Optimal Consumption and Portfolio Selection with Early Retirement Option

Abstract

In this paper we propose an approach to investigate a model of consumption and investment with a mandatory retirement date and early retirement option; we analyze properties of the optimal strategy and thereby contribute to understanding the interaction between retirement, consumption, and portfolio decisions in the presence of both the important features of retirement. In particular, we provide a characterization of the threshold of wealth as a function of time, and we show that it is strictly decreasing near the mandatory retirement date. The threshold is similar to the early exercise boundary of an American option in the sense that if the agent’s wealth is above or equal to the threshold level, then the agent immediately retires. We also provide comparative static analysis.

Finite horizon portfolio selection problems with stochastic borrowing constraints

“…The problem is similar to the standard American put option problem treated in the literature (see e.g., Myneni [23] and Peskir [25]), but different from the latter in the sense that the payoff is time dependent. In this section we provide a complete self-contained derivation of the solution to the problem, following the ideas and proofs in Yang and Koo [26].…”

“…Using the standard techniques for an optimal stopping problem, g(t, z) is a solution of the following variational inequality (VI) (see e.g., Ch. 2 of Karatzas and Shreve [18] or Yang and Koo [26]):…”

“…By analyzing the variational inequality (26), we derive the following integral representation of the value function g(t, z) of Problem 3 (In Appendix B we provide all properties for variational inequality (26) arising from Problem 3)…”

“…We will prove the existence and uniqueness of W 1,2 p,loc solution to VI (26) and describe properties of the solution in the lemmas below.…”

“…1. First, we transform the degenerate parabolic problem (26) into the following non-degenerate parabolic problem. Setting…”