Investment-consumption Model Research Articles

An optimal investment consumption model for retirees with no health insurance

Retirees meet a number of problems as they are growing older which needs persistent attention. Hence, without a doubt, the outcomes of the financial markets influence the choices that people make when nearing retirement. In our model, the stock price dynamics follow Geometric Brownian motion (GBM) and our goal was to optimize the expected discounted utility of consumption and terminal wealth whilst considering health expenses. The investment return process comprises risk free asset and risky assets, and the health expenses. We choose power utility functions where comprehensive solutions for Hyperbolic Absolute Risk Aversion (HARA) utility functions are obtained and optimal investment, consumption and health expenditure strategies are derived by applying dynamic programming and variable change technique on the Hamilton-Jacobi-Bellman (HJB) equations. In our numerical results it showed various effects of some economic and market parameters on the optimal investment, consumption and health expense strategies. The inflation price market risk governs the amount invested in stock, bond and also how much to be put in health to sustain a given period of the retiree's lifetime. As the health welfare rate R increases, the proportion of wealth invested in the stock increases. We also investigated the effects of the high correlation coefficients and low correlation coefficients on consumption and income rate respectively. As the constant variance discounting coefficient increases, seasoned enterprise annuity retirees decrease their allocation to the risky assets. Finally, a numerical example is presented to depict the effects of financial parameters on the optimal investment strategy with health expenditure.

Existence, Blow-Up, and Blow-Up Rate of Weak Solution to Fourth-Order Non-Newtonian Polytropic Variation-Inequality Arising from Consumption-Investment Models

This article obtains the conditions for the existence and nonexistence of weak solutions for a variation-inequality problem. This variational inequality is constructed by a fourth-order non-Newtonian polytropic operator which is receiving much attention recently. Under the proper condition of the parameter, the existence of a solution is proved by constructing the initial boundary value problem of an ellipse by time discretization and some elliptic equation theory. Under the opposite parameter condition, we analyze the nonexistence of the solution. The results show that the weak solution will blow up in finite time. Finally, we give the blow-up rate and the upper bound of the blow-up time.

Research and Strategy Analysis of Asset Return Optimization in Internet-based Financial Engineering

The article introduces the application advantages of financial engineering in Internet financial risk management and control in detail, finds out the main risks in Internet financial innovation through investigation and research, and puts forward effective measures to optimize risk control, such as building a risk management and control system, enhancing the decision-making power of Internet finance, scientific application of derivatives, improving the supervision and management system and strengthening credit evaluation using the saddle method and stochastic integral theory, the optimal wealth level and the corresponding optimal asset portfolio strategy of the portfolio insurer are solved ‚ And compare the differences and similarities between the optimal investment strategy of the optimal portfolio insurance model and the optimal investment consumption model.

A consumption-investment model with state-dependent lower bound constraint on consumption

This paper studies a life-time consumption-investment problem under the Black-Scholes framework, where the consumption rate is subject to a lower bound constraint that linearly depends on the investor's wealth. It is a stochastic control problem with state-dependent control constraint to which the standard stochastic control theory cannot be directly applied. We overcome this by transforming it into an equivalent stochastic control problem in which the control constraint is state-independent so that the standard theory can be applied. We give an explicit optimal consumption-investment strategy when the constraint is homogeneous. When the constraint is non-homogeneous, it is shown that the value function is third-order continuously differentiable by a differential equation approach, and a feedback form optimal consumption-investment strategy is provided. According to our findings, if the investor is concerned with long-term more than short-term consumption, then she should always consume as few as possible; otherwise, she should consume optimally when her wealth is above a threshold, and consume as few as possible when her wealth is below the threshold.

A Stochastic Control Model of Investment and Consumption with Applications to Financial Economics

This study considers a stochastic control model in which an economic unit has productive capital and liabilities in the form of debt. The worth of capital changes over time through investment and random Brownian fluctuations in the unit price of capital. Income from production is also subject to the random Brownian fluctuations. The existence of the solutions to the associated Hamilton Jacobi Bellman equation for this model is established and the optimal policies are characterized. The optimal advertising rate as a function of the market share, the optimal consumption rate and the fraction of the wealth invested in stock at any time are obtained. The worth of the capital and the optimal consumption policy are derived for the stochastic optimal investment consumption model associated with the Hamilton Jacobi Bellman equation. Analysis and numerical simulations are then presented.

Portfolio selection with transaction costs and default risk

PurposeThe purpose of this paper is to investigate the effect of default risk and transaction costs on the investor’s asset allocation and the liquidity premium. More precisely, it aims at answering the following question: can default risk generate a first-order effect on the investor’s asset allocation and a liquidity premium of the same order of magnitude as transaction costs?Design/methodology/approachThe author proposes a very simple consumption-investment model in which an infinitely lived investor allocates her wealth between a risky asset and a riskless security, and incurs in proportional transaction costs when exchanging them. In addition, the risky asset may default at some random time, thus reducing the available wealth of the agent. Two different scenarios of default risk are considered. In the total default scenario, the value of the risky asset drops to zero when default occurs, whereas, in the partial default case, the proceeds from the liquidation of the risky asset amount to 50 percent of its value.FindingsThe paper shows that default risk can generate a first-order effect on the investor’s asset allocation. On the contrary, the liquidity premium is one order of magnitude smaller than the transaction costs, implying that the additional source of risk determined by the possibility of default is not able to generate a first-order effect on asset pricing.Originality/valueTo the author knowledge, this is the first paper that investigates the interaction of default risk and transaction costs on the investor’s asset allocation and its effects on the liquidity premium.

An optimal consumption-investment model with constraint on consumption

A continuous-time consumption-investment model with constraint is considered for a small investor whose decisions are the consumption rate and the allocation of wealth to a risk-free and a risky asset with logarithmic Brownian motion fluctuations. The consumption rate is subject to an upper bound constraint which linearly depends on the investor's wealth and bankruptcy is prohibited. The investor's objective is to maximize total expected discounted utility of consumption over an infinite trading horizon. It is shown that the value function is (second order) smooth everywhere but a unique possibility of (known) exception point and the optimal consumption-investment strategy is provided in a closed feedback form of wealth, which in contrast to the existing work does not involve the value function. According to this model, an investor should take the same optimal investment strategy as in Merton's model regardless his financial situation. By contrast, the optimal consumption strategy does depend on the investor's financial situation: he should use a similar consumption strategy as in Merton's model when he is in a bad situation, and consume as much as possible when he is in a good situation.

SI “Deterministic and Stochastic Variational Principles and Applications”. December 2015

Variational methods are important tools for deriving optimality conditions and corresponding algorithms for solving optimization problems. Ekeland’s variational principle is a deep assertion about the existence of an exact solution of a perturbed optimization problem in a neighborhood of an approximate solution of the original problem. The aim of this special issue was to discuss new variational approaches from the theoretical as well as computational point of view. Furthermore, we are interested in applications of the variational methods for deterministic and stochastic models, especially in approximation theory and optimal control. In the special issue the importance of variational methods in optimization and stochastics is described.We present newdevelopments on the field of deterministic and stochastic variational methods including applications in mathematics and economics. Most of the papers in the special issue are dealing with optimal control and state estimates problems in the deterministic as well as stochastic case and maximum principles (Abdelmadjid Abba: On Mean-field Partial Information Maximum Principle of Optimal Control for Stochastic Systems with Levy Processes, Vo Anh: Least-Squares Estimation of Multifractional Random Fields in a Hilbert-Valued Context, Brigitte Breckner: Multiple solutions of Dirichlet problems on the Sierpinski gasket, Ioana Ciotir: A variational approach to Neumann stochastic semi-linear equations modelling the thermostatic control, CsabaVarga, Hannelore Lisei: Amultiplicity result for a class of elliptic problems on a compact Riemannian manifold, Tina Engler: On Investment Consumption Modeling with Jump Process Extensions for Productive Sectors, Diana

Continuous time portfolio optimization

This paper presents dynamic portfolio model based on the Merton's optimal investment-consumption model, which combines dynamic synthetic put option using risk-free and risky assets. This paper is extended version of methodological paper published by Yuan Yao (2012). Because of the long history of the development of foreign financial market, with a variety of financial derivatives, the study on theory or empirical analysis of portfolio insurance focused on how best can portfolio strategies be used in minimizing risk and market volatility. In this paper, stock and risk-free assets are used to replicate options and to establish a new dynamic model to analyze the implementation of the dynamic process of investors' actions using dynamic replication strategy. Our results show that investors' optimal strategies of portfolio are not dependent on their wealth, but are dependent on market risk and this new methodology is broaden in compare to paper of Yuan Yao (2012).

Possibilistic individual multi-period consumption–investment models

In this paper, we consider a multi-period consumption–investment problem with partially known information for an individual. We emphasize the fact that for an individual there is one and only one result will appear for his/her consumption–investment decision in each period. Different from the existing multi-period consumption–investment models which basically maximize the expected utility, we utilize the one-shot decision theory for individual multi-period consumption–investment problems. In each period, the individual chooses one state (scenario) for each strategy with considering the satisfaction of the outcome and its possibility. The selected state is called the focus point. The multi-period consumption–investment decision models with twelve types of focus points are built to maximize the sum of discounted consumption over the whole process (lifetime) for twelve types of individuals and economic insights are gained by the theoretical analysis. The proposed models are scenario-based decision models which provide a fundamental alternative to analyze individual multi-period consumption–investment behavior.

Critical Illness Insurance in Life Cycle Portfolio Problems

I analyze a critical illness insurance in a consumption-investment model over the life cycle. I solve a model with stochastic mortality risk and health shock risk numerically. These shocks are interpreted as critical illness and can negatively affect the expected remaining lifetime, the health expenses, and the income. In order to hedge the health expense effect of a shock, the agent has the possibility to contract a critical illness insurance. My results highlight that the critical illness insurance is strongly desired by the agents. With an insurance profit of 20%, nearly all agents contract the insurance in the working stage of the life cycle and more than 50% of the agents contract the insurance during retirement. With an insurance profit of 200%, still nearly all working agents contract the insurance, whereas there is little demand in the retirement stage.

On Investment Consumption Modeling with Jump Process Extensions for Productive Sectors

We consider the optimal investment and consumption behavior of an agricultural sector that generates income from productive capital and interest payments and that has expenditures due to interest payments for debt. The productivity of capital, the interest rate, and the capital depreciation rate are uncertain. We adapt an existing model from the literature and generalize it to a stochastic depreciation of capital stock. Moreover, we assume a discontinuous evolution of the productivity of capital. The uncertainties are modeled as stochastic processes with discontinuous paths using well-known Wiener processes and simple Poisson processes. Accordingly, the investor's net wealth follows a jump diffusion stochastic differential equation. The goal of maximizing the utility of consumption over an infinite time horizon leads to a stochastic optimal control problem. We choose a hyperbolic absolute risk aversion utility function. To determine the optimal investment and consumption rate, the associated Hamilton---Jacobi---Bellman equation is solved by a separation method, and a verification theorem is applied.

Markov Decision Processes on Borel Spaces with Total Cost and Random Horizon

This paper deals with Markov Decision Processes (MDPs) on Borel spaces with possibly unbounded costs. The criterion to be optimized is the expected total cost with a random horizon of infinite support. In this paper, it is observed that this performance criterion is equivalent to the expected total discounted cost with an infinite horizon and a varying-time discount factor. Then, the optimal value function and the optimal policy are characterized through some suitable versions of the Dynamic Programming Equation. Moreover, it is proved that the optimal value function of the optimal control problem with a random horizon can be bounded from above by the optimal value function of a discounted optimal control problem with a fixed discount factor. In this case, the discount factor is defined in an adequate way by the parameters introduced for the study of the optimal control problem with a random horizon. To illustrate the theory developed, a version of the Linear-Quadratic model with a random horizon and a Logarithm Consumption-Investment model are presented.

Optimization of Dynamic Portfolio Insurance Model

This paper establishes a dynamic portfolio insurance model under the condition of continuous time, based on Meton's optimal investment-consumption model, which combined the method of replicating dynamic synthetic put option using risk-free and risk assets. And it transfers the problem of investor's individual intertemporal dynamic portfolio insurance decision into a problem of static utility maximization under condition of continuous time, and give the optimal capital combination strategies corresponding to the optimal wealth level of the portfolio insurers, and compares the difference of strategies between this model and Merton model. The conclusions show that investors' optimal strategies of portfolio insurance are not dependent on their wealth, but market risk. That is to say, the higher the risk is, the more the demand of portfolio insurance is.

Optimal portfolio model under compound jump processes

PurposeThe purpose of this paper is to research the optimal portfolio proportion for the optimal investment model and the optimal consumption investment strategies for the optimal consumption investment model under compound‐jump processes.Design/methodology/approachTraditionally, the price of risky security or asset is often modeled as geometric Brownian motion. However, the analysis of stock price evolution reveals sudden and rare breaks logically accounted for by exogenous events on information. It is natural to model such behavior by means of a point process, or, more simply, by a Poisson process, which has jumps of constant size occurring at rare and unpredictable intervals. Assume that the price of risky security stock is modeled by a compound‐jump process, the renew process theory is chosen to solve the optimal investment model, the HJB equation is chosen for the optimal consumption investment model.FindingsDerive the analytical optimal portfolio proportion for the reduction model of optimal investment. The optimal consumption investment strategies are given by some equations for the optimal consumption investment model.Research limitations/implicationsAccessibility and availability of data are the main limitations which model will be applied.Practical implicationsThe results obtained in this paper could be used as a guide to actual portfolio management.Originality/valueThe new approach for the optimal portfolio model under compound‐jump processes. The paper is aimed at actual portfolio managers.

Multi-asset investment-consumption model with transaction costs

In this paper, we consider the multi-asset optimal investment-consumption model: a riskless asset and d risky assets. when the initial time is t ⩾ 0 , for a proportional transaction costs and discount factors, we proof that the value function of the model is a unique viscosity solution of a Hamilton–Jacobi–Bellman (HJB) equations.

Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs

We investigate an infinite horizon investment-consumption model in which a single agent consumes and distributes her wealth between a risk-free asset (bank account) and several risky assets (stocks) whose prices are governed by Lévy (jump-diffusion) processes. We suppose that transactions between the assets incur a transaction cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption under Hindy-Huang-Kreps intertemporal preferences. This portfolio optimisation problem is formulated as a singular stochastic control problem and is solved using dynamic programming and the theory of viscosity solutions. The associated dynamic programming equation is a second order degenerate elliptic integro-differential variational inequality subject to a state constraint boundary condition. The main result is a characterization of the value function as the unique constrained viscosity solution of the dynamic programming equation. Emphasis is put on providing a framework that allows for a general class of Lévy processes. Owing to the complexity of our investment-consumption model, it is not possible to derive closed form solutions for the value function. Hence, the optimal policies cannot be obtained in closed form from the first order conditions for the dynamic programming equation. Therefore, we have to resort to numerical methods for computing the value function as well as the associated optimal policies. In view of the viscosity solution theory, the analysis found in this paper will ensure the convergence of a large class of numerical methods for the investment-consumption model in question.

Optimal investment consumption model with a higher interest rate for borrowing

This paper considers a consumption and investment decision problem with a higher interest rate for borrowing as well as the dividend rate. Wealth is divided into a riskless asset and risky asset with logrithmic Brownian motion price fluctuations. The stochastic control problem of maximizating expected utility from terminal wealth and consumption is studied. Equivalent conditions for optimality are obtained. By using duality methods the existence of optimal portfolio consumption is proved, and the explicit solutions leading to feedback formulae are derived for deteministic coefficients.

On an Investment-Consumption Model with Transaction Costs

This paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and n risky assets whose prices are log-normal diffusions. We suppose that transactions between the assets incur a cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption. Dynamic programming leads to a variational inequality for the value function. Existence and uniqueness of a viscosity solution are proved. The variational inequality is solved by using a numerical algorithm based on policies, iterations, and multigrid methods. Numerical results are displayed for $n = 1$ and $n = 2$.

Uncertainty, liquidity, and housing choices

This paper analyzes the housing choices of illiquid households facing uncertain future housing prices. It shows that illiquidity in the housing consumption-investment model of Henderson and Ioannides ( The American Economic Review, 1983, 73, no. 1, 98–113) causes a conflict between the housing consumption and investment motives. Consequently, (1) a higher permanent income may not make housing investment and home owning more attractive, (2) the income path affects housing choices, and (3) greater expected housing price appreciation, or lesser uncertainty as to the future price of housing, does not necessarily encourage housing investment. These results indicate the importance of controlling for household liquidity in empirical housing studies.