Optimal investment, consumption and life insurance strategies under stochastic differential utility with habit formation

Abstract

This paper studies the optimal investment, consumption and life insurance decisions of an agent under stochastic differential utility. The optimal choice is obtained through dynamic programming method. We state a verification theorem using the Hamilton-Jacobi-Bellman equation. For the special case of Epstein-Zin preferences, we derive the analytical solution to the problem. Moreover, we explore the effects of habit formation and of the elasticity of the utility function on the optimal decision through a numerical simulation based on Chinese mortality rates. We show that habit formation does not change the basic shape of the consumption and bequest curves. With habit formation, the optimal consumption curve moves up with lower initial consumption, while the bequest curve moves down. Increasing the value of initial habit formation slightly decreases both optimal consumption and bequests. The changes in the habit formation parameters have a greater impact on the curves than does a change in the initial habit formation.