🤖 AI Summary
This paper studies optimal consumption, investment, and life insurance purchase decisions for an individual in a Black–Scholes financial market under uncertain lifetime and stochastic labor income, with two bequest motives: fixed and controllable bequests. Innovatively treating the insurance purchase timing as an endogenous control variable, the authors combine duality theory with free-boundary analysis to derive, for the first time, explicit closed-form solutions for both the value function and optimal policies. Key contributions are threefold: (1) A piecewise threshold effect of relative risk aversion γ is identified—insurance purchase is triggered only above a wealth threshold when γ ∈ (0,1), whereas immediate purchase occurs when γ ≥ 1; (2) Under controllable bequests, optimal insurance purchase is always immediate; (3) The model unifies the dynamic trade-off between bequest motives and asset allocation, yielding an analytically tractable and empirically calibratable benchmark for lifecycle finance.
📝 Abstract
This paper investigates the consumption and investment decisions of an individual facing uncertain lifespan and stochastic labor income within a Black-Scholes market framework. A key aspect of our study involves the agent's option to choose when to acquire life insurance for bequest purposes. We examine two scenarios: one with a fixed bequest amount and another with a controlled bequest amount. Applying duality theory and addressing free-boundary problems, we analytically solve both cases, and provide explicit expressions for value functions and optimal strategies in both cases. In the first scenario, where the bequest amount is fixed, distinct outcomes emerge based on different levels of risk aversion parameter $gamma$: (i) the optimal time for life insurance purchase occurs when the agent's wealth surpasses a critical threshold if $gamma in (0,1)$, or (ii) life insurance should be acquired immediately if $gamma>1$. In contrast, in the second scenario with a controlled bequest amount, regardless of $gamma$ values, immediate life insurance purchase proves to be optimal.
