Optimal Insurance Demand and Investment-consumption Choices for Individuals Confronting Uninsured Risks

This paper introduces a dynamic continuous-time model that simultaneously incorporates life insurance, property insurance, and the investment-consumption problem. In our model, individuals are exposed to a series of risk events that can result in a complete loss of wealth. In addition, each occurrence of these events has a positive probability of causing a loss of life (accidental death). The individual seeks to manage these risks by procuring life insurance and property insurance. However, it is important to note that the life insurance policy provides coverage exclusively for natural causes of death, thus excluding coverage for accidental death (uninsured risk). The individual’s objective is to maximize their total discounted utilities, which encompass consumption, bequest, and retirement terminal wealth. They aim to achieve this goal by selecting an investment-consumption and insurance portfolio. The value function and optimal policies are explicitly derived for the case of constant relative risk aversion (CRRA) utility. The results demonstrate that, under the expected value premium principle, optimal property insurance follows a proportional form. In the absence of bequest utility, optimal life insurance effectively explains the demand for reverse insurance. It should be noted that uninsured risk and utility preference play a vital role in shaping the optimal insurance demand and influencing consumption choices. Finally, we present some numerical analysis to show the influence of parameters on the optimal strategies.