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Journal ArticleDOI

Optimum consumption and portfolio rules in a continuous-time model☆

01 Dec 1971-Journal of Economic Theory (Academic Press)-Vol. 3, Iss: 4, pp 373-413
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.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the authors developed an equilibrium model of convergence trading and its impact on asset prices and showed that such losses can occur in the absence of any shock and that prices of identical assets can diverge even if the constraints faced by arbitrageurs are not binding.
Abstract: I develop an equilibrium model of convergence trading and its impact on asset prices. Arbitrageurs optimally decide how to allocate their limited capital over time. Their activity reduces price discrepancies, but their activity also generates losses with positive probability, even if the trading opportunity is fundamentally riskless. Moreover, prices of identical assets can diverge even if the constraints faced by arbitrageurs are not binding. Occasionally, total losses are large, making arbitrageurs’ returns negatively skewed, consistent with the empirical evidence. The model also predicts comovement of arbitrageurs’ expected returns and market liquidity. Many hedge funds and some other financial institutions attempt to exploit the relative mispricing of assets. However, from time to time, these institutions (whom I will refer to as convergence traders or arbitrageurs) suffer spectacular losses if the prices of these assets diverge, forcing them to unwind some of their positions. The near-collapse of the Long-Term Capital Management hedge fund in 1998 is frequently cited as an example of this phenomenon. 1 To what extent can these losses be attributed to the actions of arbitrageurs as opposed to unforeseen shocks? Why do other institutions with liquid capital not eliminate the abnormal returns around these events? In this paper, I develop a theoretical model to address these questions. I show that such losses can occur in the absence of any shock and that prices of identical assets can diverge even if the constraints faced by arbitrageurs are not binding.

156 citations


Cites methods from "Optimum consumption and portfolio r..."

  • ...This intuition shows also that the driving mechanism of the model is based on the hedging component of asset demand identified in Merton (1971)....

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Journal ArticleDOI
TL;DR: In this article, the authors investigate the income drawdown option and define a stochastic optimal control problem, looking for optimal investment strategies to be adopted after retirement, when allowing for periodic fixed withdrawals from the fund.
Abstract: In defined contribution pension schemes, the financial risk is borne by the member. Financial risk occurs both during the accumulation phase (investment risk) and at retirement, when the annuity is bought (annuity risk). The annuity risk faced by the member can be reduced through the “income drawdown option”: the retiree is allowed to choose when to convert the final capital into pension within a certain period of time after retirement. In some countries, there is a limiting age when annuitization becomes compulsory (in UK this age is 75). In the interim, the member can withdraw periodic amounts of money to provide for daily life, within certain limits imposed by the scheme’s rules (or by law). In this paper, we investigate the income drawdown option and define a stochastic optimal control problem, looking for optimal investment strategies to be adopted after retirement, when allowing for periodic fixed withdrawals from the fund. The risk attitude of the member is also considered, by changing a parameter in the disutility function chosen. We find that there is a natural target level of the fund, interpretable as a safety level, which can never be exceeded when optimal control is used. Numerical examples are presented in order to analyse various indices — relevant to the pensioner — when the optimal investment allocation is adopted. These indices include, for example, the risk of outliving the assets before annuitization occurs (risk of ruin), the average time of ruin, the probability of reaching a certain pension target (that is greater than or equal to the pension that the member could buy immediately on retirement), the final outcome that can be reached (distribution of annuity that can be bought at limit age), and how the risk attitude of the member affects the key performance measures mentioned above.

156 citations

Journal ArticleDOI
TL;DR: In this article, the problem of maximizing the expected utility from consumption or terminal wealth in a market where logarithmic securities prices follow a Levy process was considered, and explicit solutions for power and exponential utility in terms of the Levy-Khintchine triplet were given.
Abstract: We consider the problem of maximizing the expected utility from consumption or terminal wealth in a market where logarithmic securities prices follow a Levy process. More specifically, we give explicit solutions for power, logarithmic and exponential utility in terms of the Levy-Khintchine triplet. In the first two cases, a constant fraction of current wealth should be invested in each of the securities, as is well-known for related discrete-time models and for Brownian motion. The situation is different for exponential utility.

155 citations

Journal ArticleDOI
TL;DR: A short survey of stochastic models of risk control and dividend optimization techniques for a financial corporation shows that in most cases the optimal dividend distribution scheme is of a barrier type, while the risk control policy depends significantly on the nature of the reinsurance available.
Abstract: The current paper presents a short survey of stochastic models of risk control and dividend optimization techniques for a financial corporation. While being close to consumption/investment models of Mathematical Finance, dividend optimization models possess special features which do not allow them to be treated as a particular case of consumption/investment models.¶ In a typical model of this sort, in the absence of control, the reserve (surplus) process, which represents the liquid assets of the company, is governed by a Brownian motion with constant drift and diffusion coefficient. This is a limiting case of the classical Cramer-Lundberg model in which the reserve is a compound Poisson process, amended by a linear term, representing a constant influx of the insurance premiums. Risk control action corresponds to reinsuring part of the claims the cedent is required to pay simultaneously diverting part of the premiums to a reinsurance company. This translates into controlling the drift and the diffusion coefficient of the approximating process. The dividend distribution policy consists of choosing the times and the amounts of dividends to be paid out to shareholders. Mathematically, the cumulative dividend process is described by an increasing functional which may or may not be continuous with respect to time.¶ The objective in the models presented here is maximization of the dividend pay-outs. We will discuss models with different types of conditions imposed upon a company and different types of reinsurances available, such as proportional, noncheap, proportional in a presence of a constant debt liability, excess-of-loss. We will show that in most cases the optimal dividend distribution scheme is of a barrier type, while the risk control policy depends significantly on the nature of the reinsurance available.

155 citations


Cites background from "Optimum consumption and portfolio r..."

  • ...During the same period of time Merton published his seminal paper on the optimal consumption/investment strategy for a small investor [ 45 ]....

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Journal ArticleDOI
TL;DR: The solution to the problem of maximizing total utility of consumption is given by way of quasi-variational inequalities for the value function of risky assets and one nonrisky asset with log-normal prices.
Abstract: An investor has the opportunity of holding shares in n risky assets and one nonrisky asset at every time in a fixed interval [t, T]. The risky assets are governed by a stochastic differential equation. At random instants of his choice he may intervene in order to rebalance his portfolio and consume a nonnegative amount of money. Fixed and variable transactions costs are incurred upon intervention. At time T all remaining wealth is consumed. The solution to the problem of maximizing total utility of consumption is given by way of quasi-variational inequalities for the value function. With probability one the investor only intervenes finitely many times. Indication of the solution of the quasi-variational inequalities in the case of one risky asset with log-normal prices is given, together with a description of a discretization procedure.

155 citations


Cites background from "Optimum consumption and portfolio r..."

  • ...From Merton's important 1971 paper [13] to a recent paper by Lehoczky, Sethi, and Shreve [10], many authors have examined various portfolio-consumption models....

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References
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Journal ArticleDOI
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

Book ChapterDOI
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