Lose oneself in comparison: An investment and consumption game between two agents

We study an investment and consumption model with two agents. Each agent may derive extra utilities (disutilities) from positive (negative) outcomes of comparisons between her and the other agent’s consumption levels. In the unique Nash equilibrium, comparison induces the more (less) risk averse agent to invest more aggressively (conservatively) and the more (less) patient agent to increase consumption earlier (later). Adopting these distorted policies can be costly when agents’ real welfare is measured by their absolute consumption levels.     

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.     

An optimal investment and consumption model with stochastic returns

We consider a financial market consisting of a risky asset and a riskless one, with a constant or random investment horizon. The interest rate from the riskless asset is constant, but the relative return rate from the risky asset is stochastic with an unknown parameter in its distribution. Following the Bayesian approach, the optimal investment and consumption problem is formulated as a Markov decision process. We incorporate the concept of risk aversion into the model and characterize the optimal strategies for both the power and logarithmic utility functions with a constant relative risk aversion (CRRA). Numerical examples are provided that support the intuition that a higher proportion of investment should be allocated to the risky asset if the mean return rate on the risky asset is higher or the risky asset return rate is less volatile. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimal consumption and portfolio policies with the consumption habit constraints and the terminal wealth downside constraints

In this paper, we consider the optimal consumption and portfolio policies with the consumption habit constraints and the terminal wealth downside constraints, that is, here the consumption rate is greater than or equal to some nonnegative process, and the terminal wealth is no less than some positive constant. Using the martingale approach, we get the optimal consumption and portfolio policies.     

Optimal strategies for asset allocation and consumption under stochastic volatility

Selecting optimal asset allocation and consumption strategies is an important, but difficult, topic in modern finance. The dynamics is governed by a nonlinear partial differential equation. Stochastic volatility adds further complication. Even to obtain a numerical solution is challenging. Here, we develop a closed-form approximate solution. We show that our theoretical predictions for the optimal asset allocation strategy and the optimal consumption strategy are in surprisingly good agreement with the results from full numerical computations.     

Optimal investment,consumption and timing of annuity purchase under a preference change

In this paper, we study the optimal investment and consumption strategies for a retired individual who has the opportunity of choosing a discretionary stopping time to purchase an annuity. We assume that the individual receives a fixed annuity income and changes his/her preference after paying a fixed cost for annuitization. By using the martingale method and the variational inequality method, we tackle this problem and obtain the optimal strategies and the value function explicitly for the case of constant force of mortality and constant relative risk aversion (CRRA) utility function.     

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.     

AN OPTIMAL INVESTMENT／CONSUMPTIONPROBLEM WITH HIGHER BORROWING RATE

In this paper, optimal investment and consumption decisions for an optimal choiceproblem in infinite borizon are considered, for an investor who has available a bank account anda stock whose price is a log normal diffusion. The bank pays at an interest rate r for any de-posit, and takes at a larger rate / for any loan. As in the paper of Xu Wensheng and ChenShuping in JAMS(B), where an analogous problem in finite horizon is studied, optimal strategies are obtained via Hamilton-Jacobi-Bellman (ladE) equation which is derived from dynamic c1-programming principle. For the specific HARA case, i.e. U(t,c)=e^-βtc^1-R/1-R, this paper getsthe optimal consumption and optimal investment in the form of c^‘1 =β -^-g/Rwi and π^‘1= b -- γ / Rσ^2wr, with γ1,=max{γ,min{γ‘,b--Rσ^2‘} },^-g=(1--R)[γ (b-γ)^2/2Rσ^2]. This result coincides with the classical one under condition γ‘ ≡γ.     

On consumption/investment problems with long-term time-average utilities

We study consumption/investment problems with long-term time-average utilities. The associated Hamilton-Jacobi-Bellman equation can be solved under some regularity conditions of utility rate function, and the optimal portfolio and consumption-rates are exhibited in explicit forms. An application to the optimization problem with finite horizon is also given     

Optimal portfolio choice for an insurer with loss aversion

The problem of optimal investment for an insurance company attracts more attention in recent years. In general, the investment decision maker of the insurance company is assumed to be rational and risk averse. This is inconsistent with non fully rational decision-making way in the real world. In this paper we investigate an optimal portfolio selection problem for the insurer. The investment decision maker is assumed to be loss averse. The surplus process of the insurer is modeled by a Lévy process. The insurer aims to maximize the expected utility when terminal wealth exceeds his aspiration level. With the help of martingale method, we translate the dynamic maximization problem into an equivalent static optimization problem. By solving the static optimization problem, we derive explicit expressions of the optimal portfolio and the optimal wealth process.     

Optimal consumption and portfolio selection with quadratic utility and a subsistence consumption constraint

In this article, we analyze the optimal consumption and investment policy of an agent who has a quadratic felicity function and faces a subsistence consumption constraint. The agent's optimal investment in the risky asset increases linearly for low wealth levels. Risk taking continues to increase at a decreasing rate for wealth levels higher than subsistence wealth until it hits a maximum at a certain wealth level, and declines for wealth levels above this threshold. Further, the agent has a bliss level of consumption, since if an agent consumes more than this level she will suffer utility loss. Eventually her risk taking becomes zero at a wealth level which supports her bliss consumption.     

Symmetry-based solution of a model for a combination of a risky investment and a riskless investment

Benth and Karlsen [F.E. Benth, K.H. Karlsen, A note on Merton's portfolio selection problem for the Schwartz mean-reversion model, Stoch. Anal. Appl. 23 (2005) 687-704] treated a problem of the optimisation of the selection of a portfolio based upon the Schwartz mean-reversion model. The resulting Hamilton-Jacobi-Bellman equation in 1+2 dimensions is quite nonlinear. The solution obtained by Benth and Karlsen was very ingenious. We provide a solution of the problem based on the application of the Lie theory of continuous groups to the partial differential equation and its associated boundary and terminal conditions.     

An optimal time-management policy for labor supply and consumption decisions

The paper studies decisions regarding labor supply, consumption and cash savings that maximize the total utility of consumption and non-labor time of an individual. A new dynamic model that studies the relationship between time and wealth, with the goal of determining the optimal fractions of time an individual should allocate to three different types of activities: labor, consumption, and recreation, is developed. We analyze the optimality conditions, construct optimal solutions, and discuss their properties with respect to the division of time for two types of individuals, who can be characterized as consumption-averse and consumption-seeking. A consumption-averse individual does not enjoy shopping time, but when shopping spends money intensively. A consumption-seeking person does enjoy shopping time and spends money slowly. The difference in decision making of the two types of individuals is illustrated by comparing the optimal division of time at a singular regime that takes place over a substantial part of the planning horizon.     

Optimal investment,consumption and proportional reinsurance under model uncertainty

This paper considers the optimal investment, consumption and proportional reinsurance strategies for an insurer under model uncertainty. The surplus process of the insurer before investment and consumption is assumed to be a general jump–diffusion process. The financial market consists of one risk-free asset and one risky asset whose price process is also a general jump–diffusion process. We transform the problem equivalently into a two-person zero-sum forward–backward stochastic differential game driven by two-dimensional Lévy noises. The maximum principles for a general form of this game are established to solve our problem. Some special interesting cases are studied by using Malliavin calculus so as to give explicit expressions of the optimal strategies.     

Longevity-linked assets and pre-retirement consumption/portfolio decisions

We solve the consumption/investment problem of an agent facing a stochastic mortality intensity. The investment set includes a longevity-linked asset, as a derivative on the force of mortality. In a complete and frictionless market, we derive a closed form solution when the agent has Hyperbolic Absolute Risk Aversion preferences and a fixed financial horizon. Our calibrated numerical analysis on US data shows that individuals optimally invest a large fraction of their wealth in longevity-linked assets in the pre-retirement phase, because of their need to hedge against stochastic fluctuations in their remaining life-time at retirement.     

Stochastic utilities with subsistence and satiation: Optimal life insurance purchase,consumption and investment

We introduce stochastic utilities such that utility of any fixed amount of interest is a stochastic process or random variable. Also, there exist stochastic (or random) subsistence and satiation levels associated with stochastic utilities. Then, we consider optimal consumption, life insurance purchase and investment strategies to maximize the expected utility of consumption, bequest and pension with respect to stochastic utilities. We use the martingale approach to solve the optimization problem in two steps. First, we solve the optimization problem with an equality constraint which requires that the present value of consumption, bequest and pension is equal to the present value of initial wealth and income stream. Second, if the optimization problem is feasible, we obtain the explicit representations of the replicating life insurance purchase and portfolio strategies. As an application of our general results, we consider a family of stochastic utilities which have hyperbolic absolute risk aversion (HARA).     

一类证券市场中投资组合及消费选择的最优控制问题

研究一类证券市场中投资组合及消费选择的最优控制问题．在随机干扰源相互关联情形下,运用动态规划方法,对一类典型的效用函数CRRA(Constant Relative Risk Aversion,常数相对风险厌恶)情形,得到了最优投资组合及消费选择的显式解,并给出了最优解的经济解释和关于部分参数的灵敏度分析．     

Optimal investment and consumption models with non-linear stock dynamics

We study a generalization of the Merton's original problem of optimal consumption and portfolio choice for a single investor in an intertemporal economy. The agent trades between a bond and a stock account and he may consume out of his bond holdings. The price of the bond is deterministic as opposed to the stock price which is modelled as a diffusion process. The main assumption is that the coefficients of the stock price diffusion are arbitrary nonlinear functions of the underlying process. The investor's goal is to maximize his expected utility from terminal wealth and/or his expected utility of intermediate consumption. The individual preferences are of Constant Relative Risk Aversion (CRRA) type for both the consumption stream and the terminal wealth. Employing a novel transformation, we are able to produce closed form solutions for the value function and the optimal policies. In the absence of intermediate consumption, the value function can be expressed in terms of a power of the solution of a homogeneous linear parabolic equation. When intermediate consumption is allowed, the value function is expressed via the solution of a non-homogeneous linear parabolic equation.     

Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem

This paper concerns optimal investment problem of a CRRA investor who faces proportional transaction costs and finite time horizon. From the angle of stochastic control, it is a singular control problem, whose value function is governed by a time-dependent HJB equation with gradient constraints. We reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. This enables us to make use of the well-developed theory of obstacle problem to attack the problem. The C2,1 regularity of the value function is proven and the behaviors of the free boundaries are completely characterized.