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 authors used a utility-maximizing framework to find the optimal currency composition of external debt to minimize exposures to external price risk in Mexico and Brazil, and showed that the low correlations between the costs of borrowings and export and import prices make currency composition a very imperfect hedging tool.
Abstract: The changes in exchange rates, interest rates, and commodity prices during the past decades have had large impacts on developing countries. Many developing countries have limited access to already incomplete international long-term hedging markets. Thus the question arises whether the currency composition of external debt can be used to minimize exposures to external price risk. Using a utility-maximizing framework, this article shows that, by choosing the optimal currency composition, a country can indeed manage its external exposure. The optimal, risk-minimizing currency composition depends on the relation between export receipts and the costs of borrowings in each currency and on the relations among the costs of borrowings in different currencies. A simple methodology can be used to derive the optimal shares of individual currencies and is applied to Mexico and Brazil. The results show that Mexico and Brazil could have lowered their external exposure to a limited degree by continuously alternating the currency composition of their debts. The low correlations between the costs of borrowings and export and import prices make the currency composition of debt a very imperfect hedging tool, and it is likely that hedging instruments directly linked to prices are preferable.
37 citations
TL;DR: In this article, the authors introduce three strategies for the analysis of financial time series based on time averaged observables, which include the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the time series.
Abstract: We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black–Scholes–Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics.
37 citations
01 Jan 2004
TL;DR: In this article, the Lundberg risk model is used to estimate the risk of an insurance policy, and a simple insurance model with Dividend payments is proposed to solve the problem.
Abstract: 1. Preface
2. Introduction Into Insurance Risk
2.1. The Lundberg Risk Model
2.2. Alternatives
2.3. Ruin Probability
2.4. Asymptotic Behavior For Ruin Probabilities
3. Possible Control Variables and Stochastic Control
3.1. Possible Control Variables
3.2. Stochastic Control
4. Optimal Investment for Insurers
4.1. HJB and its Handy Form
4.2. Existence of a Solution
4.3. Exponential Claim Sizes
4.4. Two or More Risky Assets
5. Optimal Reinsurance and Optimal New Business
5.1. Optimal Proportional Reinsurance
5.2. Optimal Unlimited XL Reinsurance
5.3. Optimal XL Reinsurance
5.4. Optimal New Business
6. Asymptotic Behavior for Value Function and Strategies
6.1. Optimal Investment: Exponential Claims
6.2. Optimal Investment: Small Claims
6.3. Optimal Investment: Large Claims
6.4. Optimal Reinsurance
7. A Control Problem with Constraint: Dividends and Ruin
7.1. A Simple Insurance Model with Dividend Payments
7.2. Modified HJB Equation
7.3. Numerical Example and Conjectures
7.4. Earlier and Further Work
8. Conclusions
References
37 citations
Cites background from "Optimum consumption and portfolio r..."
...Stochastic control is well established in the finance world since the seminal papers of Merton ([33] and [ 34 ])....
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TL;DR: In this article, a new quantitative model was proposed to investigate the problem of optimal life insurance purchase, consumption and portfolio investment strategies for a wage earner under an uncertain lifetime, and two approaches, namely, the dynamic programming technique and the martingale technique, were used to analyze the model.
Abstract: In this dissertation, we set up a new quantitative model to investigate the problem of optimal life insurance purchase, consumption and portfolio investment strategies for a wage earner under an uncertain lifetime. We use two approaches, namely, the dynamic programming technique and the martingale technique, to analyze our model. First, by using the dynamic programming technique, we derive HJB equations for our problem and obtain explicit solutions for CRRA utility functions with or without subsistence requirements. Then we study the demand of life insurance by examining our explicit solutions and doing numerical experiments. Since explicit solutions to HJB equations are very rare, we develop a numerical method which is Makov Chain Approximation combining with logarithmic transformations. This numerical method is stable and easy-to-implement for our problem. Second, by using the martingale technique, we set up the mathematical framework for our model in a very general setting. A series of fundamental results are found, for example, the bankruptcy condition, the characteristic of admissible consumption, insurance premium and portfolio strategies, existence of optimality, etc.
37 citations
TL;DR: In this article, the authors consider the problem of minimizing both fixed and proportional transaction costs while simultaneously minimizing the tracking error with respect to a specified, target asset mix, and derive an explicit solution for the two-asset case, and use this to provide a sensitivity analysis.
Abstract: This paper studies the asset allocation problem of optimally tracking a target mix of asset categories when there are transaction costs. We consider the trading strategy for an investor who is trying to minimize both fixed and proportional transaction costs while simultaneously minimizing the tracking error with respect to a specified, target asset mix. We use imupulse control theory in a continuous-time, dynamic setting to deal with this problem in a general and analytical way, showing that the optimal trading strategy can be characterized in terms of a quasi-variational inequality. We derive an explicit solution for the two-asset case, and we use this to provide a sensitivity analysis, showing how the optimal strategy depends upon individual input parameters. We also use some theory for one-dimensional diffusion processes to derive analytical expressions for various measures of performance such as the average time between transactions.
37 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