Optimal portfolio and insurance strategy with biometric risks, habit formation and smooth ambiguity

This paper studies the optimal consumption, investment, health insurance and life insurance strategy for a wage earner with smooth ambiguity, habit formation and biometric risks. The individual can invest in the financial market composed of a risk-free asset and a risky asset whose unknown market price results in ambiguity. The habit formation depends on historical consumption and satisfies an ordinary differential equation. Moreover, the biometric risks, which consist of health shock risk and mortality risk, can impact the individual's income and health state. The individual can purchase health insurance and life insurance to respectively deal with health shock risk and mortality risk, and aims at maximizing the total expected utility of consumption, legacy and terminal wealth. Using the dynamic programming technique, we derive the corresponding Hamilton-Jacobi-Bellman equation in the states of health and critical illness respectively, prove the verification theorem and obtain closed-form solutions for the optimal strategies. Finally, numerical experiments are carried out to illustrate the impact of risk aversion, ambiguity aversion, health shock and habit formation on the optimal strategy. The results reveal that the wage earner with different utility functions and different health states will show different behaviors in consumption, investment and insurance purchase.