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An optimal consumption and investment problem with quadratic utility and negative wealth constraints
- Kum-Hwan Roh^{1},
- Ji Yeoun Kim^{2} and
- Yong Hyun Shin^{2}Email author
https://doi.org/10.1186/s13660-017-1469-x
© The Author(s) 2017
- Received: 22 May 2017
- Accepted: 3 August 2017
- Published: 15 August 2017
Abstract
In this paper, we investigate the optimal consumption and portfolio selection problem with negative wealth constraints for an economic agent who has a quadratic utility function of consumption and receives a constant labor income. Due to the property of the quadratic utility function, we separate our problem into two cases and derive the closed-form solutions for each case. We also illustrate some numerical implications of the optimal consumption and portfolio.
Keywords
- consumption
- portfolio selection
- quadratic utility
- negative wealth constraints
- martingale method
1 Introduction
We provide an optimal consumption and portfolio decision with negative wealth constraints for an economic agent who has a quadratic utility function of consumption. A bliss level of consumption is an import feature of the quadratic utility. It means that an agent’s risk taking becomes zero at a wealth level for some bliss point of consumption. When the wealth level exceeds a bliss point, her consumption does not increase. In this model, we derive an analytic solution with the negative wealth constraint. And we check some properties of optimal consumption and portfolio with a constant labor income.
After Merton’s seminal works [1, 2], many researchers have studied an optimal consumption and portfolio selection problem with various realistic constraints. Merton [1, 2] solved the portfolio optimization problem of an agent who has a Hyperbolic Absolute Risk Aversion (HARA) type utility function. However, he has not considered the labor income of an agent. Park and Jang [3] studied the optimal consumption, investment and retirement strategies with negative wealth constraints. But they considered the agent whose utility function is Constant Relative Risk Aversion (CRRA).
Koo et al. [4] and Shin et al. [5] considered an optimal consumption and portfolio selection problem of an agent who has a quadratic utility function and faces a subsistence consumption constraint. However, they did not consider the agent’s labor income and the negative wealth constraint. Here, negative wealth constraints contain the borrowing constraints which mean the restriction of a loan. Lim and Shin [6] and Shim [7] derived the closed-form solutions of optimal consumption and portfolio with general utility. Since the quadratic utility does not satisfy the strictly concave property, our results are different from theirs.
The rest of the paper is organized as follows. In Section 2, we illustrate the financial market setup. In Section 3, we obtain the optimal policies of our optimization problem with two cases (\(\hat{y}>1\) or \(\hat{y}<1\)). Concluding remarks are given in Section 4.
2 The financial market setup
3 The optimization problem
Remark 3.1
Remark 3.2
We define \(\hat{y}>0\) as the level of a dual variable of the negative wealth constraint level \(-\nu{I}/{r}\).
Remark 3.3
By comparison with the range from wealth constraint to the bliss level of wealth and K, we can check that \(\hat{y}>1\) or \(\hat{y}<1\). And we will solve ODEs (3.5) with two cases: one is \(\hat {y}>1\), and the other is \(\hat{y}<1\).
3.1 In the case of \(\hat{y}>1\)
Proposition 3.1
Proof
Theorem 3.1
Proof
When \(\hat{y}>1\), we can provide the optimal strategy \((c,\pi)\).
Theorem 3.2
Proof
Proposition 3.2
Proof
Proposition 3.3
In Proposition 3.1, \(C_{2}\geq0\).
Proof
Refer to Koo et al. [4]. □
3.2 In the case of \(\hat{y}<1\)
Proposition 3.4
Proof
By the duality of value function \(V(x)\), we derive the value function \(V(\cdot)\).
Theorem 3.3
Theorem 3.4
Proposition 3.5
Proof
Remark 3.4
Basically, we obtain similar results to those of Koo et al. [4], that is, if \(\hat{y}>1\), then the optimal consumption is zero until wealth reaches the threshold wealth level corresponding to the dual variable \(y=1\). After wealth reaches this level, the optimal consumption increases as wealth increases. But after the bliss level x̄, the optimal consumption stays at \(\bar{c}=1/(2R)\). For the optimal portfolio, it increases from zero to the certain maximum until wealth reaches a certain wealth level. The optimal portfolio decreases above this level and becomes zero above the bliss level x̄.
If \(0<\hat{y}<1\), however, there is no zero consumption region. So the optimal consumption increases as wealth increases. But after the bliss level x̄, the optimal consumption stays at \(\bar{c}=1/(2R)\). For the optimal portfolio, we see behavior similar to the case of \(\hat{y}>1\).
4 Concluding remarks
We solve the optimal consumption and investment problem with negative wealth constraints. Negative wealth constraints are the general borrowing constraints against future labor income. We consider the optimization problem when an agent receives a constant labor income and has quadratic utility. We use the martingale duality approach to obtain the closed-form solutions and illustrate the effects of the proportion ν of the wealth constraint on the optimal consumption and portfolio.
Declarations
Acknowledgements
We are indebted to an anonymous referee for valuable advice and useful comments, which have improved our paper essentially. Prof. Roh gratefully acknowledges the support of Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2015R1C1A1A02037039), and Ms. Kim and Prof. Shin gratefully acknowledge the support of the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2016R1A2B4008240).
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
References
- Merton, RC: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51, 247-257 (1969) View ArticleGoogle Scholar
- Merton, RC: Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3, 373-413 (1971) MathSciNetView ArticleMATHGoogle Scholar
- Park, SY, Jang, B: Optimal retirement strategy with a negative wealth constrain. Oper. Res. Lett. 42, 208-212 (2014) MathSciNetView ArticleGoogle Scholar
- Koo, JL, Ahn, SR, Koo, BL, Koo, HK, Shin, YH: Optimal consumption and portfolio selection with quadratic utility and a subsistence consumption constraint. Stoch. Anal. Appl. 34, 165-177 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Shin, YH, Koo, JL, Roh, K: Portfolio selection with quadratic utility and subsistence consumption constraints: dynamic programming approaches (submitted) Google Scholar
- Lim, BH, Shin, YH: Optimal investment, consumption and retirement decision with disutility and borrowing constraints. Quant. Finance 11, 165-177 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Shim, G: Degree of borrowing constraints and optimal consumption and investment under a general utility function. Korean Manag. Sci. Rev. 33, 77-87 (2016) View ArticleGoogle Scholar