Self-protection and insurance demand under time-inconsistent mean–variance framework

This paper investigates the equilibrium risk mitigation strategies involving both self-protection efforts and market insurance within a mean–variance framework. We formulate a time-inconsistent optimal control problem and solve it using extended Hamilton–Jacobi-Bellman (HJB) equations. We theoretically identify a substitution effect between self-protection and insurance demand in response to changes in premium loading, under an equilibrium insurance strategy, regardless of whether effort is reflected in premium pricing. In contrast, when the market offers only proportional insurance and decision-makers’ efforts are observable and fully priced into premiums, we demonstrate the existence of a complementary effect. We further explore the impact of risk preferences and loss distribution. Numerical results show that less risk-averse DMs tend to rely more on self-protection, supplemented by insurance with a high deductible. Conversely, highly risk-averse DMs prefer greater insurance coverage. To address the sensitivity of equilibrium strategies to the specification of the loss distribution, we extend our model to incorporate ambiguity aversion and find that introducing ambiguity aversion reinforces the effect of risk aversion. Finally, we provide theoretical connections between our results and existing literature on optimal self-protection strategies.