An Integral Equation Representation for Optimal Retirement Strategies in Portfolio Selection Problem
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An Integral Equation Representation for Optimal Retirement Strategies in Portfolio Selection Problem Junkee Jeon1 · Hyeng Keun Koo2 · Yong Hyun Shin3 · Zhou Yang4 Accepted: 4 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we study the consumption and portfolio selection problem of a finitelylived economic agent with an early retirement option, that is, the agent can choose her/his early retirement time before a mandatory retirement time. Based on the theoretical results in Yang and Koo (Math Oper Res, 43(4):1378–1404, 2018), we derive an integral equation satisfied by the optimal retirement boundary or free boundary using the Mellin transform technique. We also derive integral equation representations for the optimal consumption-portfolio strategies and the optimal wealth process. By using the recursive integration method, we obtain the numerical solutions for the integral equations and discuss economic implications for the optimal retirement strategies by using numerical solutions. Keywords Portfolio selection · Mandatory retirement · Early retirement · Free boundary · Mellin transform · Integral equation
1 Introduction In this paper we study the optimal retirement decision of an agent in a continuous time model. We derive an integral equation for optimal policies and provide numerical schemes for the theoretical model proposed by Yang and Koo (2018). Currently, life expectancy is increasing and population aging is prevalent, and hence retirement is a crucially important issue from a social as well as an individual Junkee Jeon is supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (Grant No. NRF-2020R1C1C1A01007313). Hyeng Keun Koo is supported by NRF Grant (Grant Nos. NRF-2019S1A5A2A03054249, NRF-2020R1A2C1A01006134). Yong Hyun Shin is supported by NRF Grant (Grant Nos. NRF-2019R1H1A2079177, NRF2020R1A2C1A01006134). Zhou Yang is supported by NNSF of China (Grant Nos. 11771158, 11801091) and Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515011338) * Hyeng Keun Koo [email protected] Extended author information available on the last page of the article
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perspective. Understanding the joint problem of optimal retirement and consumption and portfolio decision is a first step to tackle the issue. In the literature Choi and Shim (2006) initiated investigation into the joint problem by studying a model where an agent chooses her/his retirement time as well as a portfolio of assets and consumption. In their model the decision to retire comes from the trade-off between labor income and the utility cost of labor. Farhi and Panageas (2007) and Choi et al. (2008) have proposed a model where an agent chooses labor, leisure and the retirement time. Dybvig and Liu (2010) and Lim and Shin (2011) have studied a model with borrowing constraints. The time horizon of these models, however, is infinite, and hence, the models do not allow investigation of the effects
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