# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2022023
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## Time-consistent lifetime portfolio selection under smooth ambiguity

 1 China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China 2 Department of Statistics and Actuarial Science, University of Waterloo, Canada 3 School of Statistics, Renmin University of China, Beijing 100874, China

*Corresponding author: Jingzhen Liu

Received  November 2021 Revised  April 2022 Early access May 2022

Fund Project: The corresponding author is supported by National Natural Science Foundation of China (11771466, 11901574)

This paper studies the optimal consumption, life insurance and investment problem for an income earner with uncertain lifetime under smooth ambiguity model. We assume that risky assets have unknown market prices that result in ambiguity. The individual forms his belief, that is, the distribution of market prices, according to available information. His ambiguity attitude, which is similar to the risk attitude described by utility function $U$, is represented by an ambiguity preference function $\phi$. Under the smooth ambiguity model, the problem becomes time-inconsistent. We derive the extended Hamilton-Jacobi-Bellman (HJB) equation for the equilibrium value function and equilibrium strategy. Then, we obtain the explicit solution for the equilibrium strategy when both $U$ and $\phi$ are power functions. We find that a more risk- or ambiguity-averse individual will consume less, buy more life insurance and invest less. Moreover, we find that the Tobin-Markowitz separation theorem is no longer applicable when ambiguity attitude is taken into consideration. The investment strategy will change with the characteristics of the decision maker, such as risk attitude, ambiguity attitude and age.

Citation: Luyang Yu, Liyuan Lin, Guohui Guan, Jingzhen Liu. Time-consistent lifetime portfolio selection under smooth ambiguity. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022023
##### References:
 [1] T. Björk, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance Stoch., 21 2017,331–360. doi: 10.1007/s00780-017-0327-5. [2] H. Chen, M. Sherris, T. Sun and W. G. Zhu, Living with ambiguity: Pricing mortality-linked securites with smooth ambiguity preferences, The Journal of Risk and Insurance, 80 (2013), 705-732. [3] I. Ekeland, O. Mbodji and T. A. Pirvu, Time-consistent portfolio management, SIAM Journal of Financial Math., 3 (2012), 1-32.  doi: 10.1137/100810034. [4] D. Ellsberg, Risk, ambiguity, and the savage axioms, Quart. J. Econom., 75 (1961), 643-669.  doi: 10.2307/1884324. [5] W. Fei, Optimal consumption and portfolio choice with ambiguity and anticipation, Inform. Sci., 177 (2007), 5178-5190.  doi: 10.1016/j.ins.2006.07.028. [6] W. Fei, Optimal portfolio choice based on alpha-MEU under ambiguity, Stochastic Model, 25 (2009), 455-482.  doi: 10.1080/15326340903088826. [7] P. Ghirardato, F. Maccheroni and M. Marinacci, Ambiguity from the Differential Viewpoint, Discussion Paper, ICER, 2002. [8] I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, J. Math. Econom., 18 (1989), 141-153.  doi: 10.1016/0304-4068(89)90018-9. [9] G. Guan, Z. Liang and J. Feng, Time-consistent proportional and investment strategies under ambiguity environment, Insurance Math. Econom., 83 (2018), 122-183.  doi: 10.1016/j.insmatheco.2018.09.007. [10] L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, NJ, 2008.  doi: 10.1515/9781400829385. [11] D. Hu, S. Chen and H. Wang, Robust reinsurance contracts with uncertainty about jump risk, European J. Oper. Res., 266 (2018), 1175-1188.  doi: 10.1016/j.ejor.2017.10.061. [12] D. Hu and H. Wang, Reinsurance contract design when the insurer is ambiguity-averse, European J. Oper. Res., 86 (2019), 241-255.  doi: 10.1016/j.insmatheco.2019.03.007. [13] N. Jensen, Life insurance decisions under recursive utility, Scand. Actuar. J., (2019), 204–227. doi: 10.1080/03461238.2018.1541025. [14] N. Ju and J. Miao, Ambiguity, learning, and asset returns, Econometrica, 80 (2012), 559-591.  doi: 10.3982/ECTA7618. [15] P. Klibanoff, M. Marinacci and S. Mukerji, A smooth model of decision making under ambiguity, Econometrica, 73 (2005), 1849-1892.  doi: 10.1111/j.1468-0262.2005.00640.x. [16] F. Knight, Risk, Uncertainty and Profit, Boston: Houghton Mifflin, 1921. doi: 10.1017/CBO9780511817410.005. [17] B. Li, P. Luo and D. Xiong, Equilibrium strategies for alpha-maxmin expected utility maximization, SIAM J. Financial Math., 10 (2019), 394-429.  doi: 10.1137/18M1178542. [18] Z. X. Liang and X. Y. Zhao, Optimal investment, consumption and life insurance under stochastic framework, Scientia Sinica Mathematica, 46 (2016), 1863-1882. [19] 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. [20] P. J. Maenhout, Robust portfolio rules and asset pricing, The Peview of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003. [21] 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. [22] 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. [23] S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of itô type, Stochastic Analysis and Applications, Abel Symp., Springer, Berlin, 2 (2007), 541-567.  doi: 10.1007/978-3-540-70847-6_25. [24] S. R. Pliska and J. Ye, Optimal life insurance purchase and consumption/investment under uncertain lifetime, Journal of Banking and Finance, (2007), 1307–1319. [25] S. F. Richard, Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model, Journal of Financial Economics, 1975 (1975), 187-203.  doi: 10.1016/0304-405X(75)90004-5. [26] F. T. Seifried, Optimal investment for worst-case crash scenarios: A martingale approach, Math. Oper. Res., 35 (2010), 559-579.  doi: 10.1287/moor.1100.0459. [27] Y. Shen and J. Wei, Optimal investment-consumption-insurance with random parameters, Scand. Actuar. J., (2016), 37–62. doi: 10.1080/03461238.2014.900518. [28] M. Taboga, Portfolio selection with two-stage preferences, Financial Research Letters, 2 (2005), 152-164.  doi: 10.1016/j.frl.2005.06.003.

show all references

##### References:
 [1] T. Björk, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance Stoch., 21 2017,331–360. doi: 10.1007/s00780-017-0327-5. [2] H. Chen, M. Sherris, T. Sun and W. G. Zhu, Living with ambiguity: Pricing mortality-linked securites with smooth ambiguity preferences, The Journal of Risk and Insurance, 80 (2013), 705-732. [3] I. Ekeland, O. Mbodji and T. A. Pirvu, Time-consistent portfolio management, SIAM Journal of Financial Math., 3 (2012), 1-32.  doi: 10.1137/100810034. [4] D. Ellsberg, Risk, ambiguity, and the savage axioms, Quart. J. Econom., 75 (1961), 643-669.  doi: 10.2307/1884324. [5] W. Fei, Optimal consumption and portfolio choice with ambiguity and anticipation, Inform. Sci., 177 (2007), 5178-5190.  doi: 10.1016/j.ins.2006.07.028. [6] W. Fei, Optimal portfolio choice based on alpha-MEU under ambiguity, Stochastic Model, 25 (2009), 455-482.  doi: 10.1080/15326340903088826. [7] P. Ghirardato, F. Maccheroni and M. Marinacci, Ambiguity from the Differential Viewpoint, Discussion Paper, ICER, 2002. [8] I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, J. Math. Econom., 18 (1989), 141-153.  doi: 10.1016/0304-4068(89)90018-9. [9] G. Guan, Z. Liang and J. Feng, Time-consistent proportional and investment strategies under ambiguity environment, Insurance Math. Econom., 83 (2018), 122-183.  doi: 10.1016/j.insmatheco.2018.09.007. [10] L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, NJ, 2008.  doi: 10.1515/9781400829385. [11] D. Hu, S. Chen and H. Wang, Robust reinsurance contracts with uncertainty about jump risk, European J. Oper. Res., 266 (2018), 1175-1188.  doi: 10.1016/j.ejor.2017.10.061. [12] D. Hu and H. Wang, Reinsurance contract design when the insurer is ambiguity-averse, European J. Oper. Res., 86 (2019), 241-255.  doi: 10.1016/j.insmatheco.2019.03.007. [13] N. Jensen, Life insurance decisions under recursive utility, Scand. Actuar. J., (2019), 204–227. doi: 10.1080/03461238.2018.1541025. [14] N. Ju and J. Miao, Ambiguity, learning, and asset returns, Econometrica, 80 (2012), 559-591.  doi: 10.3982/ECTA7618. [15] P. Klibanoff, M. Marinacci and S. Mukerji, A smooth model of decision making under ambiguity, Econometrica, 73 (2005), 1849-1892.  doi: 10.1111/j.1468-0262.2005.00640.x. [16] F. Knight, Risk, Uncertainty and Profit, Boston: Houghton Mifflin, 1921. doi: 10.1017/CBO9780511817410.005. [17] B. Li, P. Luo and D. Xiong, Equilibrium strategies for alpha-maxmin expected utility maximization, SIAM J. Financial Math., 10 (2019), 394-429.  doi: 10.1137/18M1178542. [18] Z. X. Liang and X. Y. Zhao, Optimal investment, consumption and life insurance under stochastic framework, Scientia Sinica Mathematica, 46 (2016), 1863-1882. [19] 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. [20] P. J. Maenhout, Robust portfolio rules and asset pricing, The Peview of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003. [21] 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. [22] 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. [23] S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of itô type, Stochastic Analysis and Applications, Abel Symp., Springer, Berlin, 2 (2007), 541-567.  doi: 10.1007/978-3-540-70847-6_25. [24] S. R. Pliska and J. Ye, Optimal life insurance purchase and consumption/investment under uncertain lifetime, Journal of Banking and Finance, (2007), 1307–1319. [25] S. F. Richard, Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model, Journal of Financial Economics, 1975 (1975), 187-203.  doi: 10.1016/0304-405X(75)90004-5. [26] F. T. Seifried, Optimal investment for worst-case crash scenarios: A martingale approach, Math. Oper. Res., 35 (2010), 559-579.  doi: 10.1287/moor.1100.0459. [27] Y. Shen and J. Wei, Optimal investment-consumption-insurance with random parameters, Scand. Actuar. J., (2016), 37–62. doi: 10.1080/03461238.2014.900518. [28] M. Taboga, Portfolio selection with two-stage preferences, Financial Research Letters, 2 (2005), 152-164.  doi: 10.1016/j.frl.2005.06.003.
Effect of $\alpha$ on consumption
Effect of $\alpha$ on premium
Effect of $\alpha$ on investment
Effect of $\alpha$ on investment share
Effect of $\gamma$ on consumption
Effect of $\gamma$ on premium
Effect of $\gamma$ on investment
Effect of $\gamma$ on investment share
Effects of $\sigma$ and $\gamma$ on risk investment
Effects of $\sigma$ and $\alpha$ on risk investment
Effects of $\mu$ and $\gamma$ on risk investment
Effects of $\mu$ and $\alpha$ on risk investment
Values of the parameters
 Text interpretation Symbol Value risk-free interest rate $r$ 0.03 discount rate $\beta$ 0.04 mortality hazard function $\mu(t)$ $\frac{e^{\frac{t-86.3}{9.5}}}{9.5}$ premium rate $\eta(t)$ $1.1\mu(t)$ income $i(t)$ $50000e^{0.04t}$ retirement time $T$ 40 volatility of risky assets $\sigma_1$, $\sigma_2$ 2.5% distribution of $\lambda_1$ $N(u_1, \Sigma_1)$ $N(0.1, 0.3)$ distribution of $\lambda_2$ $N(u_2, \Sigma_2)$ $N(0.1, 0.5)$ initial wealth $X(0)$ 20000 risk aversion $\gamma$ 2 ambiguity aversion $\alpha$ 2
 Text interpretation Symbol Value risk-free interest rate $r$ 0.03 discount rate $\beta$ 0.04 mortality hazard function $\mu(t)$ $\frac{e^{\frac{t-86.3}{9.5}}}{9.5}$ premium rate $\eta(t)$ $1.1\mu(t)$ income $i(t)$ $50000e^{0.04t}$ retirement time $T$ 40 volatility of risky assets $\sigma_1$, $\sigma_2$ 2.5% distribution of $\lambda_1$ $N(u_1, \Sigma_1)$ $N(0.1, 0.3)$ distribution of $\lambda_2$ $N(u_2, \Sigma_2)$ $N(0.1, 0.5)$ initial wealth $X(0)$ 20000 risk aversion $\gamma$ 2 ambiguity aversion $\alpha$ 2
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