The impact of a partial borrowing limit on financial decisions

Abstract

We consider a consumption, investment, life insurance, and retirement decision problem in which an economic agent is allowed to borrow against only a part of future income. The closed-form solution is attained by applying a dual approach that directly imposes the conditions for the borrowing limit on a dual value function. We provide analytic comparative statics for optimal strategies with rigorous proofs. It is confirmed that a more stringent borrowing limit leads to less consumption and less life insurance purchase. However, even with a tighter borrowing limit, an agent with weak incentive to retire can invest more when the wealth level is high enough. We also show that a more stringent borrowing limit can delay or hasten the optimal retirement timing depending on the agent's current wealth level.

Keywords:

• Partial borrowing limit
• Consumption and investment
• Life insurance
• Voluntary retirement
• Duality approach

JEL Classification:

• C61
• E21
• G11
• G22

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Byung Hwa Lim http://orcid.org/0000-0003-4342-1895

Minsuk Kwak http://orcid.org/0000-0003-4787-0698

Notes

† Zeng et al. (2016) also consider utility maximization with non-negative wealth constraint, but they do not consider voluntary retirement.

† It is more natural to assume that the mortality intensity varies over time. However, if the mortality intensity is not constant, then our problem becomes hardly tractable. In particular, we have to find time-varying free boundary for the optimal retirement decision. Hence it makes difficult to obtain closed-form solution and analyse the properties of the optimal policies. Since we focus on analysing the effect of borrowing limit on the optimal policies in detail, we assume constant mortality intensity as Dybvig and Liu (2010, 2011), Lim and Kwak (2016) that consider mortality risk and retirement decision together.

‡ This is the reason why we consider the constant labour income. For constant relative risk aversion (CRRA) utility function without the borrowing limit, the problem with perfectly correlated income risk can be reduced to a one-dimensional problem by using a homogeneity property, which gives the closed-form solution. Even if there is a borrowing limit of L=0 as Duffie et al. (1997), El Karoui and Jeanblanc-Picqué (1998), Koo (1998), and Dybvig and Liu (2010), the borrowing limit can be interpreted as no investment condition at a zero wealth level, which also provides a closed-form solution for the perfectly correlated income case. Under PBL with $L\ne 0$, however, no investment condition at the borrowing limit does not hold anymore, and it is impossible to characterize the minimum wealth condition with a reduced state variable only. Thus, we need an additional state variable under PBL with $L\ne 0$.

† The reason why the agent purchases less life insurance as the wealth increases is the gap between the optimal bequest level $M_t=X_t+p_t/\lambda$ and the current wealth $X_t$ becomes smaller as the wealth level increases. This happens when the bequest motive is weak and the wealth level is high enough. If the agent has very strong bequest motive, then it might be optimal to purchase more life insurance as the wealth level increases.

† The convexity of risky investment near the retirement wealth threshold means that the agent with retirement option increases risky investment more as the wealth approaches the retirement threshold so as to retire as soon as possible. This is also reported in the previous studies such as Farhi and Panageas (2007), Dybvig and Liu (2010), and Lim and Shin (2011). Moreover, we can analytically prove that ${\mathrm{\partial }}^2{\pi }_t^{PBL}/\mathrm{\partial }{X}_t^2>0$ when $X_t$ is above certain threshold, although the proof is omitted in the paper. The proof can be provided upon request.

† The baseline parameters for the figures in Section 5 are $\beta =0.01,r=0.01,\lambda =0.015,\gamma =3,w=0.5,\mu =0.05,\sigma =0.2,\delta =0.5,l_0=1,l_1=9$.

‡ More precisely, low enough $X_t$ means that $X_t\le -v^{\prime }\left(\hat{y}\right)-w/\left(\lambda +r\right)$ where $\hat{y}$ is given in (36(36) $\hat{y}\triangleq {\left(\frac{\begin{array}{c}(l_0^{\left({\gamma }_1-\gamma \right)/{\gamma }_1}+\lambda k_M^{1/{\gamma }_1})\\ \{n_{-}(1-{\gamma }_1+{\gamma }_1{n}_{-})-\\ n_{+}(1-{\gamma }_1+{\gamma }_1{n}_{+}){\tilde{y}}^{n_{-}-n_{+}}\}\end{array}}{n_{+}{n}_{-}{\gamma }_1^2(n_{-}-n_{+})(\frac{l_1^{\left({\gamma }_1-\gamma \right)/{\gamma }_1}-l_0^{\left({\gamma }_1-\gamma \right)/{\gamma }_1}}{1-{\gamma }_1})}\right)}^{1/\left(n_{+}-1+1/{\gamma }_1\right)}\tilde{y}.$(36) ). In other words, from Proposition 6 we can show that $\mathrm{\partial }{\pi }^{PBL}/\mathrm{\partial }L>0$ when $X_t\le -v^{\prime }\left(\hat{y}\right)-w/\left(\lambda +r\right)$.

Additional information

Funding

Byung Hwa Lim acknowledges that this work was supported by the research grant of the University of Suwon in 2017. Minsuk Kwak acknowledges that this work was supported by Hankuk University of Foreign Studies Research Fund of 2018. Minsuk Kwak also acknowledges that this research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (NRF-2016R1D1A1B03931314).