Optimal investment, consumption and life insurance strategies under stochastic differential utility with habit formation

Jingzhen Liu; Shiqi Yan; Shan Jiang; Jiaqin Wei

doi:10.3934/jimo.2022040

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2023, Volume 19, Issue 3: 2226-2250. Doi: 10.3934/jimo.2022040

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Optimal investment, consumption and life insurance strategies under stochastic differential utility with habit formation

1.
China Institute for Actuarial Science, Central University of Finance and Economics, Beijing, 100081, China
2.
Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, Shanghai, 200241, China

^*Corresponding author: Shiqi Yan
^*Corresponding author: Shiqi Yan

Received: September 30, 2021

Revised: January 31, 2022

Early access: March 2022

Published: March 2023

This research was supported by National Natural Science Foundation of China (Grant No. 11771466 and No. 12071146) and CUFE Postgraduate students support program for the integration of research and teaching (Grant No. 202117)

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Abstract

This paper studies the optimal investment, consumption and life insurance decisions of an agent under stochastic differential utility. The optimal choice is obtained through dynamic programming method. We state a verification theorem using the Hamilton-Jacobi-Bellman equation. For the special case of Epstein-Zin preferences, we derive the analytical solution to the problem. Moreover, we explore the effects of habit formation and of the elasticity of the utility function on the optimal decision through a numerical simulation based on Chinese mortality rates. We show that habit formation does not change the basic shape of the consumption and bequest curves. With habit formation, the optimal consumption curve moves up with lower initial consumption, while the bequest curve moves down. Increasing the value of initial habit formation slightly decreases both optimal consumption and bequests. The changes in the habit formation parameters have a greater impact on the curves than does a change in the initial habit formation.

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Mathematics Subject Classification: Primary: 91G05, 49L20; Secondary: 93E20.

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Figure 1. Comparison of the expected optimal consumption rate with and without habit formation

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Figure 2. Comparison of the bequests with and without habit formation

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Figure 3. Comparison of actual consumption with and without habit formation

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Figure 4. Effect of $ \beta $ on consumption

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Figure 5. Effect of $ \beta $ on bequest

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Figure 6. Effect of $ \beta $ on habit formation

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Figure 7. Effect of $ \beta $ on actual consumption

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Figure 8. Effect of $ \alpha $ on consumption

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Figure 9. Effect of $ \alpha $ on bequest

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Figure 10. Effect of $ \alpha $ on habit formation

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Figure 11. Effect of $ \alpha $ on actual consumption

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Figure 12. Effect of $ h_{0} $ on consumption

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Figure 13. Effect of $ h_{0} $ on bequest

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Figure 14. Effect of $ h_{0} $ on habit formation

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Figure 15. Effect of $ h_{0} $ on actual consumption

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Table 1. Values of the parameters

$ r $	$ \sigma $	$ \lambda $	$ \delta $	$ \rho $	$ \phi $	$ \alpha $	$ \beta $	$ h_{0} $
$ 0.05 $	$ 0.2 $	$ 0.07 $	$ 0.08 $	$ 2 $	$ 8 $	$ 0.3 $	$ 0.4 $	$ 400 $

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