# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021228
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## Optimal portfolios for the DC pension fund with mispricing under the HARA utility framework

 1 School of Business, Hunan Normal University, Changsha 410081, China 2 MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China 3 Key Laboratory of Applied Statistics and Data Science, College of Hunan Province, Hunan Normal University, Changsha 410081, China

*Corresponding author: Ya Huang and Jieming Zhou

Received  July 2021 Revised  October 2021 Early access January 2022

Fund Project: The research is supported by National Social Science Foundation of China (no. 20BJY264)

This paper studies the optimal portfolio selection for defined contribution (DC) pension fund with mispricing. We adopt the general hyperbolic absolute risk averse (HARA) utility to describe the risk performance of the pension fund managers. The financial market comprises a risk-free asset, a pair of mispriced stocks, and the market index. Using the dynamic programming approach, we construct the Hamilton-Jacobi-Bellman (HJB) equation and obtain the explicit expressions for optimal portfolio choices with two methods. Finally, numerical analysis is presented to illustrate the sensitivity of the optimal portfolios to parameters of the financial market and contribution process. 200 words.

Citation: Zilan Liu, Yijun Wang, Ya Huang, Jieming Zhou. Optimal portfolios for the DC pension fund with mispricing under the HARA utility framework. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021228
##### References:
 [1] P. Battocchio and F. Menoncin, Optimal pension management in a stochastic framework, Insurance Math. Econom., 34 (2004), 79-95.  doi: 10.1016/j.insmatheco.2003.11.001. [2] L. H. Bian, Z. F. Li and H. X. Yao, Pre-commitment and equilibrium investment strategies for the DC pension plan with regime switching and a return of premiums clause, Insurance Math. Econom., 81 (2018), 78-94.  doi: 10.1016/j.insmatheco.2018.05.005. [3] J. Boulier, S. J. Huang and G. Taillard, Optimal management under stochatic interest rates: The case of a protected defined contribution pension fund, Insurance Math. Econom., 28 (2001), 173-189.  doi: 10.1016/S0167-6687(00)00073-1. [4] H. Chang and K. Chang, Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform, Insurance Math. Econom., 72 (2017), 215-227.  doi: 10.1016/j.insmatheco.2016.10.014. [5] H. Chang, J. A. Li and H. Zhao, Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria, Journal of Industrial and Management Optimization, early access, 2021. doi: 10.3934/jimo.2021025. [6] H. Chang, C. F. Wang, Z. M. Fang and D. Ma, Defined contribution pension planning with a stochastic interest rate and mean-reverting returns under the hyperbolic absolute risk aversion preference, IMA J. Manag. Math., 31 (2020), 167-189.  doi: 10.1093/imaman/dpz009. [7] Z. Chen, Z. F. Li, Y. Zeng and J. Y. 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. [8] Y. H. Dong and H. Zheng, Optimal investment of DC pension plan under short-selling constraints and portfolio insurance, Insurance Math. Econom., 85 (2019), 47-59.  doi: 10.1016/j.insmatheco.2018.12.005. [9] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2$^{nd}$ edition, Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. doi: 10.1007/0-387-31071-1. [10] J. W. Gao, Stochastic optimal control of DC pension funds, Insurance Math. Econom., 42 (2008), 1159-1164.  doi: 10.1016/j.insmatheco.2008.03.004. [11] R. Gerrard, S. Haberman and E. Vigna, Optimal investment choices post-retirement in a defined contribution pension scheme, Insurance Math. Econom., 35 (2004), 321-342.  doi: 10.1016/j.insmatheco.2004.06.002. [12] A. L. Gu, F. G. Viens and H. X. Yao, Optimal robust reinsurance-investment strategies for insurers with mean reversion and mispricing, Insurance Math. Econom., 80 (2018), 93-109.  doi: 10.1016/j.insmatheco.2018.03.004. [13] A. L. Gu, F. G. Viens and B. Yi, Optimal reinsurance and investment strategies for insurers with mispricing and model ambiguity, Insurance Math. Econom., 72 (2017), 235-249.  doi: 10.1016/j.insmatheco.2016.11.007. [14] C. Guambe, R. Kufakunesu, G. V. Zyl and C. Beyers, Optimal asset allocation for a DC plan with partial information under inflation and mortality risks, Comm. Statist. Theory Methods, 50 (2021), 2048-2061.  doi: 10.1080/03610926.2019.1657458. [15] G. H. Guan and Z. X. Liang, Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework, Insurance Math. Econom., 57 (2014), 58-66.  doi: 10.1016/j.insmatheco.2014.05.004. [16] G. H. Guan and Z. X. Liang, Mean-variance efficiency of DC pension plan under stochastic interest rate and mean-reverting returns, Insurance Math. Econom., 61 (2015), 99-109.  doi: 10.1016/j.insmatheco.2014.12.006. [17] N. W. Han and M. W. Hung, Optimal asset allocation for DC pension plans under inflation, Insurance Math. Econom., 51 (2012), 172-181.  doi: 10.1016/j.insmatheco.2012.03.003. [18] L. He and Z. X. Liang, Optimal investment strategy for the DC plan with the return of premiums clauses in a mean-variance framework, Insurance Math. Econom., 53 (2013), 643-649.  doi: 10.1016/j.insmatheco.2013.09.002. [19] M. Jonsson and R. Sircar, Optimal investment problems and volatility homogenization approximations, Modern Methods in Scientific Computing and Applications, 75 (2002), 255-281.  doi: 10.1007/978-94-010-0510-4. [20] E. J. Jung and J. H. Kim, Optimal investment strategies for the HARA utility under the constant elasticity of variance model, Insurance Math. Econom., 51 (2012), 667-673.  doi: 10.1016/j.insmatheco.2012.09.009. [21] O. A. Lamont and R. H. Thaler, Anomalies: The law of one price in financial markets, Journal of Economic Perspectives, 17 (2003), 191-202.  doi: 10.1257/089533003772034952. [22] D. P. Li, X. M. Rong, H. Zhao and B. Yi, Equilibrium investment strategy for DC pension plan with default risk and return of premiums clauses under CEV model, Insurance Math. Econom., 72 (2017), 6-20.  doi: 10.1016/j.insmatheco.2016.10.007. [23] Y. W. Li, S. Y. Wang, Y. Zeng and H. Qiao, Equilibrium investment strategy for a DC plan with partial information and mean-variance criterion, IEEE Systems Journal, 11 (2017), 1492-1504.  doi: 10.1109/JSYST.2016.2533920. [24] J. Liu and F. A. Longstaff, Losing money on arbitrage: Optimal dynamic portfolio choice in markets with arbitrage opportunities, The Review of Financial Studies, 17 (2003), 611-641.  doi: 10.1093/rfs/hhg029. [25] J. Liu and A. Timmermann, Optimal convergence trade strategies, The Review of Financial Studies, 26 (2013), 1048-1086.  doi: 10.1093/rfs/hhs130. [26] J. Ma, H. Zhao and X. M. Rong, Optimal investment strategy for a DC pension plan with mispricing under the Heston model, Comm. Statist. Theory Methods, 49 (2020), 3168-3183.  doi: 10.1080/03610926.2019.1586938. [27] J. Y. Sun, Y. J. Li and L. Zhang, Robust portfolio choice for a defined contribution pension plan with stochastic income and interest rate, Comm. Statist. Theory Methods, 47 (2018), 4106-4130.  doi: 10.1080/03610926.2017.1367815. [28] L. Y. Wang, Z. P. Chen and P. Yang, Robust equilibrium control-measure policy for a DC pension plan with state-dependent risk aversion under mean-variance criterion, J. Ind. Manag. Optim., 17 (2021), 1203-1233.  doi: 10.3934/jimo.2020018. [29] P. Wang, L. Zhang and Z. F. Li, Asset allocation for a DC pension plan with learning about stock return predictability, Journal of Industrial and Management Optimization, early access, 2021. doi: 10.3934/jimo.2021138. [30] P. Q. Wang, X. M. Rong, H. Zhao and Y. J. Wang, Robust optimal insurance and investment strategies for the government and the insurance company under mispricing phenomenon, Comm. Statist. Theory Methods, 50 (2021), 993-1017.  doi: 10.1080/03610926.2019.1646765. [31] Y. J. Wang, Y. C. Deng, Y. Huang, J. M. Zhou and X. Y. Xiang, Optimal reinsurance-investment policies for insurers with mispricing under mean-variance criterion, Communications in Statistics-Theory and Methods, early access, 2020. doi: 10.1080/03610926.2020.1844239. [32] H. L. Wu, X. G. Wang, Y. Y. Liu and L. Zeng, Multi-period optimal investment choice post-retirement with inter-temporal restrictions in a defined contribution pension plan, J. Ind. Manag. Optim., 16 (2020), 2857-2890.  doi: 10.3934/jimo.2019084. [33] H. L. Wu and Y. Zeng, Equilibrium investment strategy for defined-contribution pension schemes with generalized mean-variance criterion and mortality risk, Insurance Math. Econom., 64 (2015), 396-408.  doi: 10.1016/j.insmatheco.2015.07.007. [34] M. Yan, Z. Cao, T. Wang and S. H. Zhang, Robust optimal investment strategy of DC pension plans with stochastic salary and a return of premiums clause, Communications in Statistics-Theory and Methods, early access, 2021. doi: 10.1080/03610926.2021.1887236. [35] H. X. 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. [36] B. Yi, F. Viens, B. Law and Z. F. Li, Dynamic portfolio selection with mispricing and model ambiguity, Ann. Finance, 11 (2015), 37-75.  doi: 10.1007/s10436-014-0252-y. [37] C. B. Zhang, X. M. Rong, H. Zhao and R. J. Hou, Optimal investment for the defined-contribution pension with stochastic salary under a CEV model, Appl. Math. Ser. B, 28 (2013), 187-203.  doi: 10.1007/s11766-013-3087-9. [38] H. Zhao and X. M. Rong, Portfolio selection problem with multiple risky assets under the constant elasticity of variance model, Insurance Math. Econom., 50 (2012), 179-190.  doi: 10.1016/j.insmatheco.2011.10.013. [39] Q. Zhao, Y. Shen and J. Q. Wei, Mean-variance investment and contribution decisions for defined benefit pension plans in a stochastic framework, J. Ind. Manag. Optim., 17 (2021), 1147-1171.  doi: 10.3934/jimo.2020015.

show all references

##### References:
 [1] P. Battocchio and F. Menoncin, Optimal pension management in a stochastic framework, Insurance Math. Econom., 34 (2004), 79-95.  doi: 10.1016/j.insmatheco.2003.11.001. [2] L. H. Bian, Z. F. Li and H. X. Yao, Pre-commitment and equilibrium investment strategies for the DC pension plan with regime switching and a return of premiums clause, Insurance Math. Econom., 81 (2018), 78-94.  doi: 10.1016/j.insmatheco.2018.05.005. [3] J. Boulier, S. J. Huang and G. Taillard, Optimal management under stochatic interest rates: The case of a protected defined contribution pension fund, Insurance Math. Econom., 28 (2001), 173-189.  doi: 10.1016/S0167-6687(00)00073-1. [4] H. Chang and K. Chang, Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform, Insurance Math. Econom., 72 (2017), 215-227.  doi: 10.1016/j.insmatheco.2016.10.014. [5] H. Chang, J. A. Li and H. Zhao, Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria, Journal of Industrial and Management Optimization, early access, 2021. doi: 10.3934/jimo.2021025. [6] H. Chang, C. F. Wang, Z. M. Fang and D. Ma, Defined contribution pension planning with a stochastic interest rate and mean-reverting returns under the hyperbolic absolute risk aversion preference, IMA J. Manag. Math., 31 (2020), 167-189.  doi: 10.1093/imaman/dpz009. [7] Z. Chen, Z. F. Li, Y. Zeng and J. Y. 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. [8] Y. H. Dong and H. Zheng, Optimal investment of DC pension plan under short-selling constraints and portfolio insurance, Insurance Math. Econom., 85 (2019), 47-59.  doi: 10.1016/j.insmatheco.2018.12.005. [9] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2$^{nd}$ edition, Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. doi: 10.1007/0-387-31071-1. [10] J. W. Gao, Stochastic optimal control of DC pension funds, Insurance Math. Econom., 42 (2008), 1159-1164.  doi: 10.1016/j.insmatheco.2008.03.004. [11] R. Gerrard, S. Haberman and E. Vigna, Optimal investment choices post-retirement in a defined contribution pension scheme, Insurance Math. Econom., 35 (2004), 321-342.  doi: 10.1016/j.insmatheco.2004.06.002. [12] A. L. Gu, F. G. Viens and H. X. Yao, Optimal robust reinsurance-investment strategies for insurers with mean reversion and mispricing, Insurance Math. Econom., 80 (2018), 93-109.  doi: 10.1016/j.insmatheco.2018.03.004. [13] A. L. Gu, F. G. Viens and B. Yi, Optimal reinsurance and investment strategies for insurers with mispricing and model ambiguity, Insurance Math. Econom., 72 (2017), 235-249.  doi: 10.1016/j.insmatheco.2016.11.007. [14] C. Guambe, R. Kufakunesu, G. V. Zyl and C. Beyers, Optimal asset allocation for a DC plan with partial information under inflation and mortality risks, Comm. Statist. Theory Methods, 50 (2021), 2048-2061.  doi: 10.1080/03610926.2019.1657458. [15] G. H. Guan and Z. X. Liang, Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework, Insurance Math. Econom., 57 (2014), 58-66.  doi: 10.1016/j.insmatheco.2014.05.004. [16] G. H. Guan and Z. X. Liang, Mean-variance efficiency of DC pension plan under stochastic interest rate and mean-reverting returns, Insurance Math. Econom., 61 (2015), 99-109.  doi: 10.1016/j.insmatheco.2014.12.006. [17] N. W. Han and M. W. Hung, Optimal asset allocation for DC pension plans under inflation, Insurance Math. Econom., 51 (2012), 172-181.  doi: 10.1016/j.insmatheco.2012.03.003. [18] L. He and Z. X. Liang, Optimal investment strategy for the DC plan with the return of premiums clauses in a mean-variance framework, Insurance Math. Econom., 53 (2013), 643-649.  doi: 10.1016/j.insmatheco.2013.09.002. [19] M. Jonsson and R. Sircar, Optimal investment problems and volatility homogenization approximations, Modern Methods in Scientific Computing and Applications, 75 (2002), 255-281.  doi: 10.1007/978-94-010-0510-4. [20] E. J. Jung and J. H. Kim, Optimal investment strategies for the HARA utility under the constant elasticity of variance model, Insurance Math. Econom., 51 (2012), 667-673.  doi: 10.1016/j.insmatheco.2012.09.009. [21] O. A. Lamont and R. H. Thaler, Anomalies: The law of one price in financial markets, Journal of Economic Perspectives, 17 (2003), 191-202.  doi: 10.1257/089533003772034952. [22] D. P. Li, X. M. Rong, H. Zhao and B. Yi, Equilibrium investment strategy for DC pension plan with default risk and return of premiums clauses under CEV model, Insurance Math. Econom., 72 (2017), 6-20.  doi: 10.1016/j.insmatheco.2016.10.007. [23] Y. W. Li, S. Y. Wang, Y. Zeng and H. Qiao, Equilibrium investment strategy for a DC plan with partial information and mean-variance criterion, IEEE Systems Journal, 11 (2017), 1492-1504.  doi: 10.1109/JSYST.2016.2533920. [24] J. Liu and F. A. Longstaff, Losing money on arbitrage: Optimal dynamic portfolio choice in markets with arbitrage opportunities, The Review of Financial Studies, 17 (2003), 611-641.  doi: 10.1093/rfs/hhg029. [25] J. Liu and A. Timmermann, Optimal convergence trade strategies, The Review of Financial Studies, 26 (2013), 1048-1086.  doi: 10.1093/rfs/hhs130. [26] J. Ma, H. Zhao and X. M. Rong, Optimal investment strategy for a DC pension plan with mispricing under the Heston model, Comm. Statist. Theory Methods, 49 (2020), 3168-3183.  doi: 10.1080/03610926.2019.1586938. [27] J. Y. Sun, Y. J. Li and L. Zhang, Robust portfolio choice for a defined contribution pension plan with stochastic income and interest rate, Comm. Statist. Theory Methods, 47 (2018), 4106-4130.  doi: 10.1080/03610926.2017.1367815. [28] L. Y. Wang, Z. P. Chen and P. Yang, Robust equilibrium control-measure policy for a DC pension plan with state-dependent risk aversion under mean-variance criterion, J. Ind. Manag. Optim., 17 (2021), 1203-1233.  doi: 10.3934/jimo.2020018. [29] P. Wang, L. Zhang and Z. F. Li, Asset allocation for a DC pension plan with learning about stock return predictability, Journal of Industrial and Management Optimization, early access, 2021. doi: 10.3934/jimo.2021138. [30] P. Q. Wang, X. M. Rong, H. Zhao and Y. J. Wang, Robust optimal insurance and investment strategies for the government and the insurance company under mispricing phenomenon, Comm. Statist. Theory Methods, 50 (2021), 993-1017.  doi: 10.1080/03610926.2019.1646765. [31] Y. J. Wang, Y. C. Deng, Y. Huang, J. M. Zhou and X. Y. Xiang, Optimal reinsurance-investment policies for insurers with mispricing under mean-variance criterion, Communications in Statistics-Theory and Methods, early access, 2020. doi: 10.1080/03610926.2020.1844239. [32] H. L. Wu, X. G. Wang, Y. Y. Liu and L. Zeng, Multi-period optimal investment choice post-retirement with inter-temporal restrictions in a defined contribution pension plan, J. Ind. Manag. Optim., 16 (2020), 2857-2890.  doi: 10.3934/jimo.2019084. [33] H. L. Wu and Y. Zeng, Equilibrium investment strategy for defined-contribution pension schemes with generalized mean-variance criterion and mortality risk, Insurance Math. Econom., 64 (2015), 396-408.  doi: 10.1016/j.insmatheco.2015.07.007. [34] M. Yan, Z. Cao, T. Wang and S. H. Zhang, Robust optimal investment strategy of DC pension plans with stochastic salary and a return of premiums clause, Communications in Statistics-Theory and Methods, early access, 2021. doi: 10.1080/03610926.2021.1887236. [35] H. X. 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. [36] B. Yi, F. Viens, B. Law and Z. F. Li, Dynamic portfolio selection with mispricing and model ambiguity, Ann. Finance, 11 (2015), 37-75.  doi: 10.1007/s10436-014-0252-y. [37] C. B. Zhang, X. M. Rong, H. Zhao and R. J. Hou, Optimal investment for the defined-contribution pension with stochastic salary under a CEV model, Appl. Math. Ser. B, 28 (2013), 187-203.  doi: 10.1007/s11766-013-3087-9. [38] H. Zhao and X. M. Rong, Portfolio selection problem with multiple risky assets under the constant elasticity of variance model, Insurance Math. Econom., 50 (2012), 179-190.  doi: 10.1016/j.insmatheco.2011.10.013. [39] Q. Zhao, Y. Shen and J. Q. Wei, Mean-variance investment and contribution decisions for defined benefit pension plans in a stochastic framework, J. Ind. Manag. Optim., 17 (2021), 1147-1171.  doi: 10.3934/jimo.2020015.
Evolution of the risky assets' price
The effect of $p$ and $b$ on $\pi^{*}_{1}(t), \pi^{*}_{2}(t)$
a. Evolution of optimal investment strategy $\pi^*(t)$ over time; b. The effect of $x$ and $T$ on the utility deviation
The effect of $p$ and $b$ on $\pi_1^*(t)$ and $\pi_2^*(t)$ when $l_{1}\neq l_{2}$
The parameter values of the model in sensitivity analysis
 $r$ $b$ $c$ $\omega$ $\sigma$ $\beta$ $\eta$ $\mu_m$ $\sigma_m$ 0.03 0.3 0.1 2 0.3 1.1 0.02 0.02 0.35 $l_1$ $l_2$ $\mu_c$ $\sigma_c$ $p$ $q$ $t$ $T$ 0.1 0.1 0.05 0.1 0.6 2 2 4
 $r$ $b$ $c$ $\omega$ $\sigma$ $\beta$ $\eta$ $\mu_m$ $\sigma_m$ 0.03 0.3 0.1 2 0.3 1.1 0.02 0.02 0.35 $l_1$ $l_2$ $\mu_c$ $\sigma_c$ $p$ $q$ $t$ $T$ 0.1 0.1 0.05 0.1 0.6 2 2 4
The effect of $\mu_c$ and $\sigma_c$ on $\pi^{\ast}(t)$
 parameter optimal investment strategy $\pi^{\ast}(t)$ parameter optimal investment strategy $\pi^{\ast}(t)$ $\mu_c$ $\pi_{1}^{\ast}(t)$ $\pi_{2}^{\ast}(t)$ $\pi_{m}^{\ast}(t)$ $\sigma_c$ $\pi_{1}^{\ast}(t)$ $\pi_{2}^{\ast}(t)$ $\pi_{m}^{\ast}(t)$ 0.03 -0.8783 0.8783 1.9654 0.1 -0.8799 0.8799 1.9679 0.05 -0.8799 0.8799 1.9679 0.15 -0.8793 0.8793 1.9387 0.07 -0.8815 0.8815 1.9705 0.2 -0.8787 0.8787 1.9094 0.09 -0.8832 0.8832 1.9731 0.25 -0.8782 0.8782 1.8808
 parameter optimal investment strategy $\pi^{\ast}(t)$ parameter optimal investment strategy $\pi^{\ast}(t)$ $\mu_c$ $\pi_{1}^{\ast}(t)$ $\pi_{2}^{\ast}(t)$ $\pi_{m}^{\ast}(t)$ $\sigma_c$ $\pi_{1}^{\ast}(t)$ $\pi_{2}^{\ast}(t)$ $\pi_{m}^{\ast}(t)$ 0.03 -0.8783 0.8783 1.9654 0.1 -0.8799 0.8799 1.9679 0.05 -0.8799 0.8799 1.9679 0.15 -0.8793 0.8793 1.9387 0.07 -0.8815 0.8815 1.9705 0.2 -0.8787 0.8787 1.9094 0.09 -0.8832 0.8832 1.9731 0.25 -0.8782 0.8782 1.8808
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