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 paper, the authors investigated how purchasing power risks affect the determination of exchange rates in a fairly general optimizing model and used techniques developed in the finance literature which build on Merton's (1973) asset-pricing model.
Abstract: MOST RECENT RESEARCH on the theory of exchange rate determination treats the exchange rate as the relative price of two assets: domestic money and foreign money. The value of an asset depends on the distribution of its return. However, most of the research has dealt with models in which there is no uncertainty. The present paper focuses on questions that cannot be addressed in a world of certainty, because in such a world an asset is never risky. It investigates how purchasing power risks affect the determination of exchange rates. This investigation is pursued in a fairly general optimizing model. The present research is related to some recent work in international economics and in financial economics. It follows the approach of Obstfeld (1981) and Stockman (1980) in that it uses an optimizing model and takes into account the role of government transfers in the flow budget constraint of individuals. As in Calvo and Rodriguez (1977), Stockman (1980), and others, the fact that holdings of foreign monies are useful to domestic individuals is taken into account. Finally, the paper uses techniques developed in papers in the finance literature which build on Merton's (1973) asset-pricing model and use a concept of equilibrium developed in
82 citations
TL;DR: This paper proposes and study a continuous-time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee and shows that the value function of the problem is a regular solution of the associated Hamilton–Jacobi–Bellman equation.
Abstract: In this paper we propose and study a continuous time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee. Usually, portfolio selection models for pension funds maximize the expected utility from final wealth over a finite horizon (the retirement time), whereas our target is to maximize the expected utility from current wealth over an infinite horizon since we adopt the point of view of the fund manager. In our model the dynamics of wealth takes directly into account the flows of contributions and benefits and the level of wealth is constrained to stay above a solvency level. The fund manager can invest in a riskless asset and in a risky asset but borrowing and short selling are prohibited. We concentrate the analysis on the effect of the solvency constraint, analyzing in particular what happens when the fund wealth reaches the allowed minimum value represented by the solvency level. The model is naturally formulated as an optimal stochastic control problem and is treated by the dynamic programming approach. We show that the value function of the problem is a regular solution of the associated Hamilton-Jacobi-Bellman equation. Then we apply verification techniques to get the optimal allocation strategy in feedback form and to study its properties. We finally give a special example with explicit solution.
82 citations
Cites background from "Optimum consumption and portfolio r..."
...Maximizing expected utility from consumption and from final wealth, he proved in [Merton, 1969] that explicit solutions exist if the individual utility function belongs to the CRRA (constant relative risk aversion) family, and in [Merton, 1971] if it belongs to the HARA (hyperbolic absolute risk aversion) family....
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...Following [Merton, 1969] and [Merton, 1971], we look for a solution of HJB equation (24) of the form v (x) = C (x− l)γ γ , γ ∈ (−∞, 0) ∪ (0, 1), (31) for a suitable constant C. Substituting into HJB equation (24) we see that it must be C = ( ρ− γr − λ 2γ 2 (1− γ) )−1 , (32) under the conditions ρ >…...
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Posted Content•
TL;DR: In this article, the optimal portfolio choice over the life cycle for households facing labor income, capital market, and mortality risk was derived, and a considerable fraction of wealth should be annuitized to skim the return enhancing mortality credit, while the remaining liquid wealth is used to hedge labor income risk during work life, to earn the equity premium, and to ensure estate for the heirs.
Abstract: We derive the optimal portfolio choice over the life-cycle for households facing labor income, capital market, and mortality risk. In addition to stocks and bonds, households also have access to incomplete annuity markets offering a hedge against mortality risk. We show that a considerable fraction of wealth should be annuitized to skim the return enhancing mortality credit. The remaining liquid wealth (stocks and bonds) is used to hedge labor income risk during work life, to earn the equity premium, and to ensure estate for the heirs. Furthermore, we assess the importance of common explanations for limited participation in annuity markets.
82 citations
TL;DR: In this article, advances in financial science have made possible an improved menu of life-cycle investment products, and they have been used to improve the quality of life cycle investment products.
Abstract: Advances in financial science have made possible an improved menu of life-cycle investment products.
82 citations
TL;DR: This paper developed the properties of optimal allocations to labor, leisure, and education over the life cycle for the individual facing uncertainty from several soul-ces Results are reported for both a simple stochastic model and a more general adaptive model in which the individual, by investing in education, not only accumulates huan capital but also gradually learns about his ability to acquire additional huanman capital.
Abstract: Models of the accumulation of human capital over the life cycle have recently received widespread attention in the literature on labor economics The seminal work by Becker (1964) and Ben-Porath (1967) characterized optimal investments in education and the resulting lifetime profiles for both human capital and labor income More recent refinements by Ghez and Becker (1975), Blinder and Weiss (1976), and Heckman (1976) added consumption and leisure, thereby altering predicted patterns for labor supply and labor earnings Simultaneously, empirical stLudies by Mincer (1974), Haley (1976), Rosen (1976), and others used the current theory to estimate relevant parameters affecting labor earnings over the life cycle Despite this lengthy literature on human capital, existing research has largely ignored all significant sources of uncertainty affecting observed behavior over the life cycle' This is unfortunate if only because the introduction of uncertainty significantly alters predicted patterns for both labor supply and labor earnings For Existing models of the accumulation of human capital over the life cycle ignore uncertainty In this paper proper-ties of optimal allocations to labor, leisure, and education over the life cycle are developed in detail for the individual facing uncertainty from several soul-ces Results are reported for both a simple stochastic model and a more general adaptive model in which the individual, by investing in education, not only accumulates hUman capital but also gradually learns about his ability to acquire additional huLman capital
81 citations
Cites background or methods from "Optimum consumption and portfolio r..."
...This accumulation equation is derived in detail in Merton (1971)....
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...See Merton (1973) and the references cited therein....
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...Without additional restrictions on preferences, it is impossible to solve (A3) explicitly for the indirect utility function J (see Merton 1971). 18. Properties of HARA utility functions are described in Merton (1971)....
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...*1 am grateful to Edward Lazear, Merton Miller, Melvin Reder, and an anonymous referee for helpful suggestions on previous drafts. This paper was written while I was at the Graduate School of Business, University of Chicago. 1. Uncertainty appears in the two-period models of Levhafi and Weiss (1974) and Williams (1978)....
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...Recent empirical studies by Mincer (1974), Haley (1976), Rosen (1976), and others use the current theory to estimate parameters affecting observed labor income over the life cycle....
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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