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 article, the authors propose a generalization of the conventional design that has some interesting features, such as guaranteed minimum rate of return and new constraint, and present the explicit form of the optimal contract assuming both constraints apply.
Abstract: There is a rich variety of tailored investment products available to the retail investor in every developed economy. These contracts combine upside participation in bull markets with downside protection in bear markets. Examples include equity-linked contracts and other types of structured products. This paper analyzes these contracts from the investor’s perspective rather than the issuer’s using concepts and tools from financial economics. We analyze and critique their current design and examine their valuation from the investor’s perspective. We propose a generalization of the conventional design that has some interesting features. The generalized contract specifications are obtained by assuming that the investor wishes to maximize end of period expected utility of wealth subject to certain constraints. The first constraint is a guaranteed minimum rate of return which is a common feature of conventional contracts. The second constraint is new. It provides the investor with the opportunity to outperform a benchmark portfolio with some probability. We present the explicit form of the optimal contract assuming both constraints apply and we illustrate the nature of the solution using specific examples. The paper focusses on equity-indexed annuities as a representative type of such contracts but our approach is applicable to other types of equity-linked contracts and structured products.
62 citations
TL;DR: In this paper, it is shown that the risk sensitive criterion amounts to maximizing a portfolio's risk adjusted growth rate, which is essentially the same as what is commonly done in practice.
Abstract: The idea of using stochastic control methods for theoretical studies of portfolio management has long been standard, with maximum expected utility criteria commonly being used. But in recent years a new kind of criterion, the risk sensitive criterion, has emerged from the control theory literature and been applied to portfolio management. This paper studies various economic properties of this criterion for portfolio management, thereby providing justification for its theoretical and practical use. In particular, it is shown that the risk sensitive criterion amounts to maximizing a portfolio's risk adjusted growth rate. In other words, it is essentially the same as what is commonly done in practice: find the best trade‐off between a portfolio's average return and its average volatility.
62 citations
TL;DR: In this article, the authors trace the impact of formative experiences on portfolio choice, and find that adversely affected workers are less likely to invest in risky assets than those whose income, employment, and wealth were unaffected.
Abstract: We trace the impact of formative experiences on portfolio choice. Plausibly exogenous variation in workers’ exposure to a depression allows us to identify the effects and a new estimation approach makes addressing wealth and income effects possible. We find that adversely affected workers are less likely to invest in risky assets. This result is robust to a number of control variables and it holds for individuals whose income, employment, and wealth were unaffected. The effects travel through social networks: individuals whose neighbors and family members experienced adverse circumstances also avoid risky investments.
61 citations
TL;DR: This paper studies the time consistent strategies in the mean-variance portfolio selection with short-selling prohibition in both discrete and continuous time settings and eliminates the chanc...
Abstract: In this paper, we study the time consistent strategies in the mean-variance portfolio selection with short-selling prohibition in both discrete and continuous time settings. Recently, [T. Bjork, A. Murgoci, and X. Y. Zhou, Math. Finance, 24 (2014), pp. 1--24] considered the problem with state dependent risk aversion in the sense that the risk aversion is inversely proportional to the current wealth, and they showed that the time consistent control is linear in wealth. Considering the counterpart of their continuous time equilibrium control in the discrete time framework, the corresponding “optimal” wealth process can take negative values; and this negativity in wealth will lead the investor to a risk seeker which results in an unbounded value function that is economically unsound; even more, the limiting of the discrete solutions has shown to be their obtained continuous solution in [T. Bjork, A. Murgoci, and X. Y. Zhou, Math. Finance, 24 (2014), pp. 1--24]. To deal this limitation, we eliminate the chanc...
61 citations
TL;DR: In this paper, 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.
61 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