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Consumption-portfolio choice with subsistence consumption and risk aversion change at retirement
- Ho-Seok Lee^{1}Email author
https://doi.org/10.1186/s13660-018-1756-1
© The Author(s) 2018
Received: 1 March 2018
Accepted: 30 June 2018
Published: 6 July 2018
Abstract
This paper considers subsistence consumption of an economic agent both before and after retirement in analyzing the optimal consumption, portfolio, and retirement problem. We allow the relative risk aversion of the economic agent to make a one-off jump at retirement. With a Cobb–Douglas utility function, we obtain explicit expressions for the optimal policies. Numerical results show that, whereas post-retirement subsistence consumption tends to delay retirement, pre-retirement subsistence consumption and the magnitude of jump in relative risk aversion may stimulate early retirement. Also, the consumption drop at retirement deepens as post-retirement subsistence consumption increases, but it weakens as pre-retirement subsistence consumption increases.
Keywords
1 Introduction
In the present paper, we study lifetime consumption, portfolio, and retirement of an economic agent whose relative risk aversion changes at retirement, with the assumption that the agent faces both pre- and post-retirement subsistence consumption. Recently, Lim et al. [17] also considered both pre- and post-retirement subsistence consumption, but they did not consider risk aversion change at retirement. Also, our study and that of Lim et al. [17] are methodologically different: we use a dynamic programming method, whereas Lim et al. [17] relied on the martingale and duality approach. Risk aversion change at retirement has been well addressed in empirical studies (e.g., Yoo [23], Riley and Chow [20], and Halek and Eisenhauer [8]). According to the results of the above studies, risk aversion tends to increase substantially at retirement. Kwak et al. [13] considered risk aversion change at retirement, but did not impose any constraints (e.g., subsistence consumption constraint or borrowing constraint). Jang and Lee [10] investigated the combined effects of risk aversion change at retirement and borrowing constraints on the optimal consumption, portfolio, and retirement strategies. We allow the relative risk aversion of the economic agent to make a one-off jump at retirement as in Jang and Lee [10]. In addition, we assume a Cobb–Douglas utility function of consumption and leisure because this function is appropriate for capturing empirical observations that consumption drops significantly at retirement (for the consumption retirement puzzle: see Hurd and Rohwedder [9], Haider and Stephens [7], and Aguila et al. [1]).
This paper is an extension of consumption-portfolio choice literature that examines quantitatively the combined effects of subsistence consumption and risk aversion change at retirement. Merton’s [19] original problem on the optimal consumption and portfolio was generalized by Karatzas et al. [12], who transformed the relevant Bellman equation to a linear ordinary differential equation, which is more tractable than the original. Sethi et al. [21] extended Karatzas et al. [12] to the case with subsistence consumption, and exploited a different transformation to linearize the relevant Bellman equation. Shin et al. [22] used the Martingale and duality method to explore a similar problem. Lakner and Nygren [14] used Malliavin calculus to solve the portfolio optimization problem with subsistence consumption. Labor supply flexibility was firstly incorporated into lifetime consumption and portfolio selection problem by Bodie et al. [3]. Choi and Shim [4] and Farhi and Panageas [6] developed a more realistic model on labor supply flexibility through investigating the optimal retirement problem as an optimal stopping problem. Lim et al. [18] and Lee and Shin [15] considered the disutility from labor to extend these ideas that include subsistence consumption. Assuming a trade-off between income and leisure, Lee and Shin [16] studied the optimal consumption, portfolio, and retirement problem of an economic agent with a subsistence consumption. Most similar work to ours is by Lim et al. [17] and Lee and Shin [16], but they did not consider risk aversion change nor investigate the consumption drop at retirement. Restrictions on consumption level and limited access to credit market are realistic assumptions when we investigate lifetime optimal consumption and portfolio selection problem. Other than these constraints, individual’s uninsurable income risk or incomplete market are of crucial importance for studying lifetime optimal strategies. For example, Jang et al. [11] and Bensoussan et al. [2] developed optimal retirement rules of an individual with an exogenous unemployment risk.
Our numerical results show that pre-retirement subsistence consumption may urge early retirement, whereas post-retirement subsistence consumption may delay retirement. The wealth accumulation for retirement is likely to decrease with the magnitude of jump in relative risk aversion at retirement. We also investigate the effects of subsistence consumption on consumption drop at retirement if the relative risk aversion changes at retirement. Post-retirement subsistence consumption may intensify consumption drop at retirement, whereas pre-retirement subsistence consumption may weaken it. The remainder of this paper proceeds as follows. Section 2 introduces the financial market and constructs our model. Section 3 solves the economic agent’s optimization problem using dynamic programming method and obtains analytic expressions for the optimal policies. Some numerical illustrations and their implications are given in Sect. 4, and Sect. 5 summarizes the results.
2 The model
Suppose that the economic agent is allowed to invest in the financial market that consists of two investment opportunities: a riskless asset \(M_{t}\) with price process \(dM_{t}/M_{t}=r\,dt\), where r is the constant rate of return of the riskless asset; and a risky asset \(S_{t}\) with price process \(dS_{t}/S_{t}=\mu \,dt+\sigma dB_{t}\), where μ is the constant rate of return of the risky asset, σ is the constant volatility of the risky asset, and \(B_{t}\) is the standard Brownian motion on a probability space \((\Omega ,\mathcal{F}, \mathbb{P})\). Denote by \(\{ \mathcal{F}_{t} \} _{t \geq 0}\) the \(\mathbb{P}\)-augmentation of the filtration generated by the standard Brownian motion \(\{ B_{t} \} _{t \geq 0}\). Then we can define a portfolio process \(\boldsymbol{\pi }\triangleq \{ \pi_{t} \} _{t\geq 0}\), i.e., the amount of money invested in the risky asset, which is a measurable process that is adapted to \(\{ \mathcal{F}_{t} \}_{t \geq 0}\) and that satisfies \(\int_{0}^{t} \pi^{2}_{s}\,ds<\infty\), for all \(t\geq 0\) a.s. A consumption rate process \(\mathbf{c}\triangleq \{ c_{t} \} _{t \geq 0}\) in (2.1) is a measurable nonnegative process that is adapted to \(\{ \mathcal{F}_{t} \} _{t \geq 0}\) and satisfies \(\int_{0}^{t} c_{s}\,ds<\infty\), for all \(t\geq 0\) a.s. The retirement time τ is an \(\mathcal{F}_{t}\)-stopping time.
3 The optimization problems and the solutions
The purpose of our study is to examine combined effects of the subsistence consumption and risk aversion change at retirement on the optimal policies of an economic agent. We first obtain the value function and the optimal policies when the economic agent is retired and then obtain those when she is working, by utilizing the smooth-pasting condition of post-retirement value function and pre-retirement value function at the retirement wealth level.
Assumption 3.1
3.1 Post-retirement optimization problem
Definition 3.1
Proposition 3.1
The value function \(V_{p}(x)\) defined (3.1) is strictly concave and strictly increasing on \(x\in ((R_{2}-I_{2})/r, \infty )\).
Proof
If we follow similar lines to the proof of Proposition 2.1 in Zariphopoulou [24], we arrive at the results. □
Using the dynamic programming method, we obtain the value function \(V_{p}(\cdot )\in C^{2}((R_{2}-I_{2})/r, \infty )\) and the related optimal consumption and portfolio.
Proposition 3.2
Proof
3.2 Pre-retirement optimization problem
Theorem 3.1
Suppose that a strictly concave and strictly increasing function \(v(\cdot )\in C^{2}((R_{1}-I_{1})/r, \bar{x})\) solves the Bellman equation (3.16) and satisfies the smooth-pasting (continuous differentiability) condition with \(V_{p}(x)\) at \(x=\bar{x}\). Then \(V(x)=v(x)\) and the optimal consumption \(c^{*}\) and portfolio \(\pi^{*}\) are the maximizer of the Bellman equation (3.16) and the optimal retirement time \(\tau^{*}\) is given by \(\tau^{*} = \inf \{t \geq 0: X_{t}\geq \bar{x}\}\).
Proof
The proof follows similar lines to those of the proof of Theorem 4.1 in Lee and Shin [15]. □
Theorem 3.2
Proof
The following pre-retirement optimal strategies are immediate consequences of Theorem 3.1 and Theorem 3.2.
Proposition 3.3
4 Numerical examples
In this section, we use reasonable parameters to explore some numerical results and their implications. To represent empirical observations that at retirement, consumption drops substantially and relative risk aversion increases significantly, we make the following assumption.
Assumption 4.1
\(1<\bar{\gamma }_{1}<\bar{\gamma }_{2}\).
This assumption was also made in Jang and Lee [10] (see detailed justification therein). By the definitions of \(\gamma_{1}\) and \(\gamma_{2}\), it easily follows that \(1<\gamma_{1}<\gamma_{2}\).
Lemma 4.1
Suppose that \(B_{1}>L^{-n_{1}(\gamma _{2}-\bar{\gamma }_{2})}B_{p}\). Then \(G'(x)=0\) has a unique solution \(x_{e}\in (0,\infty )\) and \(G''(x)<0\) for all \(x>0\). Furthermore, if \(G(x_{e})>0\), \(G(x)=0\) has two distinct roots.
Proof
To examine some examples, we use parameters that satisfy the conditions of Lemma 4.1. For such parameters, we are obliged to solve \(G(x)=0\) and choose the appropriate one of the two roots, say \(x_{m}\) and \(x_{M}\) (\(x_{m}< x_{e}< x_{M}\)). Similar arguments of the proof of the Proposition 4.2 in Lee and Shin [15] lead us to choose \(x_{M}\) as the appropriate root to the equation \(G(x)=0\).
5 Conclusion
This paper aims to quantify the effects of subsistence consumption and risk aversion change at retirement on optimal consumption, portfolio, and retirement. Lifetime relative risk aversion is assumed to be constant and to make a one-off jump at retirement. A Cobb–Douglas utility function, which is appropriate for capturing consumption drop at retirement, is employed. The main findings from our analytic solution are as follows. The wealth accumulation for retirement decreases as pre-retirement subsistence consumption increases, and increases as post-retirement subsistence consumption increases. Regardless of the subsistence consumption, a large magnitude of jump in relative risk aversion tends to urge retirement. An interesting finding is that subsistence consumption constraints can affect consumption drop at retirement if the relative risk aversion changes at retirement. Whereas post-retirement subsistence consumption may intensify consumption drop at retirement, pre-retirement subsistence consumption weakens it. We also found that increase in pre-retirement subsistence consumption may lead to decrease in both consumption and investment in the risky asset. In contrast, the magnitude of the jump in relative risk aversion may lead the economic agent to decrease consumption, but to raise investment in the risky asset.
Other than imposing subsistence consumption constraint, considering an incomplete market, for example unhedgeable income risk, is a way to increase realism in investigating lifetime optimal consumption, portfolio, and retirement rules. In this respect, to incorporate income risk in the present paper is a meaningful future research.
Declarations
Acknowledgements
I highly appreciate anonymous reviewers for helpful comments and valuable suggestions.
Funding
This work is supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2016R1D1A1B03933406) and by the Research Grant of Kwangwoon University in 2017.
Authors’ contributions
The author was the only one to contribute to the writing of this paper. The author read and approved the final manuscript.
Competing interests
The author declares to have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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