Optimal portfolio selection in mutual aid insurance under smooth ambiguity and habit formation

In this article, we explore the optimal portfolio selection problem for an individual who participates in mutual aid insurance and purchases life insurance under an environment where smooth ambiguity and habit formation coexist. The individual invests in a market with one risk-free and two risky assets whose unknown market prices result in ambiguity. Habit formation depends on historical consumption and is state-dependent. We construct a mathematical model for mutual aid insurance based on a three-state (good, bad, and death) process driven by an inhomogeneous Markov chain. Meanwhile, the individual hedges mortality risk via life insurance. We apply an extended Hamilton–Jacobi–Bellman equation with regime switching to solve for the equilibrium strategies that maximize the individual’s lifetime utility. Subsequently, we analyze three combinations of utility and ambiguity preference functions, power–power, exponential–power, and exponential–exponential, and obtain equilibrium strategies under different functional forms. A key finding shows that the Tobin–Markowitz separation theorem is no longer applicable when ambiguity aversion is considered. Finally, we conduct numerical simulations to demonstrate the significance of mutual aid insurance, measure parametric impacts on equilibrium strategies, and provide theoretical guidance for mutual aid product design.