Optimal Consumption and Portfolio Choice with No-Borrowing Constraint in the Kim-Omberg Model

Abstract

In this paper, we study an intertemporal utility maximization problem in which an investor chooses consumption and portfolio strategies in the presence of a stochastic factor and a no-borrowing constraint. In the spirit of the Kim-Omberg model, the stochastic factor represents the excess return of the risky asset and follows an Ornstein-Uhlenbeck process, capturing the mean reversion of expected excess returns-a feature well supported by empirical evidence in financial markets. The investor seeks to maximize expected utility from consumption, subject to the constraint that wealth remains nonnegative at all times. To address the dynamic no-borrowing constraint, we use Lagrange duality to transform the primal problem into a singular control problem in the dual space. We then characterize the solution to the dual singular control problem via an auxiliary two-dimensional optimal stopping problem featuring stochastic volatility, and subsequently retrieve the primal value function as well as the optimal portfolio and consumption plans. Finally, a numerical study is conducted to derive economic and financial implications.

Figures (6)

• Figure 1: Simulation of optimal state processes.
• Figure 2: Comparison of optimal wealth trajectories between the Stochastic Agent and the Constant Agent ($\beta_t \equiv \bar{\beta}$) for different values of $\rho$. The Brownian path of $(W_t)_t$ is fixed across both cases, while the Stochastic Agent's wealth is averaged over 10,000 Brownian paths of $(W^\beta_t)_t$.
• Figure 3: Optimal policies for different values of labor income $\ell$.
• Figure 4: Optimal policies for different values of risk aversion $\gamma$.
• Figure 5: Optimal policies for different values of correlation $\rho$.

Theorems & Definitions (44)

• Definition 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Remark 2.5
• Proposition 3.1
• proof
• Proposition 3.3
• proof
• Proposition 3.4