Portfolio Choice with Competing Precautionary and Accumulation Goals

Technical Summary: Portfolio Choice with Competing Precautionary and Accumulation Goals

Problem Formulation
This paper addresses the optimal portfolio choice problem for a household simultaneously managing two structurally distinct financial goals: a fixed-deadline goal (e.g., retirement or college tuition) with a known maturity $T$ and a random amount $G$, and a random-deadline goal (e.g., medical emergency or job loss) arriving at a stochastic time $\tau$ with a random amount $R$. The authors model the random deadline $\tau$ as an exponential random variable with intensity $\lambda$, independent of market movements and goal amounts.

The household operates under an "all-or-nothing" funding rule. When a goal arrives, it is paid in full if the current wealth is sufficient; otherwise, the goal is missed entirely, and no partial payment is made. The objective is to maximize a weighted sum of the probabilities of fully funding both goals:
$$J(x; \pi) = \alpha_R P(\text{random goal funded}) + \alpha_D P(\text{fixed goal funded})$$
where $\alpha_R + \alpha_D = 1$. The market is modeled as a Black–Scholes environment with $n$ risky assets and a risk-free rate. The authors impose leverage and short-sale constraints ($\pi \in [-K, K]^n$) to ensure well-posedness and prevent the unbounded risk-taking often found in unconstrained probability-maximization problems.

Methodology
The authors employ stochastic control theory and viscosity solution techniques to solve the problem. The methodology proceeds in three stages:

1. Single-Goal Sub-Problems: The authors first solve two benchmark problems in isolation.

   • Fixed-Deadline: A finite-horizon problem maximizing the probability of reaching a target $G$ by time $T$. The value function is characterized as the unique viscosity solution to a Hamilton–Jacobi–Bellman (HJB) equation with terminal condition $FG(w)$.
   • Random-Deadline: An infinite-horizon stationary problem maximizing the probability of reaching a target $R$ at a random time $\tau$. The value function satisfies a stationary HJB equation.
   • Both sub-problems utilize clipped controls to handle constraints, ensuring the existence of bounded, continuous viscosity solutions.

2. Dual-Goal Synthesis: The core of the analysis combines these sub-problems. The authors demonstrate that the dual-goal problem possesses a "triangular structure." Once the first goal is resolved (either the random shock arrives or the fixed deadline passes), the household faces a standard single-goal continuation problem.

   • The dual-goal value function is constructed using continuation operators. If the random goal arrives at time $t < T$, the household receives an immediate reward if affordable and then faces the fixed-deadline problem with reduced wealth. If the fixed deadline $T$ arrives first, the household faces the random-deadline problem with remaining wealth.
   • The authors derive a coupled HJB equation where the single-goal value functions appear as endogenous boundary conditions (specifically, as source terms in the running reward and terminal conditions).
   • Existence and uniqueness of the solution are established using viscosity solution theory, specifically leveraging comparison principles for degenerate elliptic operators.

3. Optional Funding Variant: The paper extends the model to a variant where the household can decline the fixed-deadline goal at time $T$ if funding it would severely compromise the random-deadline goal. This introduces an optimal stopping/decision node at $T$, characterized by a modified terminal operator that compares the utility of funding the fixed goal versus preserving wealth for the random goal.

Key Contributions and Results

• Growth Crowding-Out Effect: The authors identify a novel non-monotonicity in the value function with respect to wealth. Under forced funding, a household with wealth just above the random-goal threshold $R$ may be worse off than a slightly poorer household. The wealthier household is forced to pay the random goal when it arrives, depleting the capital needed for the fixed goal. The poorer household, unable to pay the random goal, retains its capital and may have a higher probability of meeting the fixed goal. This effect arises purely from the interaction of the two goals under the all-or-nothing rule.
• Deadline Pressure Effect: The analysis quantifies the impact of a compressed saving horizon. Households with shorter horizons (e.g., starting to save later) are forced to maintain maximum risk exposure over a wider range of wealth levels and exhibit sharper spikes in risk-taking near the goal threshold to compensate for lost compounding time.
• Non-Monotonicity and Policy Implications: The optimal portfolio weight is not necessarily monotone in wealth. In the "crowding-out" region, the optimal strategy may involve reducing risk exposure to avoid crossing the random-goal threshold, thereby preserving wealth for the fixed goal.
• Option Value of Flexibility: In the optional funding variant, the authors characterize the ex ante and terminal option values. They find that the value of the option to decline the fixed goal is hump-shaped in wealth. It is most valuable at intermediate wealth levels where funding the fixed goal is affordable but would significantly reduce the continuation value of the random-deadline problem.
• Numerical Calibration: The model is calibrated to empirical data, including job separation rates for $\lambda$, and cost-of-attendance data for college and retirement targets. Numerical results confirm the economic significance of the identified effects. For instance, delaying retirement savings from age 25 to 50 requires a tenfold increase in initial wealth to achieve the same success probability.

Significance
The paper claims to provide the first framework that simultaneously addresses fixed-deadline accumulation and random-deadline precautionary goals within a unified stochastic control setting. By moving beyond the isolation of these problems found in prior literature (e.g., Browne [1999] for fixed deadlines, Bayraktar and Young [2016] for random deadlines), the authors reveal tradeoffs that are invisible in single-goal models.

The authors position their work as a theoretical foundation for "goals-based investing," particularly for robo-advisory platforms that currently manage goals in siloed accounts. The identification of the growth crowding-out effect suggests that treating emergency funds and long-term savings as independent problems may lead to suboptimal portfolio allocations. The paper demonstrates that optimal portfolio choice requires a dynamic balance between maintaining a precautionary buffer and pursuing growth, where the optimal strategy depends critically on the relative timing, size, and priority of the competing goals. The rigorous characterization of the value function via viscosity solutions provides a robust mathematical basis for these economic insights.