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.
Citations
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TL;DR: The value functions derived for geometric Brownian motion are shown to provide upper and lower bounds on the value functions in the more general context and are extended to consider more general risky investments.
Abstract: This paper solves a general consumption and investment decision problem in closed form. An investor seeks to maximize total expected discounted utility of consumption. There are N distinct risky investments, modelled by dependent geometric Brownian motion processes, and one riskless deterministic investment. The analysis allows for a general utility function and general rates of return. The model and analysis take into consideration the inherent nonnegativity of consumption and consider bankruptcy, so this paper generalizes many of the results of Lehoczky, Sethi, and Shreve Lehoczky, J., S. Sethi, S. Shreva. 1983. Optimal consumption and investment policies allowing consumption constraints and bankruptcy. Math. Oper. Res.8 613--636.. The value function is determined explicitly, as are the optimal consumption and investment policies. The analysis is extended to consider more general risky investments. Under certain conditions, the value functions derived for geometric Brownian motion are shown to provide upper and lower bounds on the value functions in the more general context.
394 citations
TL;DR: The interplay between objective and constraints in a number of single-period variants, including semivariance models are described, revealing the possibility of removing surplus money in future decisions, yielding approximate downside risk minimization.
Abstract: Mean-variance portfolio analysis provided the first quantitative treatment of the tradeoff between profit and risk. We describe in detail the interplay between objective and constraints in a number of single-period variants, including semivariance models. Particular emphasis is laid on avoiding the penalization of overperformance. The results are then used as building blocks in the development and theoretical analysis of multiperiod models based on scenario trees. A key property is the possibility of removing surplus money in future decisions, yielding approximate downside risk minimization.
387 citations
TL;DR: In this article, the authors study the implications of jumps in prices and volatility on investment strategies and provide an analytical solution to the optimal portfolio problem, finding that event risk dramatically affects the optimal strategy.
Abstract: An inherent risk facing investors in financial markets is that a major event may trigger a large abrupt change in stock prices and market volatility. This paper studies the implications of jumps in prices and volatility on investment strategies. Using the event-risk framework of Duffie, Pan, and Singleton, we provide an analytical solution to the optimal portfolio problem. We find that event risk dramatically affects the optimal strategy. An investor facing event risk is less willing to take leveraged or short positions. In addition, the investor acts as if some portion of his wealth may become illiquid and the optimal strategy blends elements of both dynamic and buy-and-hold portfolio strategies. Jumps in prices and volatility both have an important influence on the optimal strategy.
387 citations
TL;DR: In this paper, the authors developed a model of international equities using a multivariate system of jump-diffusion processes where the arrival of jumps is simultaneous across assets and determined an investor's optimal portfolio for this model of returns.
Abstract: Returns on international equities are characterized by jumps; moreover, these jumps tend to occur at the same time across countries leading to systemic risk. In this paper, we evaluate whether systemic risk reduces substantially the gains from international diversification. First, in order to capture these stylized facts, we develop a model of international equity returns using a multivariate system of jump-diffusion processes where the arrival of jumps is simultaneous across assets. Second, we determine an investor's optimal portfolio for this model of returns. Third, we show how one can estimate the model using the method of moments. Finally, we illustrate our portfolio optimization and estimation procedure by analyzing portfolio choice across a riskless asset, the US equity index, and five international indexes. Our main finding is that, while systemic risk affects the allocation of wealth between the riskless and risky assets, it has a small effect on the composition of the portfolio of only-risky assets, and reduces marginally the gains to a US investor from international diversification: For an investor with a relative risk aversion of 3 and a horizon of one year, the certainty-equivalent cost of ignoring systemic risk is of the order $1 for every $1000 of initial investment. These results are robust to whether the international indexes are for developed or emerging countries, to constraints on borrowing and shortselling, and to reasonable deviations in the value of the parameters around their point estimates; the cost increases with the investment horizon and decreases with risk aversion.
386 citations
TL;DR: In this article, the authors analyzed the effect of uncertainty about the mean return on the risky asset on the portfolio decisions of an investor who has a long investment horizon and showed that the possibility of future learning induces the investor to take a larger or smaller position in a risky asset than she would if there were no learning, the direction of the effect depending on whether the investor is more or less risk tolerant than the logarithmic investor whose portfolio decisions are unaffected by the possibility.
Abstract: This paper analyzes the effect of uncertainty about the mean return on the risky asset on the portfolio decisions of an investor who has a long investment horizon. Building on the earlier work of Detemple (1986), Dothan and Feldman (1986), and Gennotte (1986), it is shown that the possibility of future learning about the mean return on the risky asset induces the investor to take a larger or smaller position in the risky asset than she would if there were no learning, the direction of the effect depending on whether the investor is more or less risk tolerant than the logarithmic investor whose portfolio decisions are unaffected by the possibility of future learning. Numerical calculations show that uncertainty about the mean return on the market portfolio has a significant effect on the portfolio decision of an investor with a 20 year horizon if her assessment of the market risk premium is based solely on the Ibbotson and Sinquefield (1995) data.
386 citations
Cites background from "Optimum consumption and portfolio r..."
...Thus, the basic single-period theory was extended by Hakansson (1970), Merton (1971), Samuelson (1969), Breeden (1979) and others to allow for a multi-period horizon in which investment opportunities might either be constant, time-dependent, or even stochastic – in the latter case, these early…...
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...9 See Feldman (1992)....
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...(1970), Merton (1971) , Samuelson (1969), Breeden (1979) and others to allow for a multi-period...
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...8 See Merton (1971)....
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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