# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022051
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## Optimal portfolio selection with life insurance under subjective survival belief and habit formation

 a. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China b. School of Education, Lanzhou University of Arts and Science, Lanzhou 730000, China c. School of Business, Sun Yat-sen University, Guangzhou 510275, China d. Department of Finance, Business School, Southern University of Science and Technology, Shenzhen 518055, China

*Corresponding author: Zhongfei Li

Received  October 2021 Revised  February 2022 Early access April 2022

Fund Project: This research is supported by the National Natural Science Foundation of China (Nos. 71721001, 71991474)

This paper studies the optimal consumption, investment, and life insurance choices for a wage earner with subjective survival beliefs and habit formation types. The wage-earner has access to a risk-free asset, an index bond, and a stock in a financial market. Introducing subjective survival beliefs describes the wage earner's optimistic or pessimistic attitude towards the longevity risk. Two types of habit formation are considered: one depends on real past consumption; the other is incited by money illusion and depends on nominal past consumption. The aim is to maximize the expected utility of real consumption, total legacy, and terminal wealth, where the utility of consumption comes from the part of real consumption that exceeds real habit level. Using the dynamic programming method, we provide and prove a verification theorem and obtain the closed-form solution of the optimization problem. Numerical results reveal that subjective survival beliefs and habit formation types play important roles in the financial behaviors of the wage-earner. Detailed results are exhibited. Especially, subjective survival beliefs, the existence of habit formation, and the relative risk aversion coefficient affect the demand for life insurance.

Citation: Ailing Shi, Xingyi Li, Zhongfei Li. Optimal portfolio selection with life insurance under subjective survival belief and habit formation. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022051
##### References:
 [1] C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance Math. Econom., 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005. [2] S. Benartzi, A. Previtero and R. H. Thaler, Annuitization puzzles, Journal of Economic Perspectives, 25 (2011), 143-164.  doi: 10.1257/jep.25.4.143. [3] P. Boyle, K. S. Tan, P. Wei and S. C. Zhuang, Annuity and insurance choice under habit formation, Available at SSRN, 2020. doi: 10.2139/ssrn.3570066. [4] M. Brennan and Y. Xia, Dynamic asset allocation under inflation, Journal of Finance, 57 (2002), 1201-1238. [5] M. Browning and M. D. Collado, Habits and heterogeneity in demands: A panel data analysis, J. Appl. Econometrics, 22 (2007), 625-640.  doi: 10.1002/jae.952. [6] A. Chen, P. Hieber and M. Rach, Optimal retirement products under subjective mortality beliefs, Insurance Math. Econom., 101 (2021), 55-69.  doi: 10.1016/j.insmatheco.2020.07.002. [7] R. Chen, K. A. Wong and H. C. Lee, Age, period, and cohort effects on life insurance purchases in the U.S, Journal of Risk and Insurance, 68 (2001), 303-327.  doi: 10.2307/2678104. [8] Z. Chen, Z. Li, Y. Zeng and J. Sun, Asset allocation under loss aversion and minimum performance constraint in a DC pension plan with inflation risk, Insurance Math. Econom., 75 (2017), 137-150.  doi: 10.1016/j.insmatheco.2017.05.009. [9] G. M. Constantinides, Habit formation: A resolution of the equity premium puzzle, Journal of Political Economy, 98 (1990), 519-543.  doi: 10.1086/261693. [10] J. B. Detemple and F. Zapatero, Asset prices in an exchange economy with habit formation, Econometrica, 59 (1991), 1633-1657.  doi: 10.2307/2938283. [11] P. H. Dybvig and H. Liu, Lifetime consumption and investment: Retirement and constrained borrowing, J. Econom. Theory, 145 (2010), 885-907.  doi: 10.1016/j.jet.2009.08.003. [12] L. Eeckhoudt and H. Schlesinger, Increases in prudence and increases in risk aversion, Economics Letters, 45 (1994), 51-53.  doi: 10.1016/0165-1765(94)90056-6. [13] J. G. Eisenhauer and M. Halek, Prudence, risk aversion, and the demand for life insurance, Applied Economic Letters, 6 (1999), 239-242.  doi: 10.1080/135048599353429. [14] A. Gu, X. Guo, Z. Li and Y. Zeng, Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance Math. Econom., 51 (2012), 674-684.  doi: 10.1016/j.insmatheco.2012.09.003. [15] R. Z. Heimer, K. O. R. Myrseth and R. S. Schoenle, YOLO: Mortality beliefs and household finance puzzles, Journal of Finance, 74 (2019), 2957-2996. [16] F. D. Jong and Z. Yang, Portfolio and consumption choice with habit formation under inflation, SSRN Electronic Journal. [17] N. Keyfitz and H. Caswell, Applied mathematical demography, 3$^{nd}$ edition, Keyfitz, N., Caswell, H., Springer-Verlag, New York, 2005. [18] H. Kraft, C. Munk and S. Wagner, Housing habits and their implications for life-cycle consumption and investment, Review of Finance, 22 (2018), 1737-1762. [19] M. Kwak and B. H. Lim, Optimal portfolio selection with life insurance under inflation risk, Journal of Banking and Finance, 46 (2014), 59-71. [20] 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. [21] J. Liu, L. Lin, K. F. C. Yiu and J. Wei, Non-exponential discounting portfolio management with habit formation, Math. Control Relat. Fields, 10 (2020), 761-783.  doi: 10.3934/mcrf.2020019. [22] J. Liu, Y. Wang and Z. Ming, Utility maximization with habit formation of interaction, J. Ind. Manag. Optim., 17 (2021), 1451-1469.  doi: 10.3934/jimo.2020029. [23] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560. [24] C. Munk, Portfolio and consumption choice with stochastic investment opportunities and habit formation in preferences, J. Econom. Dynam. Control, 32 (2008), 3560-3589.  doi: 10.1016/j.jedc.2008.02.005. [25] B. Øksendal, Stochastic Differntial Equations, 6$^{th}$ edition, Springer-verlag, Berlin, 2006. doi: 10.1007/978-3-642-14394-6. [26] S. R. Pliska and J. Ye, Optimal life insurance purchase and consumption/investment under uncertain lifetime, Journal of Banking & Finance, 31 (2007), 1307-1319. [27] G. Rabitti and E. Borgonovo, Is mortality or interest rate the most important risk in annuity models? A comparison of sensitivity analysis methods, Insurance Math. Econom., 95 (2020), 48-58.  doi: 10.1016/j.insmatheco.2020.09.001. [28] 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. [29] P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, The Review of Economics and Statistics, 51 (1969), 239-246. [30] H. Schlesinger, Insurance demand without the expected-utility paradigm, Journal of Risk and Insurance, 64 (1997), 19-39.  doi: 10.2307/253910. [31] S. M. Sundaresan, Intertemporally dependent preferences and the volatility of consumption and wealth, Review of Financial Studies, 2 (1989), 73-89.  doi: 10.1093/rfs/2.1.73. [32] P. Wang and Z. Li, Robust optimal investment strategy for an AAM of DC pension plans with stochastic interest rate and stochastic volatility, Insurance Math. Econom., 80 (2018), 67-83.  doi: 10.1016/j.insmatheco.2018.03.003. [33] J. Wei, X. Cheng, Z. Jin and H. Wang, Optimal consumption investment and life-insurance purchase strategy for couples with correlated lifetimes, Insurance Math. Econom., 91 (2020), 244-256.  doi: 10.1016/j.insmatheco.2020.02.006. [34] H. Yao, P. Chen, M. Zhang and X. Li, Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk, J. Ind. Manag. Optim., 18 (2022), 511-540.  doi: 10.3934/jimo.2020166. [35] J. Ye, Stochastic utilities with subsistence and satiation: Optimal life insurance purchase, consumption and investment, Insurance Math. Econom., 89 (2019), 193-212.  doi: 10.1016/j.insmatheco.2019.10.008. [36] J. Zhang, S. Purcal and J. Wei, Optimal life insurance and annuity demand under hyperbolic discounting when bequests are luxury goods, Insurance Math. Econom., 101 (2021), 80-90.  doi: 10.1016/j.insmatheco.2020.07.003.

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##### References:
 [1] C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance Math. Econom., 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005. [2] S. Benartzi, A. Previtero and R. H. Thaler, Annuitization puzzles, Journal of Economic Perspectives, 25 (2011), 143-164.  doi: 10.1257/jep.25.4.143. [3] P. Boyle, K. S. Tan, P. Wei and S. C. Zhuang, Annuity and insurance choice under habit formation, Available at SSRN, 2020. doi: 10.2139/ssrn.3570066. [4] M. Brennan and Y. Xia, Dynamic asset allocation under inflation, Journal of Finance, 57 (2002), 1201-1238. [5] M. Browning and M. D. Collado, Habits and heterogeneity in demands: A panel data analysis, J. Appl. Econometrics, 22 (2007), 625-640.  doi: 10.1002/jae.952. [6] A. Chen, P. Hieber and M. Rach, Optimal retirement products under subjective mortality beliefs, Insurance Math. Econom., 101 (2021), 55-69.  doi: 10.1016/j.insmatheco.2020.07.002. [7] R. Chen, K. A. Wong and H. C. Lee, Age, period, and cohort effects on life insurance purchases in the U.S, Journal of Risk and Insurance, 68 (2001), 303-327.  doi: 10.2307/2678104. [8] Z. Chen, Z. Li, Y. Zeng and J. Sun, Asset allocation under loss aversion and minimum performance constraint in a DC pension plan with inflation risk, Insurance Math. Econom., 75 (2017), 137-150.  doi: 10.1016/j.insmatheco.2017.05.009. [9] G. M. Constantinides, Habit formation: A resolution of the equity premium puzzle, Journal of Political Economy, 98 (1990), 519-543.  doi: 10.1086/261693. [10] J. B. Detemple and F. Zapatero, Asset prices in an exchange economy with habit formation, Econometrica, 59 (1991), 1633-1657.  doi: 10.2307/2938283. [11] P. H. Dybvig and H. Liu, Lifetime consumption and investment: Retirement and constrained borrowing, J. Econom. Theory, 145 (2010), 885-907.  doi: 10.1016/j.jet.2009.08.003. [12] L. Eeckhoudt and H. Schlesinger, Increases in prudence and increases in risk aversion, Economics Letters, 45 (1994), 51-53.  doi: 10.1016/0165-1765(94)90056-6. [13] J. G. Eisenhauer and M. Halek, Prudence, risk aversion, and the demand for life insurance, Applied Economic Letters, 6 (1999), 239-242.  doi: 10.1080/135048599353429. [14] A. Gu, X. Guo, Z. Li and Y. Zeng, Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance Math. Econom., 51 (2012), 674-684.  doi: 10.1016/j.insmatheco.2012.09.003. [15] R. Z. Heimer, K. O. R. Myrseth and R. S. Schoenle, YOLO: Mortality beliefs and household finance puzzles, Journal of Finance, 74 (2019), 2957-2996. [16] F. D. Jong and Z. Yang, Portfolio and consumption choice with habit formation under inflation, SSRN Electronic Journal. [17] N. Keyfitz and H. Caswell, Applied mathematical demography, 3$^{nd}$ edition, Keyfitz, N., Caswell, H., Springer-Verlag, New York, 2005. [18] H. Kraft, C. Munk and S. Wagner, Housing habits and their implications for life-cycle consumption and investment, Review of Finance, 22 (2018), 1737-1762. [19] M. Kwak and B. H. Lim, Optimal portfolio selection with life insurance under inflation risk, Journal of Banking and Finance, 46 (2014), 59-71. [20] 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. [21] J. Liu, L. Lin, K. F. C. Yiu and J. Wei, Non-exponential discounting portfolio management with habit formation, Math. Control Relat. Fields, 10 (2020), 761-783.  doi: 10.3934/mcrf.2020019. [22] J. Liu, Y. Wang and Z. Ming, Utility maximization with habit formation of interaction, J. Ind. Manag. Optim., 17 (2021), 1451-1469.  doi: 10.3934/jimo.2020029. [23] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560. [24] C. Munk, Portfolio and consumption choice with stochastic investment opportunities and habit formation in preferences, J. Econom. Dynam. Control, 32 (2008), 3560-3589.  doi: 10.1016/j.jedc.2008.02.005. [25] B. Øksendal, Stochastic Differntial Equations, 6$^{th}$ edition, Springer-verlag, Berlin, 2006. doi: 10.1007/978-3-642-14394-6. [26] S. R. Pliska and J. Ye, Optimal life insurance purchase and consumption/investment under uncertain lifetime, Journal of Banking & Finance, 31 (2007), 1307-1319. [27] G. Rabitti and E. Borgonovo, Is mortality or interest rate the most important risk in annuity models? A comparison of sensitivity analysis methods, Insurance Math. Econom., 95 (2020), 48-58.  doi: 10.1016/j.insmatheco.2020.09.001. [28] 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. [29] P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, The Review of Economics and Statistics, 51 (1969), 239-246. [30] H. Schlesinger, Insurance demand without the expected-utility paradigm, Journal of Risk and Insurance, 64 (1997), 19-39.  doi: 10.2307/253910. [31] S. M. Sundaresan, Intertemporally dependent preferences and the volatility of consumption and wealth, Review of Financial Studies, 2 (1989), 73-89.  doi: 10.1093/rfs/2.1.73. [32] P. Wang and Z. Li, Robust optimal investment strategy for an AAM of DC pension plans with stochastic interest rate and stochastic volatility, Insurance Math. Econom., 80 (2018), 67-83.  doi: 10.1016/j.insmatheco.2018.03.003. [33] J. Wei, X. Cheng, Z. Jin and H. Wang, Optimal consumption investment and life-insurance purchase strategy for couples with correlated lifetimes, Insurance Math. Econom., 91 (2020), 244-256.  doi: 10.1016/j.insmatheco.2020.02.006. [34] H. Yao, P. Chen, M. Zhang and X. Li, Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk, J. Ind. Manag. Optim., 18 (2022), 511-540.  doi: 10.3934/jimo.2020166. [35] J. Ye, Stochastic utilities with subsistence and satiation: Optimal life insurance purchase, consumption and investment, Insurance Math. Econom., 89 (2019), 193-212.  doi: 10.1016/j.insmatheco.2019.10.008. [36] J. Zhang, S. Purcal and J. Wei, Optimal life insurance and annuity demand under hyperbolic discounting when bequests are luxury goods, Insurance Math. Econom., 101 (2021), 80-90.  doi: 10.1016/j.insmatheco.2020.07.003.
Effects of subjective survival beliefs on the optimal strategies
The effects of habit formation types on the optimal strategies. 'NHF', 'RHF', and 'NMHF' denote no habit formation, real habit formation, and nominal habit formation respectively
Under different subjective survival beliefs, the effects of the relative risk aversion coefficient on the optimal strategies
Under different habit formation types, the effects of the relative risk aversion coefficient on the optimal strategies
Values of parameters
 $T$ $\rho_{IS}$ $\mu_I$ $\sigma_I$ $r$ $R$ $\mu_S$ $\sigma_S$ $\alpha_1$ 30 -0.07 0.023 0.05 0.02 0.04 0.08 0.2 1 $\alpha_2$ $\alpha_3$ $\gamma$ $\rho$ $\xi_1$ $\xi_2$ $X(0)$ $h(0)$ 300 300 3 0.07 0.1 0.174 10 0.253
 $T$ $\rho_{IS}$ $\mu_I$ $\sigma_I$ $r$ $R$ $\mu_S$ $\sigma_S$ $\alpha_1$ 30 -0.07 0.023 0.05 0.02 0.04 0.08 0.2 1 $\alpha_2$ $\alpha_3$ $\gamma$ $\rho$ $\xi_1$ $\xi_2$ $X(0)$ $h(0)$ 300 300 3 0.07 0.1 0.174 10 0.253
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