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doi: 10.3934/jimo.2020018

Robust equilibrium control-measure policy for a DC pension plan with state-dependent risk aversion under mean-variance criterion

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Shaanxi 710049, China

2. 

Center for Optimization Technique and Quantitative Finance, Xi'an International Academy for Mathematics and Mathematical Technology, Shaanxi 710049, China

3. 

School of Science, Xijing University, Xi'an, Shaanxi 710123, China

* Corresponding author: Zhiping Chen

Received  February 2019 Revised  July 2019 Published  January 2020

Fund Project: This research was supported by the National Natural Science Foundation of China under Grant Numbers 11571270 and 11735011, and the World-Class Universities (Disciplines) and the Characteristic Development Guidance Funds for the Central Universities under Grant Number PY3A058.

In reality, when facing a defined contribution (DC) pension fund investment problem, the fund manager may not have sufficient confidence in the reference model and rather considers some similar alternative models. In this paper, we investigate the robust equilibrium control-measure policy for an ambiguity-averse and risk-averse fund manger under the mean-variance (MV) criterion. The ambiguity aversion is introduced by adopting the model uncertainty robustness framework developed by Anderson. The risk aversion model is state-dependent, and takes a linear form of the current wealth level after contribution. Moreover, the fund manager faces stochastic labor income risk and allocates his wealth among a risk-free asset and a risky asset. We also propose two complicated ambiguity preference functions which are economically meaningful and facilitate analytical tractability. Due to the time-inconsistency of the resulting stochastic control problem, we attack it by using the game theoretical framework and the concept of subgame perfect Nash equilibrium. The extended Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations and the verification theorem for our problem are established. The explicit expressions for the robust equilibrium policy and the corresponding robust equilibrium value function are derived by stochastic control technique. In addition, we discuss two special cases of our model, which shows that our results extend some existing works in the literature. Finally, some numerical experiments are conducted to demonstrate the effects of model parameters on our robust equilibrium policy.

Citation: Liyuan Wang, Zhiping Chen, Peng Yang. Robust equilibrium control-measure policy for a DC pension plan with state-dependent risk aversion under mean-variance criterion. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020018
References:
[1]

E. W. Anderson, L. P. Hansen and T. J. Sargent, Robustness, detection and the price of risk, 1999. Available from: https://www.researchgate.net/profile/Lars_Hansen/publication/2637084_Robustness_Detection_and_the_Price_of_Risk/links/0deec51f6c2524ada9000000/Robustness-Detection-and-the-Price-of-Risk.pdf. Google Scholar

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E. W. AndersonL. 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.   Google Scholar

[3]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, The Review of Financial Studies, 23 (2010), 2970-3016.   Google Scholar

[4]

P. Battocchio and F. Menoncin, Optimal pension management in a stochastic framework, Insurance: Mathematics and Economics, 34 (2004), 79-95.  doi: 10.1016/j.insmatheco.2003.11.001.  Google Scholar

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T. BjörkA. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Mathematical Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x.  Google Scholar

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T. BjörkM. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance and Stochastics, 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.  Google Scholar

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D. BlakeD. Wright and Y. M. Zhang, Target-driven investing: Optimal investment strategies in defined contribution pension plans under loss aversion, Journal of Economic Dynamics and Control, 37 (2013), 195-209.  doi: 10.1016/j.jedc.2012.08.001.  Google Scholar

[8]

Z. BodieJ. B. DetempleS. Otruba and S. Walter, Optimal consumption-portfolio choices and retirement planning, Journal of Economic Dynamics and Control, 28 (2004), 1115-1148.  doi: 10.1016/S0165-1889(03)00068-X.  Google Scholar

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A. J. G. CairnsD. Blake and K. Dowd, Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans, Journal of Economic Dynamics and Control, 30 (2006), 843-877.  doi: 10.1016/j.jedc.2005.03.009.  Google Scholar

[10]

Z. ChenZ. F. LiY. Zeng and J. Y. Sun, Asset allocation under loss aversion and minimum performance constraint in a DC pension plan with inflation risk, Insurance: Mathematics and Economics, 75 (2017), 137-150.  doi: 10.1016/j.insmatheco.2017.05.009.  Google Scholar

[11]

X. Y. CuiD. LiS. Y. Wang and S. S. Zhu, Better than dynamic mean-variance: Time inconsistency and free cash flow stream, Mathematical Finance, 22 (2012), 346-378.  doi: 10.1111/j.1467-9965.2010.00461.x.  Google Scholar

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X. Y. CuiL. Xu and Y. Zeng, Continuous time mean-variance portfolio optimization with piecewise state-dependent risk aversion, Optimization Letters, 10 (2016), 1681-1691.  doi: 10.1007/s11590-015-0970-8.  Google Scholar

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X. Y. CuiX. LiD. Li and Y. Shi, Time consistent behavioral portfolio policy for dynamic mean-variance formulation, Journal of the Operational Research Society, 68 (2017), 1647-1660.  doi: 10.1057/s41274-017-0179-6.  Google Scholar

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G. DeelstraM. 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.  Google Scholar

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C. R. Flor and L. S. Larsen, Robust portfolio choice with stochastic interest rates, Annals of Finance, 10 (2014), 243-265.  doi: 10.1007/s10436-013-0234-5.  Google Scholar

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G. H. Guan and Z. X. Liang, Mean-variance efficiency of DC pension plan under stochastic interest rate and mean-reverting returns, Insurance: Mathematics and Economics, 61 (2015), 99-109.  doi: 10.1016/j.insmatheco.2014.12.006.  Google Scholar

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L. P. HansenT. J.SargentG. Turmuhambetova and N. Williams, Robust control and model misspecification, Journal of Economic Theory, 128 (2006), 45-90.  doi: 10.1016/j.jet.2004.12.006.  Google Scholar

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Y. HuH. Q. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM Journal on Control and Optimization, 50 (2012), 1548-1572.  doi: 10.1137/110853960.  Google Scholar

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R. KornO. Menkens and M. Steffensen, Worst-case-optimal dynamic reinsurance for large claims, European Actuarial Journal, 2 (2012), 21-48.  doi: 10.1007/s13385-012-0050-8.  Google Scholar

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Z. X. Liang and M. Song, Time-consistent reinsurance and investment strategies for mean-variance insurer under partial information, Insurance: Mathematics and Economics, 65 (2015), 66-76.  doi: 10.1016/j.insmatheco.2015.08.008.  Google Scholar

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Y. W. Li and Z. F. Li, Optimal time-consistent investment and reinsurance strategies for mean-variance insurers with state dependent risk aversion, Insurance: Mathematics and Economics, 53 (2013), 86-97.  doi: 10.1016/j.insmatheco.2013.03.008.  Google Scholar

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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.  Google Scholar

[25]

X. LinC. H. Zhang and T. K. Siu, Stochastic differential portfolio games for an insurer in a jump-diffusion risk process, Mathematical Methods of Operations Research, 75 (2012), 83-100.  doi: 10.1007/s00186-011-0376-z.  Google Scholar

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J. LiuJ. Pan and T. Wang, An equilibrium model of rare-event premia and its implication for option smirks, The Review of Financial Studies, 18 (2005), 131-164.  doi: 10.1093/rfs/hhi011.  Google Scholar

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H. Liu, Robust consumption and portfolio choice for time varying investment opportunities, Annals of Finance, 6 (2010), 435-454.   Google Scholar

[28]

Y. L. LiuM. Y. YangJ. Zhai and M. Y. Bai, Portfolio selection of the defined contribution pension fund with uncertain return and salary: A multi-period mean-variance model, Journal of Intelligent and Fuzzy Systems, 34 (2018), 2363-2371.  doi: 10.3233/JIFS-171440.  Google Scholar

[29]

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[30]

P. J. Maenhout, Robust portfolio rules and asset pricing, The Review of Financial Studies, 17 (2004), 951-983.   Google Scholar

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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.  Google Scholar

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J. PoterbaJ. RauhS. Venti and D. Wise, Defined contribution plans, defined benefit plans, and the accumulation of retirement wealth, Journal of Public Economics, 91 (2007), 2062-2086.  doi: 10.3386/w12597.  Google Scholar

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C. S. Pun and H. Y. Wong, Robust investment-reinsurance optimization with multiscale stochastic volatility, Insurance: Mathematics and Economics, 62 (2015), 245-256.  doi: 10.1016/j.insmatheco.2015.03.030.  Google Scholar

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C. S. Pun, Robust time-inconsistent stochastic control problems, Automatica, 94 (2018), 249-257.  doi: 10.1016/j.automatica.2018.04.038.  Google Scholar

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C. S. Pun, Robust time-inconsistent stochastic control problems (extended version), Working paper, 2018. Available from: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3035656. Google Scholar

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show all references

References:
[1]

E. W. Anderson, L. P. Hansen and T. J. Sargent, Robustness, detection and the price of risk, 1999. Available from: https://www.researchgate.net/profile/Lars_Hansen/publication/2637084_Robustness_Detection_and_the_Price_of_Risk/links/0deec51f6c2524ada9000000/Robustness-Detection-and-the-Price-of-Risk.pdf. Google Scholar

[2]

E. W. AndersonL. 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.   Google Scholar

[3]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, The Review of Financial Studies, 23 (2010), 2970-3016.   Google Scholar

[4]

P. Battocchio and F. Menoncin, Optimal pension management in a stochastic framework, Insurance: Mathematics and Economics, 34 (2004), 79-95.  doi: 10.1016/j.insmatheco.2003.11.001.  Google Scholar

[5]

T. BjörkA. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Mathematical Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x.  Google Scholar

[6]

T. BjörkM. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance and Stochastics, 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.  Google Scholar

[7]

D. BlakeD. Wright and Y. M. Zhang, Target-driven investing: Optimal investment strategies in defined contribution pension plans under loss aversion, Journal of Economic Dynamics and Control, 37 (2013), 195-209.  doi: 10.1016/j.jedc.2012.08.001.  Google Scholar

[8]

Z. BodieJ. B. DetempleS. Otruba and S. Walter, Optimal consumption-portfolio choices and retirement planning, Journal of Economic Dynamics and Control, 28 (2004), 1115-1148.  doi: 10.1016/S0165-1889(03)00068-X.  Google Scholar

[9]

A. J. G. CairnsD. Blake and K. Dowd, Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans, Journal of Economic Dynamics and Control, 30 (2006), 843-877.  doi: 10.1016/j.jedc.2005.03.009.  Google Scholar

[10]

Z. ChenZ. F. LiY. Zeng and J. Y. Sun, Asset allocation under loss aversion and minimum performance constraint in a DC pension plan with inflation risk, Insurance: Mathematics and Economics, 75 (2017), 137-150.  doi: 10.1016/j.insmatheco.2017.05.009.  Google Scholar

[11]

X. Y. CuiD. LiS. Y. Wang and S. S. Zhu, Better than dynamic mean-variance: Time inconsistency and free cash flow stream, Mathematical Finance, 22 (2012), 346-378.  doi: 10.1111/j.1467-9965.2010.00461.x.  Google Scholar

[12]

X. Y. CuiL. Xu and Y. Zeng, Continuous time mean-variance portfolio optimization with piecewise state-dependent risk aversion, Optimization Letters, 10 (2016), 1681-1691.  doi: 10.1007/s11590-015-0970-8.  Google Scholar

[13]

X. Y. CuiX. LiD. Li and Y. Shi, Time consistent behavioral portfolio policy for dynamic mean-variance formulation, Journal of the Operational Research Society, 68 (2017), 1647-1660.  doi: 10.1057/s41274-017-0179-6.  Google Scholar

[14]

G. DeelstraM. 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.  Google Scholar

[15]

C. R. Flor and L. S. Larsen, Robust portfolio choice with stochastic interest rates, Annals of Finance, 10 (2014), 243-265.  doi: 10.1007/s10436-013-0234-5.  Google Scholar

[16]

G. H. Guan and Z. X. Liang, Mean-variance efficiency of DC pension plan under stochastic interest rate and mean-reverting returns, Insurance: Mathematics and Economics, 61 (2015), 99-109.  doi: 10.1016/j.insmatheco.2014.12.006.  Google Scholar

[17]

L. P. HansenT. J.SargentG. Turmuhambetova and N. Williams, Robust control and model misspecification, Journal of Economic Theory, 128 (2006), 45-90.  doi: 10.1016/j.jet.2004.12.006.  Google Scholar

[18]

Y. HuH. Q. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM Journal on Control and Optimization, 50 (2012), 1548-1572.  doi: 10.1137/110853960.  Google Scholar

[19]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991 doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[20]

F. Knight, Risk, Uncertainty and Profit, Houghton Mifflin, New York, 1921. doi: 10.1017/CBO9780511817410.005.  Google Scholar

[21]

R. KornO. Menkens and M. Steffensen, Worst-case-optimal dynamic reinsurance for large claims, European Actuarial Journal, 2 (2012), 21-48.  doi: 10.1007/s13385-012-0050-8.  Google Scholar

[22]

Z. X. Liang and M. Song, Time-consistent reinsurance and investment strategies for mean-variance insurer under partial information, Insurance: Mathematics and Economics, 65 (2015), 66-76.  doi: 10.1016/j.insmatheco.2015.08.008.  Google Scholar

[23]

Y. W. Li and Z. F. Li, Optimal time-consistent investment and reinsurance strategies for mean-variance insurers with state dependent risk aversion, Insurance: Mathematics and Economics, 53 (2013), 86-97.  doi: 10.1016/j.insmatheco.2013.03.008.  Google Scholar

[24]

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.  Google Scholar

[25]

X. LinC. H. Zhang and T. K. Siu, Stochastic differential portfolio games for an insurer in a jump-diffusion risk process, Mathematical Methods of Operations Research, 75 (2012), 83-100.  doi: 10.1007/s00186-011-0376-z.  Google Scholar

[26]

J. LiuJ. Pan and T. Wang, An equilibrium model of rare-event premia and its implication for option smirks, The Review of Financial Studies, 18 (2005), 131-164.  doi: 10.1093/rfs/hhi011.  Google Scholar

[27]

H. Liu, Robust consumption and portfolio choice for time varying investment opportunities, Annals of Finance, 6 (2010), 435-454.   Google Scholar

[28]

Y. L. LiuM. Y. YangJ. Zhai and M. Y. Bai, Portfolio selection of the defined contribution pension fund with uncertain return and salary: A multi-period mean-variance model, Journal of Intelligent and Fuzzy Systems, 34 (2018), 2363-2371.  doi: 10.3233/JIFS-171440.  Google Scholar

[29]

Q.-P. Ma, On "optimal pension management in a stochastic framework" with exponential utility, Insurance: Mathematics and Economics, 49 (2011), 61-69.  doi: 10.1016/j.insmatheco.2011.02.003.  Google Scholar

[30]

P. J. Maenhout, Robust portfolio rules and asset pricing, The Review of Financial Studies, 17 (2004), 951-983.   Google Scholar

[31]

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.  Google Scholar

[32]

H. M. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.   Google Scholar

[33]

C. Munk and A. Rubtsov, Portfolio management with stochastic interest rates and inflation ambiguity, Annals of Finance, 10 (2014), 419-455.  doi: 10.1007/s10436-013-0238-1.  Google Scholar

[34]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Fifth edition. Universitext. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03620-4.  Google Scholar

[35]

J. PoterbaJ. RauhS. Venti and D. Wise, Defined contribution plans, defined benefit plans, and the accumulation of retirement wealth, Journal of Public Economics, 91 (2007), 2062-2086.  doi: 10.3386/w12597.  Google Scholar

[36]

C. S. Pun and H. Y. Wong, Robust investment-reinsurance optimization with multiscale stochastic volatility, Insurance: Mathematics and Economics, 62 (2015), 245-256.  doi: 10.1016/j.insmatheco.2015.03.030.  Google Scholar

[37]

C. S. Pun, Robust time-inconsistent stochastic control problems, Automatica, 94 (2018), 249-257.  doi: 10.1016/j.automatica.2018.04.038.  Google Scholar

[38]

C. S. Pun, Robust time-inconsistent stochastic control problems (extended version), Working paper, 2018. Available from: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3035656. Google Scholar

[39]

R. H. Strotz, Myopia and inconsistency in dynamic utility maximization, The Review of Economic Studies, (1973), 128–143. doi: 10.1007/978-1-349-15492-0_10.  Google Scholar

[40]

E. Vigna, On efficiency of mean-variance based portfolio selection in defined contribution pension schemes, Quantitative Finance, 14 (2014), 237-258.  doi: 10.1080/14697688.2012.708778.  Google Scholar

[41]

L. Y. Wang and Z. P. Chen, Nash equilibrium strategy for a DC pension plan with state-dependent risk aversion: A multiperiod mean-variance framework, Discrete Dynamics in Nature and Society, 2018 (2018), Art. ID 7581231, 17 pp. doi: 10.1155/2018/7581231.  Google Scholar

[42]

L. Y. Wang and Z. P. Chen, Stochastic game theoretic formulation for a multi-period DC pension plan with state-dependent risk aversion, Mathematics, 7 (2019). doi: 10.3390/math7010108.  Google Scholar

[43]

P. Wang and Z. F. 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.  Google Scholar

[44]

H. L. Wu, Time-consistent strategies for a multiperiod mean-variance portfolio selection problem, Journal of Applied Mathematics, 2013 (2013), Art. ID 841627, 13 pp. doi: 10.1155/2013/841627.  Google Scholar

[45]

H. L. WuL. Zhang and H. Chen, Nash equilibrium strategies for a defined contribution pension management, Insurance: Mathematics and Economics, 62 (2015), 202-214.  doi: 10.1016/j.insmatheco.2015.03.014.  Google Scholar

[46]

H. WuC. Weng and Y. Zeng, Equilibrium consumption and portfolio decisions with stochastic discount rate and time-varying utility functions, OR Spectrum, 40 (2018), 541-582.   Google Scholar

[47]

W. D. XuC. F. Wu and H. Y. Li, Robust general equilibrium under stochastic volatility model, Finance Research Letters, 7 (2010), 224-231.  doi: 10.1016/j.frl.2010.05.002.  Google Scholar

[48]

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Figure 1.  Effect of $ \mu $ on the robust equilibrium policy, and values of $ m(t,x,l) $ and $ f_{2}(t) $
Figure 2.  Effect of $ \sigma $ on the robust equilibrium policy, and values of $ m(t,x,l) $ and $ f_{2}(t) $
Figure 3.  Effects of $ \alpha $ on the robust equilibrium policy
Figure 4.  Effects of $ \varphi $ on the robust equilibrium policy
Figure 5.  Effects of $ \beta $ on the robust equilibrium policy
Figure 6.  Effects of $ \gamma $ on the robust equilibrium policy
Figure 7.  Effect of $ T $ on the robust equilibrium policy
Figure 8.  Effect of $ c $ on the robust equilibrium policy
Figure 9.  Effect of $ \xi $ on the robust equilibrium policy
Figure 10.  Effect of $ \xi $ on the discrepancy function
Figure 11.  Effect of $ X_0 $ on the robust equilibrium policy
Figure 12.  Effect of $ L_0 $ on the robust equilibrium policy
Table 1.  Parameter values
Parameter Symbol Value
Time horizon (retirement date) $T$ 5
Initial wealth $X_{0}$ 4
Initial labor income $L_{0}$ 1
Risk-free interest rate $r(t)$ 0.05
Appreciation rate of the risky asset $\mu(t)$ 0.15
Volatility rate of the risky asset $\sigma(t)$ 0.25
Appreciation rate of the labor income $\alpha(t)$ 0.08
Volatility rate of the labor income (hedgeable) $\varphi(t)$ 0.15
Volatility rate of the labor income (non-hedgeable) $\beta(t)$ 0.20
Contribution rate $c$ 0.2
Risk aversion coefficient $\gamma$ 2
Aggregate ambiguity aversion $\xi$ 1
Parameter Symbol Value
Time horizon (retirement date) $T$ 5
Initial wealth $X_{0}$ 4
Initial labor income $L_{0}$ 1
Risk-free interest rate $r(t)$ 0.05
Appreciation rate of the risky asset $\mu(t)$ 0.15
Volatility rate of the risky asset $\sigma(t)$ 0.25
Appreciation rate of the labor income $\alpha(t)$ 0.08
Volatility rate of the labor income (hedgeable) $\varphi(t)$ 0.15
Volatility rate of the labor income (non-hedgeable) $\beta(t)$ 0.20
Contribution rate $c$ 0.2
Risk aversion coefficient $\gamma$ 2
Aggregate ambiguity aversion $\xi$ 1
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