Optimal Consumption, Life Insurance and Investment Decision with Habit Formation

LIU Jingzhen, LIN Liyuan, MENG Hui

Acta Mathematicae Applicatae Sinica ›› 2020, Vol. 43 ›› Issue (3) : 517-534.

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Acta Mathematicae Applicatae Sinica ›› 2020, Vol. 43 ›› Issue (3) : 517-534. DOI: 10.12387/C2020040

Optimal Consumption, Life Insurance and Investment Decision with Habit Formation

  • LIU Jingzhen1, LIN Liyuan2, MENG Hui1
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Abstract

In this paper, we consider the optimal consumption, life insurance and investment problem with habit formation. We assume that the individual invests his wealth in a risk-free asset and a risky asset, and he purchases life insurance before retirement. His goal is to maximize the utility which consists of the utility from consumption before retirement or death, wealth at the retirement and heritage. From the dynamic programming principle, the problem is reduced to solve HJB equation. We obtain the explicit solutions of optimal strategy for the CRRA case. The result is that when the habit formation is included, the individual's investment amount is smaller than that without habit formation. Furthermore, the low bound of the optimal consumption depends on time, while it's zero in the classic case. If the individual's relative risk aversion is greater than 1, we also found that both the volatility of optimal consumption and the optimal insurance purchase are smaller than that in the classic case. Then we use the data in China to carry out numerical simulation in order to analyze how habit formation influences the optimal strategy. We found that when the habit formation level becomes higher, the difference of consumption between early and late periods become more significant, and the investment and the insurance purchase become less. The stronger the adaptability of the individual is, the smoother the consumption would be and the individual has stronger ability to take risks which lead to more investment in risk asset.

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habit formation / life insurance / dynamic control / optimal strategy / utility function

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LIU Jingzhen, LIN Liyuan, MENG Hui. Optimal Consumption, Life Insurance and Investment Decision with Habit Formation. Acta Mathematicae Applicatae Sinica, 2020, 43(3): 517-534 https://doi.org/10.12387/C2020040

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