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

  • *Corresponding author: Shiqi Yan

    *Corresponding author: Shiqi Yan 

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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  • 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.

    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

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

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

    Figure 4.  Effect of $ \beta $ on consumption

    Figure 5.  Effect of $ \beta $ on bequest

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

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

    Figure 8.  Effect of $ \alpha $ on consumption

    Figure 9.  Effect of $ \alpha $ on bequest

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

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

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

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

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

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

    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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