simulation-based approach for optimal portfolio allocation in realistic environments with complex dynamics for the state variables and large numbers of factors and assets. A first illustration involves a choice between equity and cash with nonlinear interest rate and market price of risk dynamics. Intertemporal hedging demands significantly increase the demand for stocks and exhibit low volatility. We then analyze settings where stock returns are also predicted by dividend yields and where investors have wealth-dependent relative risk aversion. Large-scale problems with many assets, including the Nasdaq, SP500, bonds, and cash, are also examined. The question of optimal portfolio allocation has been of long-standing interest for academics and practitioners in finance. While the mean-variance analysis of Markowitz (1952) is still commonly used among portfolio managers, it has been well understood, since Merton (1971), that long-term investors would prefer portfolios that include hedging components to protect against fluctuations in their investment opportunities. Prompted by the seminal papers of Merton (1969, 1971) and Samuelson (1969), studies have explored various aspects of the dynamic portfolio problem when asset prices follow diffusion processes (e.g., Richard (1975)). This literature has relied, for the most part, on a dynamic programming approach to the problem. More recent contributions by Pliska (1986), Karatzas, Lehoczky, and Shreve (1987), and Cox and Huang (1989) have proposed an alternative resolution method based on martingale techniques. In the context of this approach, an optimal portfolio formula was derived by Ocone and Karatzas (1991). This expression involves expectations of random variables depending on *J6erme B. Detemple is at Boston University, School of Management and CIRANO; Rene
273 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