# American Institute of Mathematical Sciences

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

Received  October 2021 Revised  February 2022 Early access March 2022

Fund Project: 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)

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.

Citation: Jingzhen Liu, Shiqi Yan, Shan Jiang, Jiaqin Wei. Optimal investment, consumption and life insurance strategies under stochastic differential utility with habit formation. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022040
##### References:
 [1] A. B. Abel, Asset prices under habit formation and catching up with the joneses, The American Economic Review, 80 (1990), 38-42.  doi: 10.3386/w3279. [2] C. D. Carroll and D. N. Weil, Saving and growth: A reinterpretation, In Carnegie-Rochester conference series on public policy, Elsevier, 40 (1994), 133–192. doi: 10.3386/w4470. [3] F. De Jong and Y. Zhou, Portfolio and consumption choice with habit formation under inflation, Working Paper. [4] D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394.  doi: 10.2307/2951600. [5] L. G. Epstein and S. E. Zin, Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis, Journal of political Economy, 99 (1991), 263-286. [6] N. Gârleanu and S. Panageas, Young, old, conservative, and bold: The implications of heterogeneity and finite lives for asset pricing, Journal of Political Economy, 123 (2015), 670-685. [7] R. E. Hall, Intertemporal substitution in consumption, Journal of political economy, 96 (1988), 339-357.  doi: 10.3386/w0720. [8] J. Hicks, Capital and growth, New York Journal of Applied Econometrics, 28 (1965), 527-550. [9] N. R. Jensen, Life insurance decisions under recursive utility, Scand. Actuar. J., 2019 (2019), 204-227.  doi: 10.1080/03461238.2018.1541025. [10] H. Kraft and F. T. Seifried, Foundations of continuous-time recursive utility: Differentiability and normalization of certainty equivalents, Math. Financ. Econ., 3 (2010), 115-138.  doi: 10.1007/s11579-010-0030-1. [11] H. Kraft and F. T. Seifried, Stochastic differential utility as the continuous-time limit of recursive utility, J. Econom. Theory, 151 (2014), 528-550.  doi: 10.1016/j.jet.2013.12.007. [12] H. Kraft, F. T. Seifried and M. Steffensen, Consumption-portfolio optimization with recursive utility in incomplete markets, Finance Stoch., 17 (2013), 161-196.  doi: 10.1007/s00780-012-0184-1. [13] D. M. Kreps and E. L. Porteus, Temporal resolution of uncertainty and dynamic choice theory, Econometrica, 46 (1978), 185-200.  doi: 10.2307/1913656. [14] N. Li, G. Wang and Z. Wu, Linear–quadratic optimal control for time-delay stochastic system with recursive utility under full and partial information, Automatica, 121 (2020), 109169, 9 pp. doi: 10.1016/j.automatica.2020.109169. [15] W. Li, K. S. Tan and P. Wei, Demand for non-life insurance under habit formation, Insurance Math. Econom., 101 (2021), 38-54.  doi: 10.1016/j.insmatheco.2020.06.012. [16] J. Liu, L. Lin and H. Meng, Optimal consumption, life insurance and investment decisions with habit formation, Acta Math. Appl. Sin., 43 (2020), 517-534. [17] R. Mehra and E. C. Prescott, The equity premium: A puzzle, Journal of Monetary Economics, 15 (1985), 145-161.  doi: 10.1016/0304-3932(85)90061-3. [18] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560. [19] S. F. Richard, Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model, Journal of Financial Economics, 2 (1975), 187-203.  doi: 10.1016/0304-405X(75)90004-5. [20] M. E. Yaari, Uncertain lifetime, life insurance, and the theory of the consumer, The Review of Economic Studies, 32 (1965), 137-150.  doi: 10.2307/2296058.

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##### References:
 [1] A. B. Abel, Asset prices under habit formation and catching up with the joneses, The American Economic Review, 80 (1990), 38-42.  doi: 10.3386/w3279. [2] C. D. Carroll and D. N. Weil, Saving and growth: A reinterpretation, In Carnegie-Rochester conference series on public policy, Elsevier, 40 (1994), 133–192. doi: 10.3386/w4470. [3] F. De Jong and Y. Zhou, Portfolio and consumption choice with habit formation under inflation, Working Paper. [4] D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394.  doi: 10.2307/2951600. [5] L. G. Epstein and S. E. Zin, Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis, Journal of political Economy, 99 (1991), 263-286. [6] N. Gârleanu and S. Panageas, Young, old, conservative, and bold: The implications of heterogeneity and finite lives for asset pricing, Journal of Political Economy, 123 (2015), 670-685. [7] R. E. Hall, Intertemporal substitution in consumption, Journal of political economy, 96 (1988), 339-357.  doi: 10.3386/w0720. [8] J. Hicks, Capital and growth, New York Journal of Applied Econometrics, 28 (1965), 527-550. [9] N. R. Jensen, Life insurance decisions under recursive utility, Scand. Actuar. J., 2019 (2019), 204-227.  doi: 10.1080/03461238.2018.1541025. [10] H. Kraft and F. T. Seifried, Foundations of continuous-time recursive utility: Differentiability and normalization of certainty equivalents, Math. Financ. Econ., 3 (2010), 115-138.  doi: 10.1007/s11579-010-0030-1. [11] H. Kraft and F. T. Seifried, Stochastic differential utility as the continuous-time limit of recursive utility, J. Econom. Theory, 151 (2014), 528-550.  doi: 10.1016/j.jet.2013.12.007. [12] H. Kraft, F. T. Seifried and M. Steffensen, Consumption-portfolio optimization with recursive utility in incomplete markets, Finance Stoch., 17 (2013), 161-196.  doi: 10.1007/s00780-012-0184-1. [13] D. M. Kreps and E. L. Porteus, Temporal resolution of uncertainty and dynamic choice theory, Econometrica, 46 (1978), 185-200.  doi: 10.2307/1913656. [14] N. Li, G. Wang and Z. Wu, Linear–quadratic optimal control for time-delay stochastic system with recursive utility under full and partial information, Automatica, 121 (2020), 109169, 9 pp. doi: 10.1016/j.automatica.2020.109169. [15] W. Li, K. S. Tan and P. Wei, Demand for non-life insurance under habit formation, Insurance Math. Econom., 101 (2021), 38-54.  doi: 10.1016/j.insmatheco.2020.06.012. [16] J. Liu, L. Lin and H. Meng, Optimal consumption, life insurance and investment decisions with habit formation, Acta Math. Appl. Sin., 43 (2020), 517-534. [17] R. Mehra and E. C. Prescott, The equity premium: A puzzle, Journal of Monetary Economics, 15 (1985), 145-161.  doi: 10.1016/0304-3932(85)90061-3. [18] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560. [19] S. F. Richard, Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model, Journal of Financial Economics, 2 (1975), 187-203.  doi: 10.1016/0304-405X(75)90004-5. [20] M. E. Yaari, Uncertain lifetime, life insurance, and the theory of the consumer, The Review of Economic Studies, 32 (1965), 137-150.  doi: 10.2307/2296058.
Comparison of the expected optimal consumption rate with and without habit formation
Comparison of the bequests with and without habit formation
Comparison of actual consumption with and without habit formation
Effect of $\beta$ on consumption
Effect of $\beta$ on bequest
Effect of $\beta$ on habit formation
Effect of $\beta$ on actual consumption
Effect of $\alpha$ on consumption
Effect of $\alpha$ on bequest
Effect of $\alpha$ on habit formation
Effect of $\alpha$ on actual consumption
Effect of $h_{0}$ on consumption
Effect of $h_{0}$ on bequest
Effect of $h_{0}$ on habit formation
Effect of $h_{0}$ on actual consumption
Values of the parameters
 $r$ $\sigma$ $\lambda$ $\delta$ $\rho$ $\phi$ $\alpha$ $\beta$ $h_{0}$ $0.05$ $0.2$ $0.07$ $0.08$ $2$ $8$ $0.3$ $0.4$ $400$
 $r$ $\sigma$ $\lambda$ $\delta$ $\rho$ $\phi$ $\alpha$ $\beta$ $h_{0}$ $0.05$ $0.2$ $0.07$ $0.08$ $2$ $8$ $0.3$ $0.4$ $400$
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