Stochastic optimization problems in financial planning

Abstract

In this dissertation, we consider two stochastic optimization problems in financial planning. Among various topics in financial planning, we focus on investment, insurance planning, and retirement planning of economic agents. First, we consider an optimal portfolio, consumption and retirement decision problem in which an economic agent can determine the discretionary stopping time as a retirement time with constant labor wage and disutility. We allow the preference of the agent to be changed before and after retirement. It is assumed that the agent`s coefficient of relative risk aversion becomes higher after retirement. Under a constant relative risk aversion(CRRA) utility function, we obtain the optimal policies in closed-forms using martingale methods and variational inequality methods. We analyze the properties of the optimal policies with numerical examples. Second, we study an optimal portfolio and consumption choice problem of a family that combines life insurance for parents who receive deterministic labor income until the fixed time T. We consider utility functions of parents and children separately and assume that parents have an uncertain lifetime. If parents die before time T, children have no labor income and they choose the optimal consumption and portfolio with remaining wealth and life insurance benefit. The object of the family is to maximize the weighted average of utility of parents and that of children. We obtain analytic solutions for the value function and the optimal policies, and then analyze how the changes of the weight of the parents` utility function and other factors affect the optimal policies.