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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Book•
17 May 2013TL;DR: A summary of research on life annuities, longevity insurance, and their role in the "optimal" retirement portfolio is provided in this article, with an overview of institutional aspects, moves on to discuss valuation issues, and concludes with a comprehensive review of the scholarly literature.
Abstract: This book provides a summary of research on life annuities, longevity insurance, and their role in the “optimal” retirement portfolio. It starts with an overview of institutional aspects, moves on to discuss valuation issues, and concludes with a comprehensive review of the scholarly literature.
35 citations
TL;DR: In this paper, the authors derive an explicit analytical solution to the dynamic portfolio problem of an individual saving for retirement or other change of status, like the purchase of a house or starting college.
Abstract: In recent years it has been shown empirically that stock returns exhibit positive or negative autocorrelation, depending on observation frequency. In this context of autocorrelated returns the present paper is the first to derive an explicit analytical solution to the dynamic portfolio problem of an individual agent saving for retirement or other change of status, like the purchase of a house or starting college. Using a normal ARMA1,1 process, dynamic programming techniques combined with the use of Stein's Lemma are employed to examine "dollar-cost-averaging" and "age effects" in intertemporal portfolio choice with CARA preferences. We show that with a positive moving average parameter and positive riskfree rates, if first-order serial correlation is nonnegative, then the expected value of the optimal risky investment is increasing over time, while if first-order serial correlation is negative this path can be increasing or decreasing over time. Thus a necessary but not sufficient condition to obtain the conventional age effect of increasing conservatism over time is that first-order serial correlation be negative. Further, dollar-cost averaging in the general sense of gradual entry into the risky asset does not emerge as an optimal policy. Simulation results for U.S. data are used to illustrate optimal portfolio paths.
35 citations
TL;DR: In this paper, the authors consider the asset allocation problem of an investor allocating his funds between several corporate bonds and a money market account, and analyze the impact of contagion on an investor's demand for corporate bonds.
35 citations
Posted Content•
TL;DR: In this article, a continuous-time portfolio optimization with defaultable bonds and stocks is proposed, where the investor has the opportunity to put her wealth into derivatives with counterparty risk or credit derivatives.
Abstract: Credit risk is an important issue of current research in finance. While there is a lot of work on modelling credit risk and on valuing credit derivatives there is no work on continuous-time portfolio optimization with defaultable securities. Therefore, in this paper we solve investment problems with defaultable bonds and stocks. Besides, our approach can be applied to portfolio problems, where the investor has the opportunity to put her wealth into derivatives with counterparty risk or credit derivatives.
35 citations
Cites background from "Optimum consumption and portfolio r..."
...every utility function considered in Merton (1969,1971) , Cox/Huang (1989, 1991)...
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...As a consequence, the results hold for every utility function considered in Merton (1969,1971), Cox/Huang (1989, 1991) or Karatzas/Lehoczky/Shreve (1987). c) By comparing the actual amount of money invested in the risky asset as computed in Proposition 1 and 2 we find π∗V ·X∗ = λ σ2 X∗ > λ σ2 B BvV…...
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...1) To our knowledge only Merton (1971) considers a portfolio problem with defaultable bonds, but he used a bond model which can be seen as a rudimentary reduced form model with deterministic interest rates....
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...In his pioneering work Merton (1969, 1971) considered an investor who allocates her wealth in stocks or a riskless money market account....
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...1) To our knowledge only Merton (1971) considers a portfolio problem with default-...
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Posted Content•
TL;DR: In this article, the authors reformulate and reduce the Hamilton-Jacobi-Bellman equation for this singular stochastic control problem to a non-standard free-boundary problem for a first-order ODE with an integral constraint.
Abstract: We revisit the optimal investment and consumption model of Davis and Norman (1990) and Shreve and Soner (1994), following a shadow-price approach similar to that of Kallsen and Muhle-Karbe (2010). Making use of the completeness of the model without transaction costs, we reformulate and reduce the Hamilton-Jacobi-Bellman equation for this singular stochastic control problem to a non-standard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. Furthermore, we provide an explicit characterization of model parameters for which the value function is finite.
35 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