# American Institute of Mathematical Sciences

March  2022, 18(2): 1393-1423. doi: 10.3934/jimo.2021025

## Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria

 1 School of Mathematical Sciences, Tiangong University, Tianjin 300387, China 2 School of Mathematical Sciences, Tianjin University, Tianjin 300072, China

* Corresponding authors: Hao Chang and Hui Zhao

Received  March 2020 Revised  December 2020 Published  March 2022 Early access  February 2021

Fund Project: This research is supported by the National Natural Science Foundation of China (Nos.71671122 and 11771329)

This paper studies a robust optimal investment problem under the mean-variance criterion for a defined contribution (DC) pension plan with an ambiguity-averse member (AAM), who worries about model misspecification and aims to find robust optimal strategy. The member has access to a risk-free asset (i.e., cash or bank account) and a risky asset (i.e., the stock) in a financial market. In order to get closer to the actual environment, we assume that both the income level and stock price are driven by Heston's stochastic volatility model. A continuous-time mean-variance model with ambiguity aversion for a DC pension plan is established. By using the Lagrangian multiplier method and stochastic optimal control theory, the closed-form expressions for robust efficient strategy and efficient frontier are derived. In addition, some special cases are derived in detail. Finally, a numerical example is presented to illustrate the effects of model parameters on the robust efficient strategy and the efficient frontier, and some economic implications have been revealed.

Citation: Hao Chang, Jiaao Li, Hui Zhao. Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1393-1423. doi: 10.3934/jimo.2021025
##### References:
 [1] E. W. Anderson, L. P. Hansen and T. J. Sargent, A quartet of semigroups for model specification, robustness, prices of risk, and model detection, Journal of the European Economic Association, 1 (2003), 68-123.  doi: 10.4324/9780203358061_chapter_16. [2] E. W. Anderson, L. P. Hansen and T. J. Sargent, Robustness, detection and the price of risk, Working paper, University of Chicago, 1999. Available from: https://files.nyu.edu/ts43/public/research/.svn/text-base/ahs3.pdf.svn-base. [3] N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1. [4] J. Bi, Z. Liang and K. C. Yuen, Optimal mean-variance investment/reinsurance with common shock in a regime-switching market, Mathematical Methods of Operations Research, 90 (2019), 109-135.  doi: 10.1007/s00186-018-00657-3. [5] H. Chang, Dynamic mean-variance portfolio selection with liability and stochastic interest rate, Economic Modelling, 51 (2015), 172-182.  doi: 10.1016/j.econmod.2015.07.017. [6] H. Chang and X. M. Rong, An investment and consumption problem with CIR interest rate and stochastic volatility, Abstract and Applied Analysis, 2013, 219397. doi: 10.1155/2013/219397. [7] P. Christoffersen, K. Jacobs and K. Mimouni, Volatility dynamics for the S & P500: Evidence from realized volatility, daily returns, and option prices, The Review of Financial Studies, 23 (2010), 3141-3189. [8] G. Deelstra, M. Grasselli and P. F. Koehl, Optimal investment strategies in the presence of a minimum guarantee, Insurance: Mathematics and Economics, 33 (2003), 189-207.  doi: 10.1016/S0167-6687(03)00153-7. [9] M. Escobar, S. Ferrando and A. Rubtsov, Robust portfolio choice with derivative trading under stochastic volatility, Journal of Banking & Finance, 61 (2015), 142-157. [10] R. Ferland and F. Watier, Mean-variance efficiency with extended CIR interest rates, Applied Stochastic Models in Business and Industry, 26 (2010), 71-84.  doi: 10.1002/asmb.767. [11] C. Fu, A. Lari-Lavassani and X. Li, Dynamic mean-variance portfolio selection with borrowing constraint, European Journal of Operational Research, 200 (2010), 312-319.  doi: 10.1016/j.ejor.2009.01.005. [12] M. D. Giacinto, S. Federico and F. Gozzi, Pension funds with a minimum guarantee: A stochastic control approach, Finance and Stochastics, 15 (2011), 297-342.  doi: 10.1007/s00780-010-0127-7. [13] G. Guan and Z. Liang, Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework, Insurance: Mathematics and Economics, 57 (2014), 58-66.  doi: 10.1016/j.insmatheco.2014.05.004. [14] S. Haberman and E. Vigna, Optimal investment strategies and risk measures in defined contribution pension schemes, Insurance: Mathematics and Economics, 31 (2002), 35-69.  doi: 10.1016/S0167-6687(02)00128-2. [15] N. W. Han and M. W. Hung, Optimal asset allocation for DC pension plans under inflation, Insurance: Mathematics and Economics, 51 (2012), 172-181.  doi: 10.1016/j.insmatheco.2012.03.003. [16] L. Hansen and T. J. Sargent, Robust control and model uncertainty, American Economic Review, 91 (2001), 60-66.  doi: 10.1142/9789814578127_0005. [17] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [18] H. Y. Kim and F. G. Viens, Portfolio optimization in discrete time with proportional transaction costs under stochastic volatility, Annals of Finance, 8 (2012), 405-425.  doi: 10.1007/s10436-010-0149-3. [19] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [20] D. Li, X. Rong and H. Zhao, Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model, Computational and Applied Mathematics, 35 (2016), 533-557.  doi: 10.1007/s40314-014-0204-1. [21] Z. Liang and M. Ma, Optimal dynamic asset allocation of pension fund in mortality and salary risks framework, Insurance: Mathematics and Economics, 64 (2015), 151-161.  doi: 10.1016/j.insmatheco.2015.05.008. [22] A. E. Lim and X. Y. Zhao, Mean-variance portfolio selection with random parameters in a complete market, Mathematics of Operations Research, 27 (2002), 101-120.  doi: 10.1287/moor.27.1.101.337. [23] P. J. Maenhout, Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium, Journal of Economic Theory, 128 (2006), 136-163.  doi: 10.1016/j.jet.2005.12.012. [24] P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003. [25] H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. [26] C. U. Okonkwo, B. O. Osu, S. A. Ihedioha and C. Chibuisi, Optimal Investment Strategy for Defined Contribution Pension Scheme under the Heston Volatility Model, Journal of Mathematical Finance, 8 (2018), 613-622.  doi: 10.4236/jmf.2018.84039. [27] R. Uppal and T. Wang, Model misspecification and underdiversification, The Journal of Finance, 58 (2003), 2465-2486.  doi: 10.1046/j.1540-6261.2003.00612.x. [28] E. Vigna and S. Haberman, Optimal investment strategy for defined contribution pension schemes, Insurance: Mathematics and Economics, 28 (2001), 233-262.  doi: 10.1016/S0167-6687(00)00077-9. [29] P. Wang and Z. Li, Robust optimal investment strategy for an AAM of DC pension plans with stochastic interest rate and stochastic volatility, Insurance: Mathematics and Economics, 80 (2018), 67-83.  doi: 10.1016/j.insmatheco.2018.03.003. [30] H. Yao, Z. Yang and P. Chen, Markowitz's mean-variance defined contribution pension fund management under inflation: A continuous-time model, Insurance: Mathematics and Economics, 53 (2013), 851-863.  doi: 10.1016/j.insmatheco.2013.10.002. [31] B. Yi, Z. Li, F. G. Viens and Y. Zeng, Robust optimal control for an insurer with reinsurance and investment under Heston's stochastic volatility model, Insurance: Mathematics and Economics, 53 (2013), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011. [32] B. Yi, F. Viens, Z. Li and Y. Zeng, Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scandinavian Actuarial Journal, 2015 (2015), 725-751.  doi: 10.1080/03461238.2014.883085. [33] Y. Zeng, D. Li, Z. Chen and Z. Yang, Ambiguity aversion and optimal derivative-based pension investment with stochastic income and volatility, Journal of Economic Dynamics and Control, 88 (2018), 70-103.  doi: 10.1016/j.jedc.2018.01.023. [34] A. Zhang and C. O. Ewald, Optimal investment for a pension fund under inflation risk, Mathematical Methods of Operations Research, 71 (2010), 353-369.  doi: 10.1007/s00186-009-0294-5. [35] H. Zhao, X. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance: Mathematics and Economics, 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004. [36] X. Zheng, J. Zhou and Z. Sun, Robust optimal portfolio and proportional reinsurance for an insurer under a CEV model, Insurance: Mathematics and Economics, 67 (2016), 77-87.  doi: 10.1016/j.insmatheco.2015.12.008. [37] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.

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##### References:
 [1] E. W. Anderson, L. P. Hansen and T. J. Sargent, A quartet of semigroups for model specification, robustness, prices of risk, and model detection, Journal of the European Economic Association, 1 (2003), 68-123.  doi: 10.4324/9780203358061_chapter_16. [2] E. W. Anderson, L. P. Hansen and T. J. Sargent, Robustness, detection and the price of risk, Working paper, University of Chicago, 1999. Available from: https://files.nyu.edu/ts43/public/research/.svn/text-base/ahs3.pdf.svn-base. [3] N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1. [4] J. Bi, Z. Liang and K. C. Yuen, Optimal mean-variance investment/reinsurance with common shock in a regime-switching market, Mathematical Methods of Operations Research, 90 (2019), 109-135.  doi: 10.1007/s00186-018-00657-3. [5] H. Chang, Dynamic mean-variance portfolio selection with liability and stochastic interest rate, Economic Modelling, 51 (2015), 172-182.  doi: 10.1016/j.econmod.2015.07.017. [6] H. Chang and X. M. Rong, An investment and consumption problem with CIR interest rate and stochastic volatility, Abstract and Applied Analysis, 2013, 219397. doi: 10.1155/2013/219397. [7] P. Christoffersen, K. Jacobs and K. Mimouni, Volatility dynamics for the S & P500: Evidence from realized volatility, daily returns, and option prices, The Review of Financial Studies, 23 (2010), 3141-3189. [8] G. Deelstra, M. Grasselli and P. F. Koehl, Optimal investment strategies in the presence of a minimum guarantee, Insurance: Mathematics and Economics, 33 (2003), 189-207.  doi: 10.1016/S0167-6687(03)00153-7. [9] M. Escobar, S. Ferrando and A. Rubtsov, Robust portfolio choice with derivative trading under stochastic volatility, Journal of Banking & Finance, 61 (2015), 142-157. [10] R. Ferland and F. Watier, Mean-variance efficiency with extended CIR interest rates, Applied Stochastic Models in Business and Industry, 26 (2010), 71-84.  doi: 10.1002/asmb.767. [11] C. Fu, A. Lari-Lavassani and X. Li, Dynamic mean-variance portfolio selection with borrowing constraint, European Journal of Operational Research, 200 (2010), 312-319.  doi: 10.1016/j.ejor.2009.01.005. [12] M. D. Giacinto, S. Federico and F. Gozzi, Pension funds with a minimum guarantee: A stochastic control approach, Finance and Stochastics, 15 (2011), 297-342.  doi: 10.1007/s00780-010-0127-7. [13] G. Guan and Z. Liang, Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework, Insurance: Mathematics and Economics, 57 (2014), 58-66.  doi: 10.1016/j.insmatheco.2014.05.004. [14] S. Haberman and E. Vigna, Optimal investment strategies and risk measures in defined contribution pension schemes, Insurance: Mathematics and Economics, 31 (2002), 35-69.  doi: 10.1016/S0167-6687(02)00128-2. [15] N. W. Han and M. W. Hung, Optimal asset allocation for DC pension plans under inflation, Insurance: Mathematics and Economics, 51 (2012), 172-181.  doi: 10.1016/j.insmatheco.2012.03.003. [16] L. Hansen and T. J. Sargent, Robust control and model uncertainty, American Economic Review, 91 (2001), 60-66.  doi: 10.1142/9789814578127_0005. [17] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [18] H. Y. Kim and F. G. Viens, Portfolio optimization in discrete time with proportional transaction costs under stochastic volatility, Annals of Finance, 8 (2012), 405-425.  doi: 10.1007/s10436-010-0149-3. [19] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [20] D. Li, X. Rong and H. Zhao, Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model, Computational and Applied Mathematics, 35 (2016), 533-557.  doi: 10.1007/s40314-014-0204-1. [21] Z. Liang and M. Ma, Optimal dynamic asset allocation of pension fund in mortality and salary risks framework, Insurance: Mathematics and Economics, 64 (2015), 151-161.  doi: 10.1016/j.insmatheco.2015.05.008. [22] A. E. Lim and X. Y. Zhao, Mean-variance portfolio selection with random parameters in a complete market, Mathematics of Operations Research, 27 (2002), 101-120.  doi: 10.1287/moor.27.1.101.337. [23] P. J. Maenhout, Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium, Journal of Economic Theory, 128 (2006), 136-163.  doi: 10.1016/j.jet.2005.12.012. [24] P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003. [25] H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. [26] C. U. Okonkwo, B. O. Osu, S. A. Ihedioha and C. Chibuisi, Optimal Investment Strategy for Defined Contribution Pension Scheme under the Heston Volatility Model, Journal of Mathematical Finance, 8 (2018), 613-622.  doi: 10.4236/jmf.2018.84039. [27] R. Uppal and T. Wang, Model misspecification and underdiversification, The Journal of Finance, 58 (2003), 2465-2486.  doi: 10.1046/j.1540-6261.2003.00612.x. [28] E. Vigna and S. Haberman, Optimal investment strategy for defined contribution pension schemes, Insurance: Mathematics and Economics, 28 (2001), 233-262.  doi: 10.1016/S0167-6687(00)00077-9. [29] P. Wang and Z. Li, Robust optimal investment strategy for an AAM of DC pension plans with stochastic interest rate and stochastic volatility, Insurance: Mathematics and Economics, 80 (2018), 67-83.  doi: 10.1016/j.insmatheco.2018.03.003. [30] H. Yao, Z. Yang and P. Chen, Markowitz's mean-variance defined contribution pension fund management under inflation: A continuous-time model, Insurance: Mathematics and Economics, 53 (2013), 851-863.  doi: 10.1016/j.insmatheco.2013.10.002. [31] B. Yi, Z. Li, F. G. Viens and Y. Zeng, Robust optimal control for an insurer with reinsurance and investment under Heston's stochastic volatility model, Insurance: Mathematics and Economics, 53 (2013), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011. [32] B. Yi, F. Viens, Z. Li and Y. Zeng, Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scandinavian Actuarial Journal, 2015 (2015), 725-751.  doi: 10.1080/03461238.2014.883085. [33] Y. Zeng, D. Li, Z. Chen and Z. Yang, Ambiguity aversion and optimal derivative-based pension investment with stochastic income and volatility, Journal of Economic Dynamics and Control, 88 (2018), 70-103.  doi: 10.1016/j.jedc.2018.01.023. [34] A. Zhang and C. O. Ewald, Optimal investment for a pension fund under inflation risk, Mathematical Methods of Operations Research, 71 (2010), 353-369.  doi: 10.1007/s00186-009-0294-5. [35] H. Zhao, X. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance: Mathematics and Economics, 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004. [36] X. Zheng, J. Zhou and Z. Sun, Robust optimal portfolio and proportional reinsurance for an insurer under a CEV model, Insurance: Mathematics and Economics, 67 (2016), 77-87.  doi: 10.1016/j.insmatheco.2015.12.008. [37] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.
The effects of volatility parameters $\alpha$, $\sigma _0$, $\lambda _S$ and $\rho$ on $\pi ^{\ast }\left( t \right)$
The effects of income parameters $\mu$, $\sigma _1$ and $k$ on $\pi ^ {\ast }\left( t \right)$
The effect of ambiguity-aversion coefficient $\beta$ on $\pi ^ {\ast }\left( t \right)$
The effects of volatility parameters $\alpha$ and $\sigma _0$ on $\sigma \left( {X\left( T \right)} \right)$
The effects of income parameters $\mu$ and $k$ on $\sigma \left( {X\left( T \right)} \right)$
The effect of ambiguity-aversion coefficient $\beta$ on $\sigma \left( {X\left( T \right)} \right)$
Comparisons of the efficient strategies and the efficient frontiers
When $\rho = -0.65$ and $\lambda _S = 1.5$, we have $\Delta>0$; when $\rho = -0.87$ and $\lambda _S = 12.56$, we get $\Delta = 0$; when $\rho = -0.9$ and $\lambda _S = 10$, we have $\Delta<0$. The effects of different symbols of $\Delta$ on $\pi ^ {\ast }(t)$ and $\sigma (X(T))$
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