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

Liyuan Wang; Zhiping Chen; Peng Yang

doi:10.3934/jimo.2020018

Article Contents

2021, Volume 17, Issue 3: 1203-1233. Doi: 10.3934/jimo.2020018

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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
^* Corresponding author: Zhiping Chen

Received: February 2019

Revised: July 2019

Early access: January 2020

Published: May 2021

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.

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Abstract

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.

Keywords:

DC pension plan,
state-dependent risk aversion,
robust equilibrium control-measure policy,
extended HJBI equation.

Mathematics Subject Classification: Primary: 90B50, 93E20; Secondary: 91G80, 91A10.

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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) $

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Figure 2. Effect of $ \sigma $ on the robust equilibrium policy, and values of $ m(t,x,l) $ and $ f_{2}(t) $

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Figure 3. Effects of $ \alpha $ on the robust equilibrium policy

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Figure 4. Effects of $ \varphi $ on the robust equilibrium policy

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Figure 5. Effects of $ \beta $ on the robust equilibrium policy

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Figure 6. Effects of $ \gamma $ on the robust equilibrium policy

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Figure 7. Effect of $ T $ on the robust equilibrium policy

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Figure 8. Effect of $ c $ on the robust equilibrium policy

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Figure 9. Effect of $ \xi $ on the robust equilibrium policy

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Figure 10. Effect of $ \xi $ on the discrepancy function

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Figure 11. Effect of $ X_0 $ on the robust equilibrium policy

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Figure 12. Effect of $ L_0 $ on the robust equilibrium policy

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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

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