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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01 Jan 1989
TL;DR: A survey of numerical methods for stochastic differential equations can be found in this article, where the authors focus on diffusion processes, but also indicate the extensions needed to handle processes with jump components.
Abstract: The development of numerical methods for stochastic differential equations has intensified over the past decade. The earliest methods were usually heuristic adaptations of deterministic methods, but were found to have limited accuracy regardless of the order of the original scheme. A stochastic counterpart of the Taylor formula now provides a framework for the systematic investigation of numerical methods for stochastic differential equations. It suggests numerical schemes, which involve multiple stochastic integrals, of higher order of convergence. We shall survey the literature on these and on the earlier schemes in this paper. Our discussion will focus on diffusion processes, but we shall also indicate the extensions needed to handle processes with jump components. In particular, we shall classify the schemes according to strong or weak convergence criteria, depending on whether the approximation of the sample paths or of the probability distribution is of main interest.
97 citations
Cites background from "Optimum consumption and portfolio r..."
...They are also starting to be used in the social sciences, such as economics (eg Karatzas and Shreve (1988); Merton (1971) ), and experimental psychology (eg Schoener et al. (1986))....
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TL;DR: In this article, the authors apply stochastic optimal control theory to find an optimal investment policy before and after retirement in a defined contribution pension plan where benefits are paid under the form of annuities.
Abstract: The purpose of this paper is to show how stochastic optimal control theory can be applied to find an optimal investment policy before and after retirement in a defined contribution pension plan where benefits are paid under the form of annuities; annuities are supposed to be guaranteed during a certain fixed period of time. Using different kinds of utility functions we try to look at different strategies on the one hand in the investment part (i.e. before retirement) and on the other hand in the payment part (i.e. after retirement). The needed change of strategy after retirement can be interpreted in this model as a typical ALM constraint taking into account a guaranteed technical interest rate used by the insurer.
97 citations
Posted Content•
TL;DR: In this paper, the decision of when to invest in an indivisible project whose value is perfectly observable but driven by a parameter that is unknown to the decision maker ex ante was studied.
Abstract: We study the decision of when to invest in an indivisible project whose value is perfectly observable but driven by a parameter that is unknown to the decision maker ex ante. This problem is equivalent to an optimal stopping problem for a bivariate Markov process. Using filtering and martingale techniques, we show that the optimal investment region is characterised by a continuous and non-decreasing boundary in the value/belief state space. This generates path-dependency in the optimal investment strategy. We further show that the decision maker always benefits from an uncertain drift relative to an 'average' drift situation. However, a local study of the investment boundary reveals that the value of the option to invest is not globally increasing with respect to the volatility of the value process.
97 citations
TL;DR: In this article, the authors compare the performance of equal-, value-, and price-weighted portfolios of stocks in the major U.S. equity indices over the last four decades.
Abstract: We compare the performance of equal-, value-, and price-weighted portfolios of stocks in the major U.S. equity indices over the last four decades. We nd that the equal-weighted portfolio with monthly rebalancing outperforms the value- and price-weighted portfolios in terms of total mean return, four factor alpha, Sharpe ratio, and certainty-equivalent return, even though the equal-weighted portfolio has greater portfolio risk. The total return of the equal-weighted portfolio exceeds that of the value- and price-weighted because the equal-weighted portfolio has both a higher return for bearing systematic risk and a higher alpha when using the fourfactor model. The nonparametric test of Patton and Timmermann (2009) indicates that the dierences in the total return of the equal-weighted portfolio and the valueand price-weighted portfolios is monotonically related to size, price, liquidity and idiosyncratic volatility; the relation with reversal is not monotonic, although the equal-weighted portfolio strongly outperforms the value- and price-weighted portfolios for the deciles with the lowest and the highest reversal characteristic. The higher systematic return of the equal-weighted portfolio arises from its higher exposure to the market, size, and value factors. The higher alpha of the equal-weighted portfolio arises from the monthly rebalancing required to maintain equal weights, which is implicitly a contrarian strategy that exploits reversal; thus, alpha depends only on the rebalancing frequency and not on the choice of initial weights.
97 citations
TL;DR: In this article, the global existence and uniqueness result for a general one-dimensional backward stochastic Riccati equation was obtained and applied to the mean-variance hedging problem.
97 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