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 optimal portfolio policy for a multi-period mean-variance investor facing multiple risky assets in the presence of general transaction costs is analyzed, and it is shown empirically that the losses associated with ignoring transaction costs and behaving myopically may be large.
Abstract: We analyze the optimal portfolio policy for a multiperiod mean–variance investor facing multiple risky assets in the presence of general transaction costs. For proportional transaction costs, we give a closed-form expression for a no-trade region, shaped as a multi-dimensional parallelogram, and show how the optimal portfolio policy can be efficiently computed for many risky assets by solving a single quadratic program. For market impact costs, we show that at each period it is optimal to trade to the boundary of a state-dependent rebalancing region. Finally, we show empirically that the losses associated with ignoring transaction costs and behaving myopically may be large.
37 citations
TL;DR: In this paper, the authors analyse the effects arising from imposing a Value-at-Risk constraint in an agent's portfolio selection problem and derive the optimal consumption and portfolio allocation policy in closed form for the case of logarithmic utility.
Abstract: In this paper we analyse the effects arising from imposing a Value-at-Risk constraint in an agent's portfolio selection problem. The financial market is incomplete and consists of multiple risky assets (stocks) plus a risk-free asset. The stocks are modelled as exponential Brownian motions with random drift and volatility. The risk of the trading portfolio is re-evaluated dynamically, hence the agent must satisfy the Value-at-Risk constraint continuously. We derive the optimal consumption and portfolio allocation policy in closed form for the case of logarithmic utility. The non-logarithmic CRRA utilities are considered as well, when the randomness of market coefficients is independent of the Brownian motion driving the stocks. The portfolio selection, a stochastic control problem, is reduced, in this context, to a deterministic control one, which is analysed, and a numerical treatment is proposed.
36 citations
TL;DR: In this article, the authors investigate the role of regulators in acting as surrogate "intelligent consumers" on behalf of plan members, in order to improve the design of their defined contribution pension plans.
Abstract: Many, if not most, individuals cannot be regarded as ‘intelligent consumers’ when it comes to understanding and assessing different investment strategies for their defined contribution pension plans. This gives very little incentive to plan providers to improve the design of their pension plans. As a consequence, pension plans and their investment strategies are still currently in a very primitive stage of their development. In particular, there is very little integration between the accumulation and decumulation stages. It is possible to produce well-designed DC plans but these need to be designed from back to front (that is, from desired outputs to required inputs) with the goal of delivering an adequate targeted pension with a high degree of probability. We use the analogy of designing a commercial aircraft to explain how this might be done. We also investigate the possible role of regulators in acting as surrogate ‘intelligent consumers’ on behalf of plan members.
36 citations
TL;DR: Conditions on the utility function, for large wealth levels, are provided, which are sufficient for the optimal portfolio policy to approximate, as the trading horizon becomes very long, the policy of investing a constant proportion of wealth in the various assets.
Abstract: We study the behavior of the optimal portfolio policy of a long-run investor in markets with stationary investment opportunity sets. We provide conditions on the utility function, for large wealth levels, which are sufficient for the optimal portfolio policy to approximate, as the trading horizon becomes very long, the policy of investing a constant proportion of wealth in the various assets. The analysis is carried out by employing the associated HJB equation and recent advances in the area of viscosity solutions.
36 citations
Cites background or methods from "Optimum consumption and portfolio r..."
...Moreover, it follows (see Merton 1971) from the dynamic programming principle that ifJ (x, τ ) is C2,1( +, +),1 it satisfies the Bellman equation: max A { 1 2 σ2A2Jxx + (µ − r )AJx } + rxJx −Jτ = 0, (x, τ ) ∈ (0,∞)× (0, T] (2.3) with initial and boundary conditions{ J (x, 0) = U (x) ∀x ≥ 0 J (0, τ…...
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...Using the properties ofU and the linearity of the state dynamics it can be shown (see, for example, Merton 1971) thatJ is strictly increasing and concave in x....
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...This is the classical optimal investment model, introduced by Merton (1971), in a market with a stationary opportunity set and a finite investment horizon....
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TL;DR: In this article, a new asset pricing model with stochastic transaction costs and investors with heterogenous horizons is developed, which generates segmentation effects in the pricing of liquid versus illiquid assets.
Abstract: We develop a new asset pricing model with stochastic transaction costs and investors with heterogenous horizons. Short-term investors hold only liquid assets in equilibrium. This generates segmentation effects in the pricing of liquid versus illiquid assets. Specifically, the liquidity (risk) premia of illiquid assets are determined by the heterogeneity in investor horizons and by the correlation between liquid and illiquid assets. We estimate our model for the cross-section of U.S. stocks and find that it fits average returns substantially better than a standard liquidity CAPM. Allowing for heterogenous horizons also leads to much larger estimates for the liquidity premia.
36 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