By Junkee Jeon, Minsuk Kwak & Kyunghyun Park

We study an optimal consumption, investment, life insurance, and retirement decision of an economic agent who has an option to retire early any time before the mandatory retirement date. We conduct a thorough theoretical analysis for the optimal retirement problem with general utility function in the presence of mandatory retirement date, which leads to the optimal stopping problem in finite horizon. Furthermore, different marginal utility of consumption before and after retirement is considered, which can provide an explanation for the retirement-consumption puzzle, while it makes the problem technically more challenging. Based on the theory of partial differential equation, we analyze the variational inequality arising from the dual problem and establish the duality theorem. We show that the optimal retirement decision is determined by the time-varying optimal retirement wealth boundary, and we provide an integral equation representation for the optimal retirement wealth boundary, which can be solved accurately and efficiently by using recursive integration method. As an extension, the case with stochastic labor income is also considered. The properties of the optimal strategies are provided with emphasis on the role of the mandatory retirement date and the impact of having a retirement option on the optimal financial decisions.

Source: SSRN