Consumption Choice Problem Research Articles

Portfolio-consumption choice problem with voluntary retirement and consumption constraints

We study the investment–consumption choice problem with voluntary retirement and upside/downside consumption constraints in infinite horizon, which can be formulated as a two-phase mixed stochastic control problem including stopping time coupled with controls. By verification theorem, we show the strong solution of a fully nonlinear differential variational inequality, satisfying some special properties, is the value function of this problem. Then we eliminate the nonlinearity through Legendre transformation, and obtain the closed-form formulas of the optimal investment–consumption feedback strategy, the optimal retirement strategy and the value function by solving the dual equation. Since the introduction of consumption constraints, these formulas are both segmented and semi-explicit, and bring certain difficulties to the analysis of the properties of the optimal strategies and the value function. The regularity of the feedback strategy and the value function are obtained, as well as their monotonicity or non-monotonicity with respect to various parameters are discussed by some stochastic analysis methods and differential equation techniques.

Optimal consumption with Hindy–Huang–Kreps preferences under nonlinear expectations

AbstractWe study an intertemporal consumption and portfolio choice problem under Knightian uncertainty in which agent’s preferences exhibit local intertemporal substitution. We also allow for market frictions in the sense that the pricing functional is nonlinear. We prove existence and uniqueness of the optimal consumption plan, and we derive a set of sufficient first-order conditions for optimality. With the help of a backward equation, we are able to determine the structure of optimal consumption plans. We obtain explicit solutions in a stationary setting in which the financial market has different risk premia for short and long positions.

Robust consumption and portfolio choice with derivatives trading

This paper analyzes the optimal consumption and portfolio choice problem for an ambiguity-averse investor with recursive preferences. The investor has access to both the stock and derivatives markets. The stock price process follows a stochastic volatility jump-diffusion model and the investor can have different levels of ambiguity about diffusion and jump risks, respectively. We obtain an exact analytical solution for investors with unit elasticity of intertemporal substitution of consumption and an approximate solution otherwise. We find that the optimal consumption policy is more sensitive to ambiguity aversion with respect to diffusion risks than the jump risk. The optimal exposures to diffusion and jump risks are significantly affected by the corresponding ambiguity aversions in the complete market; however, the optimal stock investment is relatively insensitive to jump misspecification in the incomplete market. We also show that taking into consideration ambiguity aversion to diffusion risks and participating in the derivatives market are essential to reduce the potential welfare loss, while the impact of jump misspecification is marginal.

Robust consumption portfolio optimization with stochastic differential utility

This paper examines a continuous time intertemporal consumption and portfolio choice problem with a stochastic differential utility preference of Epstein–Zin type for a robust investor, who worries about model misspecification and seeks robust decision rules. We provide a verification theorem which formulates the Hamilton–Jacobi–Bellman–Isaacs equation under a non-Lipschitz condition. Then, with the verification theorem, the explicit closed-form optimal robust consumption and portfolio solutions to a Heston model are given. Also we compare our robust solutions with the non-robust ones, and the comparisons shown in a few figures coincide with our common sense.

Dynamic Portfolio Choice with Annuities When the Interest Rate Is Stochastic

This paper studies the optimal life cycle consumption and portfolio choice problem taking into account annuity risk due to stochastic interest rates. When the purchase of annuities is restricted to the retirement date, the annuitant is exposed to the risk of meeting low interest rates at the purchase date. This annuity risk can be diversified by spreading annuity purchases over the whole pre-retirement period. The numerical results of our life cycle model show that such temporal diversification enhances welfare for retirees up to 9% of certainty equivalent consumption and that welfare gains increase with higher interest rate risk.

Robust Portfolio Choice and Consumption with Derivative Trading Under Stochastic Volatility and Jumps

This paper analyzes the optimal consumption and portfolio choice problem for an ambiguity averse investor who has access to the stock and derivatives markets with recursive preferences. The stock process follows a stochastic volatility jump-diffusion model and the investor can have different levels of ambiguity about diffusion risks and the jump risk, respectively. We obtain an analytical solution which is exact when the investor has unit elasticity of intertemporal substitution of consumption, and approximate otherwise. We find that optimal exposures to diffusion risks and to the jump risk are significantly affected by the ambiguity aversion about the corresponding risk factors in the complete market. However, the optimal stock investment is insensitive to the ambiguity aversion about the jump risk in the incomplete market. We also find that considering ambiguity aversion with respect to diffusion risks and participating in the derivatives markets are essential to reduce the potential welfare loss, while the impact of ignoring the jump ambiguity is negligible.

The effects of pre-/post-retirement downside consumption constraints on optimal consumption, portfolio, and retirement

We investigate the optimal consumption, portfolio, and voluntary retirement choice problem in which an agent faces both pre-/post-retirement downside consumption constraints. We obtain the closed-form solutions using the martingale method and analyze the effect of the constraints on the optimal policies. We observe that the retirement wealth level is monotonically decreasing with pre-retirement constraints, while it is monotonically increasing with post-retirement constraints. We also find that the post-retirement downside consumption constraint has a significant impact on pre-retirement optimal policies. However, the pre-retirement downside consumption constraint does not act on post-retirement optimal policies.

Necessary and sufficient conditions for near-optimality of stochastic delay systems

ABSTRACTIn this paper, we study a class of near-optimisation problems with controlled systems described by the stochastic differential equations with delay. Thanks to the Ekeland's variational principle, we establish both necessary and sufficient conditions for the near-optimality of this problem by several delicate estimates for the state and adjoint processes. It is also shown that the error estimate for the near-optimality is of the order exactly. Moreover, we consider a production and consumption choice problem with delay to illustrate the application of our theoretical results.

Momentum in strategic asset allocation

This paper explores a continuous-time intertemporal consumption and portfolio choice problem for an infinite horizon investor with recursive utility defined over consumption. The investor who tries to exploit momentum is assumed to have access to a risk-free asset and a risky asset whose return exhibits short run momentum. We derive an exact explicit solution and an approximate analytical solution to the dynamic asset allocation problem. We find that the optimal portfolio demand for stocks contains two components: the “momentum-adjusted” myopic demand and the intertemporal hedging demand. When the model is calibrated to Chinese stock market data, it implies that intertemporal hedging demand motives greatly decrease the portfolio demand for stocks by investors whose risk aversion coefficients exceed one when the latest levels of stock returns are non-negative or moderate negative. In addition, hedging motives increase the optimal portfolio when they are sufficiently negative. Also, we find that risk aversion is the main preference parameter in determining portfolio choice rather than the elasticity of intertemporal substitution.

Dynamic consumption and portfolio choice with permanent learning

This paper studies a continuous–time intertemporal consumption and portfolio choice problem when a long–horizon investor who has recursive preferences cannot exactly observe the expected returns of the risky asset. I contribute to belief–behavior solutions to the explicit log-utility case, and to the approximate unit-risk-aversion case. I show explicitly that her belief behavior depends on the parameters of investment opportunities and investor preferences.

An optimal consumption, investment and voluntary retirement choice problem with disutility and subsistence consumption constraints: A dynamic programming approach

In this paper, we investigate an optimal consumption/portfolio problem of an agent with voluntary retirement and subsistence consumption constraints before retirement. We assume that the agent's utility function of consumption is of CRRA type and the agent suffers a utility loss from labor before retirement. We use the dynamic programming method to obtain explicit forms of the optimal consumption/portfolio and retirement time.

Liquidation Shocks and Transaction Costs

This paper investigates an Epstein-Zin type investors' optimal consumption and portfolio choice problem in the presence of transaction costs and liquidation shocks. We model the liquidation shocks as a Poisson process, which enforces the representative investors to liquidate their wealth in an illiquid asset. We calculate an average liquidity premium to transaction cost ratio with the steady state distribution and show that the liquidation shocks can significantly amplify the effect of the transaction costs on the excess rate of return of the illiquid asset. Our further numerical analysis also demonstrates how the level of elasticity of intertemporal substitution, as well as relative risk aversion, affects the investors' optimal trading behavior.

Optimal retirement strategy with a negative wealth constraint

This paper investigates an optimal consumption, portfolio, and retirement time choice problem of an individual with a negative wealth constraint. We obtain analytical results of the optimal consumption, investment, and retirement behaviors and discuss the effect of the negative wealth constraint on the optimal behaviors. We find that, as an individual can borrow more with better credit, she is more likely to retire at a higher wealth level, to consume more, and to invest more in risky assets.

Portfolio Selection with Subsistence Consumption Constraints and CARA Utility

We consider the optimal consumption and portfolio choice problem with constant absolute risk aversion (CARA) utility and a subsistence consumption constraint. A subsistence consumption constraint means there exists a positive constant minimum level for the agent’s optimal consumption. We use the dynamic programming approach to solve the optimization problem and also give the verification theorem. We illustrate the effects of the subsistence consumption constraint on the optimal consumption and portfolio choice rules by the numerical results.

Dynamic asset allocation with ambiguous return predictability

We study an investor's optimal consumption and portfolio choice problem when he is confronted with two possibly misspecified submodels of stock returns: one with IID returns and the other with predictability. We adopt a generalized recursive ambiguity model to accommodate the investor's aversion to model uncertainty. The investor deals with specification doubts by slanting his beliefs about submodels of returns pessimistically, causing his investment strategy to be more conservative than the Bayesian strategy. Unlike in the Bayesian framework, the hedging demand against model uncertainty may cause the investor's stock allocation to decrease sharply given a small doubt of return predictability, even though the expected return according to the VAR model is large. Over much of the parameter space, the robust strategy is very close to the Bayesian strategy with Epstein–Zin preferences and risk aversion chosen to match the same average portfolio holdings. This is true in particular when the IID model is unlikely and the dividend yield is low, as in recent years. However, differences in strategies can be substantial if the IID model is unlikely and the dividend yield is high.

Optimal Volatility Timing: A Life-Cycle Perspective

We study the role of time-varying stock return volatility in a consumption and portfolio choice problem for a life-cycle investor facing short-selling and borrowing constraints. Faced with a benchmark investment strategy that conditions on age and wealth only, we find that an investor is willing to pay a fee of up to 1%-1.5% of total life time consumption in order to optimally condition on volatility. Tilts in the optimal asset allocation in response to volatility shocks are considerably more pronounced than tilts in response to wealth shocks, and almost as important as life-cycle effects. Lastly, we find that the correlation between volatility and permanent labor income shocks may explain the low equity share of young households in the data.

When Do the Unemployed Jump in the Workforce?

This paper studies an optimal consumption and portfolio choice problem for unemployed people who have an option to work. Our problem is to find optimal consumption, risky investment, and workforce re-entry strategies for the unemployed. We find a closed form of the critical wealth level to re-enter the workforce. We show that the unemployed with a higher disutility of labor or a larger relative risk aversion are more reluctant to re-enter the workforce.

Default Risk of Life Annuity and the Annuity Puzzle

This paper considers a model of optimal consumption and portfolio choice of the retired person who has partially annuitized her wealth and faces default risk of her annuity provider. We develop a new method for solving the optimal consumption and portfolio choice problem in an incomplete financial market. We find that the effects of default risk on the retiree's optimal investment strategies and annuity demand depend crucially on the level of default risk, risk aversion, wealth, and investment opportunity. Higher default risk may lead retired people to increase their stock holdings. Lower default risk can substantially increase the demand for annuities. The annuity demand increases as risk aversion increases if default risk is small, whereas the annuity demand decreases as risk aversion increases if default risk is large.

How does the no-Ponzi game condition work in an optimal consumption problem?

In an optimal consumption choice problem, in which households have assets yielding interest rates, it is difficult to guarantee the existence of a solution without some restrictions for the consumption space, if the assumed utility function is unbounded. In this article, we formally state how the no-Ponzi game condition is used to guarantee an existence of optimal solutions. Furthermore, we provide the condition in which a solution attains the finite intertemporal utility.

Optimal portfolio and consumption selection with default risk

We investigate an optimal portfolio and consumption choice problem with a defaultable security. Under the goal of maximizing the expected discounted utility of the average past consumption, a dynamic programming principle is applied to derive a pair of second-order parabolic Hamilton-Jacobi-Bellman (HJB) equations with gradient constraints. We explore these HJB equations by a viscosity solution approach and characterize the post-default and pre-default value functions as a unique pair of constrained viscosity solutions to the HJB equations.