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.     

Asset allocation,sustainable withdrawal,longevity risk and non-exponential discounting

The present paper studies an optimal withdrawal and investment problem for a retiree who is interested in sustaining her retirement consumption above a pre-specified minimum consumption level. Apparently, the withdrawal and investment policy depends substantially on the retiree’s health condition and her time preferences (subjective discount factor). We assume that the health of the retiree can worsen or improve in an unpredictable way over her lifetime and model the retiree’s mortality intensity by a stochastic process. In order to make the decision about the consumption and investment policy more realistic, we assume that the retiree applies a non-exponential discount factor (an exponential discount factor with a small amount of hyperbolic discounting) to value her future income. In other words, we consider an optimization problem by combining four important aspects: asset allocation, sustainable withdrawal, longevity risk and non-exponential discounting. Due to the non-exponential discount factor, we have to solve a time-inconsistent optimization problem. We derive a non-local HJB equation which characterizes the equilibrium optimal investment and consumption strategy. We establish the first-order expansions of the equilibrium value function and the equilibrium strategies by applying expansion techniques. The expansion is performed on the parameter controlling the degree of discounting in the hyperbolic discounting that is added to the exponential discount factors. The first-order equilibrium investment and consumption strategies can be calculated in a feasible way by solving PDEs.     

Consumption and portfolio rules for time-inconsistent investors

This paper extends the classical consumption and portfolio rules model in continuous time [Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: The continuous time case. Review of Economics and Statistics 51, 247–257, Merton, R.C., 1971. Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory 3, 373–413] to the framework of decision-makers with time-inconsistent preferences. The model is solved for different utility functions for both, naive and sophisticated agents, and the results are compared. In order to solve the problem for sophisticated agents, we derive a modified HJB (Hamilton–Jacobi–Bellman) equation. It is illustrated how for CRRA functions within the family of HARA functions (logarithmic and power utilities) the optimal portfolio rule does not depend on the discount rate, but this is not the case for a general utility function, such as the exponential (CARA) utility function.     

OPTIMAL CONTROL OF MARKOVIAN SWITCHING SYSTEMS WITH APPLICATIONS TO PORTFOLIO DECISIONS UNDER INFLATION

This article is concerned with a class of control systems with Markovian switching,in which an ltd formula for Markov-modulated processes is derived.Moreover,an optimal control law satisfying the generalized Hamilton-Jacobi-Bellman(HJB) equation with Markovian switching is characterized.Then,through the generalized HJB equation,we study an optimal consumption and portfolio problem with the financial markets of Markovian switching and inflation.Thus,we deduce the optimal policies and show that a modified Mutual Fund Theorem consisting of three funds holds.Finally,for the CRRA utility function,we explicitly give the optimal consumption and portfolio policies.Numerical examples are included to illustrate the obtained results.     

Consumption,investment and life insurance strategies with heterogeneous discounting

In this paper we analyze how the optimal consumption, investment and life insurance rules are modified by the introduction of a class of time-inconsistent preferences. In particular, we account for the fact that an agent’s preferences evolve along the planning horizon according to her increasing concern about the bequest left to her descendants and about her welfare at retirement. To this end, we consider a stochastic continuous time model with random terminal time for an agent with a known distribution of lifetime under heterogeneous discounting. In order to obtain the time-consistent solution, we solve a non-standard dynamic programming equation. For the case of CRRA and CARA utility functions we compare the explicit solutions for the time-inconsistent and the time-consistent agent. The results are illustrated numerically.     

跳扩散最优控制的随机最大值原理及在金融中的应用

讨论了由金融市场中投资组合和消费选择问题引出的一类最优控制问题,投资者的期望效用是常数相对风险厌恶(CRRA)情形.在跳扩散框架下,利用古典变分法得到了一个局部随机最大值原理.结果应用到最优投资组合和消费选择策略问题,得到了状态反馈形式的显式最优解.     

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.     

Optimal impulse control of a portfolio with a fixed transaction cost

The aim of this work is to investigate a portfolio optimization problem in presence of fixed transaction costs. We consider an economy with two assets: one risky, modeled by a geometric Brownian motion, and one risk-free which grows at a certain fixed rate. The agent is fully described by his/her utility function and the objective is to maximize the expected utility from the liquidation of wealth at a terminal date. We deal with different forms of utility functions (power, logarithmic and exponential utility), describing in each case how the fixed transaction costs influence the agent’s behavior. We show when it is optimal to recalibrate his/her portfolio and which are the best adjusted portfolios. We also analyze how the optimal strategy is influenced by the risk-aversion, as well as other model parameters.     

Optimal portfolio, consumption and retirement decision under a preference change

We investigate an optimal portfolio, consumption and retirement decision problem in which an economic agent can determine the discretionary stopping time as a retirement time with constant labor wage and disutility. We allow the preference of the agent to be changed before and after retirement. It is assumed that the agent's coefficient of relative risk aversion becomes higher after retirement. Under a constant relative risk aversion (CRRA) utility function, we obtain the optimal policies in closed-forms using martingale methods and variational inequality methods. We give some numerical results of the optimal policies. We also consider the relation between the level of disutility and the labor wage with the optimal retirement wealth level.     

Optimal investment, consumption and retirement choice problem with disutility and subsistence consumption constraints

In this paper we consider a general optimal consumption-portfolio selection problem of an infinitely-lived agent whose consumption rate process is subject to subsistence constraints before retirement. That is, her consumption rate should be greater than or equal to some positive constant before retirement. We integrate three optimal decisions which are the optimal consumption, the optimal investment choice and the optimal stopping problem in which the agent chooses her retirement time in one model. We obtain the explicit forms of optimal policies using a martingale method and a variational inequality arising from the dual function of the optimal stopping problem. We treat the optimal retirement time as the first hitting time when her wealth exceeds a certain wealth level which will be determined by a free boundary value problem and duality approaches. We also derive closed forms of the optimal wealth processes before and after retirement. Some numerical examples are presented for the case of constant relative risk aversion (CRRA) utility class.     

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.     

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.     

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．     

一类终端财富期望效用最大化问题: 通货膨胀情形

本文研究了一类受通货膨胀影响的终端财富期望效用最大化问题.对常数相对风险厌恶(CRRA) 情形的效用函数,用直接构造的方法得到了代理人的显式最优投资策略和最大期望效用,并给出其经济含义. 该思想来自线性二次最优控制问题中的完全平方技术.根据股票价格和通货膨胀率的历史数据,我们用SAS软件估计出模型中参数的近似值,并给出代理人的最优投资策略和最大期望效用.     

Dynamic portfolio selection with market impact costs

This paper concerns optimal dynamic portfolio choice with quadratic utility when there are market impact costs. The optimal policy is difficult to characterize, so we look instead for sub-optimal policies. Our proposed suboptimal policy solves a tractable dynamic portfolio choice problem where the cost of trading is captured in the objective instead of the price dynamics. A multiple time scale asymptotic expansion shows that our proposed policy has sensible structural properties, while numerical experiments show promising performance and robustness properties.     

Utility Indifference Pricing: A Time Consistent Approach

Abstract

This article considers the optimal portfolio selection problem in a dynamic multi-period stochastic framework with regime switching. The risk preferences are of exponential (CARA) type with an absolute coefficient of risk aversion that changes with the regime. The market model is incomplete and there are two risky assets: tradable and non-tradable. In this context, the optimal investment strategies are time inconsistent. Consequently, the subgame perfect equilibrium strategies are considered. The utility indifference ask price of a contingent claim written on the risky assets is computed through an indifference valuation algorithm. By running numerical experiments, we examine how this price varies in response to changes in model parameters.     

Asset allocation with distorted beliefs and transaction costs

In this paper we study the problem of the optimal portfolio selection with transaction costs for a decision-maker who is faced with Knightian uncertainty. The decision-maker’s portfolio consists of one risky and one risk-free asset, and we assume that the transaction costs are proportional to the traded volume of the risky asset. The attitude to uncertainty is modeled by the Choquet expected utility. We derive optimal strategies and bounds of the no-transaction region for both optimistic and pessimistic decision-makers. The no-transaction region of a pessimistic investor is narrower and its bounds lie closer to the origin than that of an optimistic trader. Moreover, under the Choquet expected utility the structure of the no-transaction region is not necessarily a closed interval as it is under the standard expected utility model.     

Portfolio selection in stochastic markets with exponential utility functions

We consider the optimal portfolio selection problem in a multiple period setting where the investor maximizes the expected utility of the terminal wealth in a stochastic market. The utility function has an exponential structure and the market states change according to a Markov chain. The states of the market describe the prevailing economic, financial, social and other conditions that affect the deterministic and probabilistic parameters of the model. This includes the distributions of the random asset returns as well as the utility function. The problem is solved using the dynamic programming approach to obtain the optimal solution and an explicit characterization of the optimal policy. We also discuss the stochastic structure of the wealth process under the optimal policy and determine various quantities of interest including its Fourier transform. The exponential return-risk frontier of the terminal wealth is shown to have a linear form. Special cases of multivariate normal and exponential returns are disussed together with a numerical illustration.     

A DIRECT METHOD IN OPTIMAL PORTFOLIO AND CONSUMPTION CHOICE

In this paper, we use a direct method to solve the optimal portfolio and consumption choice problem in the security market for a specific case, in which the utility function is of a given homogenous form, i.e. the so-called CRRA case. The idea comes from the completion technique ever used in LQ optimal control.     

Stochastic differential portfolio games for an insurer in a jump-diffusion risk process

We discuss an optimal portfolio selection problem of an insurer who faces model uncertainty in a jump-diffusion risk model using a game theoretic approach. In particular, the optimal portfolio selection problem is formulated as a two-person, zero-sum, stochastic differential game between the insurer and the market. There are two leader-follower games embedded in the game problem: (i) The insurer is the leader of the game and aims to select an optimal portfolio strategy by maximizing the expected utility of the terminal surplus in the “worst-case” scenario; (ii) The market acts as the leader of the game and aims to choose an optimal probability scenario to minimize the maximal expected utility of the terminal surplus. Using techniques of stochastic linear-quadratic control, we obtain closed-form solutions to the game problems in both the jump-diffusion risk process and its diffusion approximation for the case of an exponential utility.