A consumption–investment problem with heterogeneous discounting

We analyze a stochastic continuous time model in finite horizon in which the agent discounts the instantaneous utility function and the final function at constant but different discount rates of time preference. Within this framework we can model problems in which, when the time t$t$ approaches to the final time, the valuation of the final function increases compared with previous valuations. We study a consumption and portfolio rules problem for CRRA and CARA utility functions for time-consistent agents, and we compare the different equilibria with the time-inconsistent solutions. The introduction of random terminal time is also discussed. Differences with both the mathematical treatment and agent’s behavior in the case of hyperbolic discounting are stressed.     

Mean-variance versus expected utility in dynamic investment analysis

Given the existence of a Markovian state price density process, this paper extends Merton??s continuous time (instantaneous) mean-variance analysis and the mutual fund separation theory in which the risky fund can be chosen to be the growth optimal portfolio. The CAPM obtains without the assumption of log-normality for prices. The optimal investment policies for the case of a hyperbolic absolute risk aversion (HARA) utility function are derived analytically. It is proved that only the quadratic utility exhibits the global mean-variance efficiency among the family of HARA utility functions. A numerical comparison is made between the growth optimal portfolio and the mean-variance analysis for the case of log-normal prices. The optimal choice of target return which maximizes the probability that the mean-variance analysis outperforms the expected utility portfolio is discussed. Mean variance analysis is better near the mean and the expected utility maximization is better in the tails.     

Merton problem in a discrete market with frictions

We study the classical optimal investment and consumption problem of Merton in a discrete time model with frictions. Market friction causes the investor to lose wealth due to trading. This loss is modeled through a nonlinear penalty function of the portfolio adjustment. The classical transaction cost and the liquidity models are included in this abstract formulation. The investor maximizes her utility derived from consumption and the final portfolio position. The utility is modeled as the expected value of the discounted sum of the utilities from each step. At the final time, the stock positions are liquidated and a utility is obtained from the resulting cash value. The controls are the investment and the consumption decisions at each time. The utility function is maximized over all controls that keep the after liquidation value of the portfolio non-negative. A dynamic programming principle is proved and the value function is characterized as its unique solution with appropriate initial data. Optimal investment and consumption strategies are constructed as well.     

基于双曲折现的跨期消费和投资组合的一种解析

通过在默顿(1969年,1971年)的经典模型中引入Harris和Laibson(2013年)的随机双曲偏好,研究得到了针对常绝对风险厌恶效用函数的最优消费和投资组合的解析解.与默顿的结果相比,发现消费与财富尽管仍有线性关系,但其比例再也不是一个常数.投资于风险资产的比例也非固定常数,但投资于风险资产的总价值保持不变.     

An optimal job,consumption/leisure,and investment policy

In this paper we investigate an optimal job, consumption, and investment policy of an economic agent in a continuous and infinite time horizon. The agent’s preference is characterized by the Cobb–Douglas utility function whose arguments are consumption and leisure. We use the martingale method to obtain the closed-form solution for the optimal job, consumption, and portfolio policy. We compare the optimal consumption and investment policy with that in the absence of job choice opportunities.     

A stochastic portfolio optimization model with complete memory

In this article, we consider a portfolio optimization problem of the Merton’s type with complete memory over a finite time horizon. The problem is formulated as a stochastic control problem on a finite time horizon and the state evolves according to a process governed by a stochastic process with memory. The goal is to choose investment and consumption controls such that the total expected discounted utility is maximized. Under certain conditions, we derive the explicit solutions for the associated Hamilton–Jacobi–Bellman (HJB) equations in a finite-dimensional space for exponential, logarithmic, and power utility functions. For those utility functions, verification results are established to ensure that the solutions are equal to the value functions, and the optimal controls are also derived.     

Adaptive control of three continuous-time portfolio and consumption models

An economic application of adaptive control is presented using three continuous time portfolio and consumption models that are natural generalizations of a model of Merton. In these models of the wealth of an individual investor, it is assumed that the various parameters are deterministic functions of time or stochastic processes. An adaptive control problem arises for each of these models when it is assumed that the average return rate of the risky asset, which is either a deterministic function or a stochastic process, is not observed. For these models, a recursive family of estimators of the average return rate of the risky asset is given based on the observations of the wealth. These estimates are used in the control of the wealth equation.This research was partially supported by NSF Grant No. ECS-84-03286-A01 and by University of Kansas General Research Allocation No. 3806-XO-0038.     

关于无风险资产不存在时资产组合选择的研究

以往关于资产组合选择的研究大多假设市场上存在无风险资产,但无风险资产实际上是不存在的.当不存在无风险资产时,假设投资者的效用定义在消费上,消费一直是投资者财富的一个固定比例,投资者的最优资产组合由两部分组成:短视的资产组合和对冲组合.假设只有股票和债券两种风险资产,当股票和债券的风险具有负的相关性时,投资者现在会消费更多,同时也会在股票上投资更多;两者正相关时,投资者无法降低风险,会减持股票并降低当前消费;两者不相关时,投资者持有的股票权重和存在无风险资产时一样.最后,还推导出了多种资产情况下最优消费和资产组合的解析表达式.     

通胀环境下股价波动率具有奈特不确定的最优消费与投资问题

本文在通胀环境和连续时间模型假设下,研究股票价格波动率具有奈特不确定对投资者的最优消费和投资策略的影响．首先在通胀环境和股票价格波动率具有奈特不确定的条件下,建立最优消费与投资问题的随机控制数学模型,得到了最优消费与投资所满足的HJB方程,并在常相对风险厌恶效用的情形下,获得最优化问题值函数的显式解．其次在通胀环境中当股价波动率具有奈特不确定时,得到了含糊厌恶的投资者是基于股价波动率的上界作出决策,并给出了投资者的最优投资和消费策略．最后在给定参数的条件下,对所得结果进行数值模拟和经济分析．     

Optimal portfolios for DC pension plans under a CEV model

This paper studies the portfolio optimization problem for an investor who seeks to maximize the expected utility of the terminal wealth in a DC pension plan. We focus on a constant elasticity of variance (CEV) model to describe the stock price dynamics, which is an extension of geometric Brownian motion. By applying stochastic optimal control, power transform and variable change technique, we derive the explicit solutions for the CRRA and CARA utility functions, respectively. Each solution consists of a moving Merton strategy and a correction factor. The moving Merton strategy is similar to the result of Devolder et al. [Devolder, P., Bosch, P.M., Dominguez F.I., 2003. Stochastic optimal control of annunity contracts. Insurance: Math. Econom. 33, 227-238], whereas it has an updated instantaneous volatility at the current time. The correction factor denotes a supplement term to hedge the volatility risk. In order to have a better understanding of the impact of the correction factor on the optimal strategy, we analyze the property of the correction factor. Finally, we present a numerical simulation to illustrate the properties and sensitivities of the correction factor and the optimal strategy.     

Optimal consumption problems in discontinuous markets

We study an extension of Merton’s classical portfolio investment – consumption optimization problem (1969–1970) to a particular case of complete discontinuous market, with a single jump. The market consists of a non-risky asset, a ‘standard risky’ asset and a risky asset with discontinuous price dynamics (e.g. a defaultable bond or a mortality linked security). We consider three different problems of maximization of the expected utility from consumption: in the case when the investment horizon is fixed and finite, when it is finite, but possibly uncertain and when it is infinite. The innovative setting is the second one. In a general stochastic coefficients’ model, we solve the problems and we compare the three optimal consumption rates, finding quite interesting results. In the logarithmic and power utility cases, explicit solutions are provided. Furthermore, the benchmark – constant coefficients’ case is deeply investigated and a partial information setting is also studied in the uncertain time horizon case.     

一类组合受约束的最优化问题及其最优解

这篇文章中，我们建立了资产组合在受到约束时的期望效用优化问题，在我们特殊的指数效用函数下，我们发现最终的决策不依赖于具体的贴现函数，在文章的结尾部分，我们给出了几类常见约束下的最优消费和资产组合决策。     

Investment and consumption without commitment

In this paper, we investigate the Merton portfolio management problem in the context of non-exponential discounting. This gives rise to time-inconsistency of the decision-maker. If the decision-maker at time t = 0 can commit her successors, she can choose the policy that is optimal from her point of view, and constrain the others to abide by it, although they do not see it as optimal for them. If there is no commitment mechanism, one must seek a subgame-perfect equilibrium policy between the successive decision-makers. In the line of the earlier work by Ekeland and Lazrak (Preprint, 2006) we give a precise definition of equilibrium policies in the context of the portfolio management problem, with finite horizon. We characterize them by a system of partial differential equations, and establish their existence in the case of CRRA utility. An explicit solution is provided for the case of logarithmic utility. We also investigate the infinite-horizon case and provide two different equilibrium policies for CRRA utility (in contrast with the case of exponential discounting, where there is only one optimal policy). Some of our results are proved under the assumption that the discount function h(t) is a linear combination of two exponentials, or is the product of an exponential by a linear function. I. Ekeland was supported by PIMS under NSERC grant 298427-04.     

Expected utility approximation and portfolio optimisation

Classical portfolio selection problems that optimise expected utility can usually not be solved in closed form. It is natural to approximate the utility function, and we investigate the accuracy of this approximation when using Taylor polynomials. In the important case of a Merton market and power utility we show analytically that increasing the order of the polynomial does not necessarily improve the approximation of the expected utility. The proofs use methods from the theory of parabolic second-order partial differential equations. All results are illustrated by numerical examples.     

Optimal investment and consumption decision of a family with life insurance

We study an optimal portfolio and consumption choice problem of a family that combines life insurance for parents who receive deterministic labor income until the fixed time T. We consider utility functions of parents and children separately and assume that parents have an uncertain lifetime. If parents die before time T, children have no labor income and they choose the optimal consumption and portfolio with remaining wealth and life insurance benefit. The object of the family is to maximize the weighted average of utility of parents and that of children. We obtain analytic solutions for the value function and the optimal policies, and then analyze how the changes of the weight of the parents’ utility function and other factors affect the optimal policies.     

变折现率下带含糊厌恶与预期的最优投资研究

本文采用折现率为时间的函数下的递推多先验效用,研究Merton模型在带预期条件下的最优消费和投资组合决策问题,其中含糊与风险是有区别的.在幂效用函数情形下,刻画了投资者最优投资决策,表明了含糊厌恶和预期对最优投资的影响.最优投资组合决策由倒向随机微分方程和Malliavin导数导出.     

无套利期权定价模型在一般均衡框架下的一致性研究

期权定价有无套利方法和一般均衡方法两种.本文在一般均衡框架下构造了一个允许连续消费的简单经济模型,并将基于无套利方法的期权定价模型中所假定的标的证券的价格变化动态过程内生化于理性预期均衡中.在常数相对风险厌恶(CRRA)的效用函数的条件下,我们推导出Merton(1973)期权定价公式,从而证明无套利方法与均衡方法的内在一致性,而CRRA这种类型的效用函数是无套利定价模型在一般均衡框架中成立的充分条件.本文进一步将此模型在一个简单经济中扩展到m种证券的情况,也得到相似的结论.     

考虑红利支付与提前退休的最优投资组合

研究了在经济代理人通过不可逆退休时间选择来调整劳动时间框架下的最优消费和投资问题,主要考虑风险资产派发红利的情形.运用随机控制方法,求解使得消费-闲暇预期效用最大化的最优策略.最优投资组合及最优退休时刻表明,代理人在为提前退休积累财富的同时,也能最佳享受消费和闲暇所带来的快乐.     

考虑相对业绩的最优投资选择博弈问题研究

研究了具有相互作用的两个竞争机构投资者之间的离散时间最优投资选择博弈问题,每个机构投资者都考虑其竞争对手的相对业绩.机构投资者可以投资于相同的无风险资产和不同的具有相关关系的风险股票,以反映投资的资产专门化.机构投资者选择投资组合策略使得期望终端绝对财富和相对财富的效用最大.首先,定义了Nash均衡投资组合选择策略.然后,在机构投资者具有指数效用函数的假设下,得到了Nash均衡投资组合选择策略和值函数的显示表达式,分析了机构投资者之间的竞争对Nash均衡投资组合选择策略的影响.最后,通过数值计算给出了各种情况下Nash均衡投资组合选择策略和值函数与模型主要参数之间的关系.结果表明:机构投资者之间的竞争会影响其对风险的承担,投资机会集对机构投资者的Nash均衡投资组合选择策略和值函数与模型主要参数之间的关系会产生很大的影响.     

A note on the integrability of the classical portfolio selection model

We revisit the classical Merton portfolio selection model from the perspective of integrability analysis. By an application of a nonlocal transformation the nonlinear partial differential equation for the two-asset model is mapped into a linear option valuation equation with a consumption dependent source term. This result is identical to the one obtained by Cox–Huang [J.C. Cox, C.-f. Huang, Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Econom. Theory 49 (1989) 33–88], using measure theory and stochastic integrals. The nonlinear two-asset equation is then analyzed using the theory of Lie symmetry groups. We show that the linearization is directly related to the structure of the generalized symmetries.