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This paper investigates the consumption and investment decisions of anindividual facing uncertain lifespan and stochastic labor income within aBlack-Scholes market framework. A key aspect of our study involves the agent'soption to choose when to acquire life insurance for bequest purposes. Weexamine two scenarios: one with a fixed bequest amount and another with acontrolled bequest amount. Applying duality theory and addressing free-boundaryproblems, we analytically solve both cases, and provide explicit expressionsfor value functions and optimal strategies in both cases. In the firstscenario, where the bequest amount is fixed, distinct outcomes emerge based ondifferent levels of risk aversion parameter $\gamma$: (i) the optimal time forlife insurance purchase occurs when the agent's wealth surpasses a criticalthreshold if $\gamma \in (0,1)$, or (ii) life insurance should be acquiredimmediately if $\gamma>1$. In contrast, in the second scenario with acontrolled bequest amount, regardless of $\gamma$ values, immediate lifeinsurance purchase proves to be optimal.

Striking the Balance: Life Insurance Timing and Asset Allocation in Financial Planning