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doi: 10.3934/jimo.2019084

## Multi-period optimal investment choice post-retirement with inter-temporal restrictions in a defined contribution pension plan

 1 China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China 2 School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, China 3 Beijing Branch, China Minsheng Banking Corporation Limited, Beijing 100031, China 4 School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China

* Corresponding author: zengli@gdufs.edu.cn

Received  October 2018 Revised  March 2019 Published  July 2019

Fund Project: The first author is supported by grants from National Natural Science Foundation of China (Nos. 11671411, 11771465).

This paper studies a multi-period portfolio selection problem during the post-retirement phase of a defined contribution pension plan. The retiree is allowed to defer the purchase of the annuity until the time of compulsory annuitization. A series of investment targets over time are set, and restrictions on the inter-temporal expected values of the portfolio are considered. We aim to minimize the accumulated variances from the time of retirement to the time of compulsory annuitization. Using the Lagrange multiplier technique and dynamic programming, we study in detail the existence of the optimal strategy and derive its closed-form expression. For comparison purposes, the explicit solution of the classical target-based model is also provided. The properties of the optimal investment strategy, the probabilities of achieving a worse or better pension at the time of compulsory annuitization and the bankruptcy probability are compared in detail under two models. The comparison shows that our model can greatly decrease the probability of achieving a worse pension at the compulsory time and can significantly increase the probability of achieving a better pension.

Citation: Huiling Wu, Xiuguo Wang, Yuanyuan Liu, Li Zeng. Multi-period optimal investment choice post-retirement with inter-temporal restrictions in a defined contribution pension plan. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019084
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##### References:
$\eta_n- {{\rm{E}}}_{0,x_0}\left(X^{\tilde\pi}_{n}\right)$, $n = 1,2,\ldots,T$
The targets over time
 $\eta_1=198747$ $\eta_2=197494$ $\eta_3=196241$ $\eta_4=194988$ $\eta_5=193735$ $\eta_6=192482$ $\eta_7=191229$ $\eta_8=189976$ $\eta_9=188723$ $\eta_{10}=187470$ $\eta_{11}=186217$ $\eta_{12}=184964$ $\eta_{13}=183711$ $\eta_{14}=182458$ $\eta_{15}=181205$
 $\eta_1=198747$ $\eta_2=197494$ $\eta_3=196241$ $\eta_4=194988$ $\eta_5=193735$ $\eta_6=192482$ $\eta_7=191229$ $\eta_8=189976$ $\eta_9=188723$ $\eta_{10}=187470$ $\eta_{11}=186217$ $\eta_{12}=184964$ $\eta_{13}=183711$ $\eta_{14}=182458$ $\eta_{15}=181205$
The targets over time
 $\eta_1=201356$ $\eta_2=202333$ $\eta_3=202942$ $\eta_4=203160$ $\eta_5=202960$ $\eta_6=202362$ $\eta_7=201386$ $\eta_8=200057$ $\eta_9=198330$ $\eta_{10}=196284$ $\eta_{11}=193886$ $\eta_{12}=191191$ $\eta_{13}=188231$ $\eta_{14}=184908$ $\eta_{15}=181193$
 $\eta_1=201356$ $\eta_2=202333$ $\eta_3=202942$ $\eta_4=203160$ $\eta_5=202960$ $\eta_6=202362$ $\eta_7=201386$ $\eta_8=200057$ $\eta_9=198330$ $\eta_{10}=196284$ $\eta_{11}=193886$ $\eta_{12}=191191$ $\eta_{13}=188231$ $\eta_{14}=184908$ $\eta_{15}=181193$
The frequencies that $X_{T}/$ä$_{75} $$<\zeta_0  The first decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_1\; \; \text{Our model} 0.2774 0.2661 0.2535 p_2\; \; \text{Target-based model} 0.5394 0.4597 0.3975 {{\rm{Var}}}(R_n)=0.6 p_1\; \; \text{Our model} 0.3300 0.3151 0.3024 p_2\; \; \text{Target-based model} 0.6624 0.5736 0.4957 {{\rm{Var}}}(R_n)=0.8 p_1\; \; \text{Our model} 0.3661 0.3507 0.3360 p_2\; \; \text{Target-based model} 0.7428 0.6532 0.5672 The second decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_1\; \; \text{Our model} 0.2759 0.2611 0.2523 p_2\; \; \text{Target-based model} 0.5125 0.4391 0.3760 {{\rm{Var}}}(R_n)=0.6 p_1\; \; \text{Our model} 0.3302 0.3167 0.3090 p_2\; \; \text{Target-based model} 0.6395 0.5450 0.4763 {{\rm{Var}}}(R_n)=0.8 p_1\; \; \text{Our model} 0.3641 0.3545 0.3431 p_2\; \; \text{Target-based model} 0.7178 0.6313 0.5508  The first decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_1\; \; \text{Our model} 0.2774 0.2661 0.2535 p_2\; \; \text{Target-based model} 0.5394 0.4597 0.3975 {{\rm{Var}}}(R_n)=0.6 p_1\; \; \text{Our model} 0.3300 0.3151 0.3024 p_2\; \; \text{Target-based model} 0.6624 0.5736 0.4957 {{\rm{Var}}}(R_n)=0.8 p_1\; \; \text{Our model} 0.3661 0.3507 0.3360 p_2\; \; \text{Target-based model} 0.7428 0.6532 0.5672 The second decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_1\; \; \text{Our model} 0.2759 0.2611 0.2523 p_2\; \; \text{Target-based model} 0.5125 0.4391 0.3760 {{\rm{Var}}}(R_n)=0.6 p_1\; \; \text{Our model} 0.3302 0.3167 0.3090 p_2\; \; \text{Target-based model} 0.6395 0.5450 0.4763 {{\rm{Var}}}(R_n)=0.8 p_1\; \; \text{Our model} 0.3641 0.3545 0.3431 p_2\; \; \text{Target-based model} 0.7178 0.6313 0.5508 The gap between p_1 and p_2  The first decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_2-p_1 0.2620 0.1936 0.1440 {{\rm{Var}}}(R_n)=0.6 p_2-p_1 0.3324 0.2585 0.1933 {{\rm{Var}}}(R_n)=0.8 p_2-p_1 0.3767 0.3025 0.2312 The second decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_2-p_1 0.2366 0.1780 0.1237 {{\rm{Var}}}(R_n)=0.6 p_2-p_1 0.3093 0.2283 0.1673 {{\rm{Var}}}(R_n)=0.8 p_2-p_1 0.3537 0.2768 0.2077  The first decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_2-p_1 0.2620 0.1936 0.1440 {{\rm{Var}}}(R_n)=0.6 p_2-p_1 0.3324 0.2585 0.1933 {{\rm{Var}}}(R_n)=0.8 p_2-p_1 0.3767 0.3025 0.2312 The second decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_2-p_1 0.2366 0.1780 0.1237 {{\rm{Var}}}(R_n)=0.6 p_2-p_1 0.3093 0.2283 0.1673 {{\rm{Var}}}(R_n)=0.8 p_2-p_1 0.3537 0.2768 0.2077 The frequencies that X_{T}/ ä _{75}$$ \ge k\zeta_0$
 The first decision mechanism of targets Final target $1.5\zeta_0$ä$_{75}$ $1.7\zeta_0$ä$_{75}$ $2\zeta_0$ä$_{75}$ ${{\rm{Var}}}(R_n) = 0.4$ $g_1\; \; \text{Our model}$ 0.5717 0.5725 0.5762 $g_2\; \; \text{Target-based model}$ 0.0038 0.0045 0.0040 ${{\rm{Var}}}(R_n) = 0.6$ $g_1\; \; \text{Our model}$ 0.5458 0.5466 0.5493 $g_2\; \; \text{Target-based model}$ 0.0036 0.0028 0.0027 ${{\rm{Var}}}(R_n) = 0.8$ $g_1\; \; \text{Our model}$ 0.5198 0.5167 0.5250 $g_2\; \; \text{Target-based model}$ 0.0037 0.0029 0.0027 The second decision mechanism of targets Final target $1.5\zeta_0$ä$_{75}$ $1.7\zeta_0$ä$_{75}$ $2\zeta_0$ä$_{75}$ ${{\rm{Var}}}(R_n) = 0.4$ $g_1\; \; \text{Our model}$ 0.3042 0.4329 0.5427 $g_2\; \; \text{Target-based model}$ 0.0001 0.0004 0.0018 ${{\rm{Var}}}(R_n) = 0.6$ $g_1\; \; \text{Our model}$ 0.3530 0.4405 0.5220 $g_2\; \; \text{Target-based model}$ 0.0003 0.00045 0.0019 ${{\rm{Var}}}(R_n) = 0.8$ $g_1\; \; \text{Our model}$ 0.3617 0.4293 0.5033 $g_2\; \; \text{Target-based model}$ 0.00005 0.0003 0.0013
 The first decision mechanism of targets Final target $1.5\zeta_0$ä$_{75}$ $1.7\zeta_0$ä$_{75}$ $2\zeta_0$ä$_{75}$ ${{\rm{Var}}}(R_n) = 0.4$ $g_1\; \; \text{Our model}$ 0.5717 0.5725 0.5762 $g_2\; \; \text{Target-based model}$ 0.0038 0.0045 0.0040 ${{\rm{Var}}}(R_n) = 0.6$ $g_1\; \; \text{Our model}$ 0.5458 0.5466 0.5493 $g_2\; \; \text{Target-based model}$ 0.0036 0.0028 0.0027 ${{\rm{Var}}}(R_n) = 0.8$ $g_1\; \; \text{Our model}$ 0.5198 0.5167 0.5250 $g_2\; \; \text{Target-based model}$ 0.0037 0.0029 0.0027 The second decision mechanism of targets Final target $1.5\zeta_0$ä$_{75}$ $1.7\zeta_0$ä$_{75}$ $2\zeta_0$ä$_{75}$ ${{\rm{Var}}}(R_n) = 0.4$ $g_1\; \; \text{Our model}$ 0.3042 0.4329 0.5427 $g_2\; \; \text{Target-based model}$ 0.0001 0.0004 0.0018 ${{\rm{Var}}}(R_n) = 0.6$ $g_1\; \; \text{Our model}$ 0.3530 0.4405 0.5220 $g_2\; \; \text{Target-based model}$ 0.0003 0.00045 0.0019 ${{\rm{Var}}}(R_n) = 0.8$ $g_1\; \; \text{Our model}$ 0.3617 0.4293 0.5033 $g_2\; \; \text{Target-based model}$ 0.00005 0.0003 0.0013
The frequencies of bankruptcies–20000 simulations
 The first decision mechanism of targets $\eta_T=1.5\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 16355(0.8177) 19774(0.9887) $S\neq 0$ 3645(0.1823) 226(0.0113) Our model Target-based model Mean of $S$ 0.6833 0.0181 $\eta_T=2.0\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 14424(0.7212) 18909(0.9455) $S\neq 0$ 5576(0.2788) 1091(0.0546) Our model Target-based model Mean of $S$ 1.3019 0.1135 The second decision mechanism of targets $\eta_T=1.5\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 16270(0.8135) 19667(0.9833) $S\neq 0$ 3730(0.1865) 333(0.0167) Our model Target-based model Mean of $S$ 0.8719 0.0297 $\eta_T=2.0\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 14118(0.7059) 18726(0.9363) $S\neq 0$ 5882(0.2941) 1274(0.0637) Our model Target-based model Mean of $S$ 1.6798 0.1457
 The first decision mechanism of targets $\eta_T=1.5\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 16355(0.8177) 19774(0.9887) $S\neq 0$ 3645(0.1823) 226(0.0113) Our model Target-based model Mean of $S$ 0.6833 0.0181 $\eta_T=2.0\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 14424(0.7212) 18909(0.9455) $S\neq 0$ 5576(0.2788) 1091(0.0546) Our model Target-based model Mean of $S$ 1.3019 0.1135 The second decision mechanism of targets $\eta_T=1.5\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 16270(0.8135) 19667(0.9833) $S\neq 0$ 3730(0.1865) 333(0.0167) Our model Target-based model Mean of $S$ 0.8719 0.0297 $\eta_T=2.0\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 14118(0.7059) 18726(0.9363) $S\neq 0$ 5882(0.2941) 1274(0.0637) Our model Target-based model Mean of $S$ 1.6798 0.1457
Comparison summary
 The first decision mechanism of targets Comparison items Our model Target-based model Times Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=1.5\zeta_0 ä _{75} 0.2774 0.5394 0.5143 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=1.5\zeta_0$ä$_{75}$ 0.5717 0.0038 150.4 Bankruptcy probabilities $\eta_T=1.5\zeta_0$ä$_{75}$ 0.1823 0.0113 16.63 Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=2.0\zeta_0 ä _{75} 0.2535 0.3975 0.6377 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=2.0\zeta_0$ä$_{75}$ 0.5762 0.0040 144.05 Bankruptcy probabilities $\eta_T=2.0\zeta_0$ä$_{75}$ 0.2788 0.0546 5.106 The second decision mechanism of targets Comparison items Our model Target-based model Times Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=1.5\zeta_0 ä _{75} 0.2759 0.5125 0.5383 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=1.5\zeta_0$ä$_{75}$ 0.3042 0.0001 3042 Bankruptcy probabilities $\eta_T=1.5\zeta_0$ä$_{75}$ 0.1865 0.0167 11.17 Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=2.0\zeta_0 ä _{75} 0.2523 0.3760 0.6710 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=2.0\zeta_0$ä$_{75}$ 0.5427 0.0018 301.5 Bankruptcy probabilities $\eta_T=2.0\zeta_0$ä$_{75}$ 0.2941 0.0637 4.617
 The first decision mechanism of targets Comparison items Our model Target-based model Times Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=1.5\zeta_0 ä _{75} 0.2774 0.5394 0.5143 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=1.5\zeta_0$ä$_{75}$ 0.5717 0.0038 150.4 Bankruptcy probabilities $\eta_T=1.5\zeta_0$ä$_{75}$ 0.1823 0.0113 16.63 Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=2.0\zeta_0 ä _{75} 0.2535 0.3975 0.6377 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=2.0\zeta_0$ä$_{75}$ 0.5762 0.0040 144.05 Bankruptcy probabilities $\eta_T=2.0\zeta_0$ä$_{75}$ 0.2788 0.0546 5.106 The second decision mechanism of targets Comparison items Our model Target-based model Times Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=1.5\zeta_0 ä _{75} 0.2759 0.5125 0.5383 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=1.5\zeta_0$ä$_{75}$ 0.3042 0.0001 3042 Bankruptcy probabilities $\eta_T=1.5\zeta_0$ä$_{75}$ 0.1865 0.0167 11.17 Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=2.0\zeta_0 ä _{75} 0.2523 0.3760 0.6710 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=2.0\zeta_0$ä$_{75}$ 0.5427 0.0018 301.5 Bankruptcy probabilities $\eta_T=2.0\zeta_0$ä$_{75}$ 0.2941 0.0637 4.617
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