Optimal investment, consumption and life insurance decisions for households with consumption habits under the health shock risk

This paper investigates the optimal investment, consumption, and life insurance strategies for households under the impact of health shock risk. Considering the uncertainty of the future health status of family members, a non-homogeneous Markov process is used to model the health status of the breadwinner. Drawing upon the theory of habit formation, we investigate the influence of different consumption habits on households' investment, consumption, and life insurance strategies. Based on whether the breadwinner is alive or not, we formulate and solve the corresponding Hamilton-Jacobi-Bellman (HJB) equations for the two scenarios of breadwinner survival and breadwinner's demise, respectively, and obtain explicit expressions for the optimal investment, consumption, and life insurance strategies. Through sensitivity analysis, it has been shown that the presence of health shocks within households has a negative impact on investment and consumption decisions, while the formation of consumption habits increases household propensity for precautionary savings.