Figure 1.
The effects of volatility parameters $ \alpha $ , $ \sigma _0 $ , $ \lambda _S $ and $ \rho $ on $ \pi ^{\ast }\left( t \right) $
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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 |
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.
Keywords:
Heston's stochastic volatility model,
stochastic income,
DC pension plan,
mean-variance model,
ambiguity-averse member.
Mathematics Subject Classification:
Primary: 60H30, 91B28; Secondary: 93E20.
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:
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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. |
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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. |
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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. |
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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. |
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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. |
show all references
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. |
Figure 2.
The effects of income parameters $ \mu $ , $ \sigma _1 $ and $ k $ on $ \pi ^ {\ast }\left( t \right) $
Figure 3.
The effect of ambiguity-aversion coefficient $ \beta $ on $ \pi ^ {\ast }\left( t \right) $
Figure 4.
The effects of volatility parameters $ \alpha $ and $ \sigma _0 $ on $ \sigma \left( {X\left( T \right)} \right) $
Figure 5.
The effects of income parameters $ \mu $ and $ k $ on $ \sigma \left( {X\left( T \right)} \right) $
Figure 6.
The effect of ambiguity-aversion coefficient $ \beta $ on $ \sigma \left( {X\left( T \right)} \right) $
Figure 7.
Comparisons of the efficient strategies and the efficient frontiers
Figure 8.
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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