Optimal Consumption, Investment and Life Insurance Decisions under Stochastic Differential Utility with Habit Formation

This paper introduces the habit formation to stochastic differential utility which includes lifetime uncertainty and utility from bequest. We study the optimal investment, consumption and life insurance decision of an agent under stochastic differential utility and utility from terminal wealth.

We use the method of dynamic programming to get the optimal policy and state a verification theorem with a Hamilton-Jacobi-Bellman equation.

For Epstein-Zin preferences, we find out the analytical expression of the optimal investment, consumption and life insurance decision of an agent.Moreover,we investigate the effect of habit formation on the optimal decision,together with the elasticity of the utility function.

It is shown that habit formation does not change the basic shape of the curve of investment, consumption and insurance. 

With habit formation, the optimal consumption curves are lower during the first few years and higher after those years than those without habit formation, while the bequest curves is lower for the case of habit formation.

Increasing the initial value of the habit formation will decrease both the optimal consumption curves and bequest curves slightly. The parameters of the habit formation have a greater impact on the curves than the initial value of the habit formation.