# American Institute of Mathematical Sciences

January  2019, 15(1): 401-427. doi: 10.3934/jimo.2018059

## Multi-period portfolio optimization in a defined contribution pension plan during the decumulation phase

 1 Research Center for International Trade and Economic, Guangdong University of Foreign Studies, Guangzhou 510006, China 2 School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China 3 China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China

* Corresponding author: zsuzengli@126.com

Received  September 2017 Revised  December 2017 Published  April 2018

Fund Project: The first author is supported by National Natural Science Foundation of China (No. 11671411) and Innovative School Project in Higher Education of Guangdong, China (No. GWTP-SY-2014-02).

This paper studies a multi-period portfolio selection problem for retirees during the decumulation phase. We set a series of investment targets over time and aim to minimize the expected losses from the time of retirement to the time of compulsory annuitization by using a quadratic loss function. A target greater than the expected wealth is given and the corresponding explicit expressions for the optimal investment strategy are obtained. In addition, the withdrawal amount for daily life is assumed to be a linear function of the wealth level. Then according to the parameter value settings in the linear function, the withdrawal mechanism is classified as deterministic withdrawal, proportional withdrawal or combined withdrawal. The properties of the investment strategies, targets, bankruptcy probabilities and accumulated withdrawal amounts are compared under the three withdrawal mechanisms. Finally, numerical illustrations are presented to analyze the effects of the final target and the interest rate on some obtained results.

Citation: Chuangwei Lin, Li Zeng, Huiling Wu. Multi-period portfolio optimization in a defined contribution pension plan during the decumulation phase. Journal of Industrial & Management Optimization, 2019, 15 (1) : 401-427. doi: 10.3934/jimo.2018059
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##### References:
Fn over time
The targets and the expected wealths over time
${\rm{E}}_{0, x_0}\left((1-\rho_n)X_n-c_n\right)$ and ${\rm{E}}_{0, x_0}(\pi_n^*)$ over time
Accumulated withdrawal amounts over time
Expected withdrawal amounts at each time
$F_n-{\rm{E}}_{0, x_0}(X_n), ~n = 0, 1, \ldots, T$
Expectation of the accumulated withdrawal amount over time
Growth ranges of the accumulated withdrawal amounts
Frequencies of some events that $X_T$ is in the neighborhood of $F_T$
 $X_T\in[a, b)$ Deter. withdrawal Frequencies Comb. withdrawal Frequencies Prop. withdrawal Frequencies $(-\infty, F_T-6000)$ 0.3024 0.2472 0.1985 $[F_T-6000, F_T-4000)$ 0.0997 0.0885 0.0791 $[F_T-4000, F_T-2000)$ 0.1811 0.1801 0.1724 $[F_T-2000, F_T)$ 0.4168 0.4842 0.5500 $[F_T, +\infty)$ 0 0 0
 $X_T\in[a, b)$ Deter. withdrawal Frequencies Comb. withdrawal Frequencies Prop. withdrawal Frequencies $(-\infty, F_T-6000)$ 0.3024 0.2472 0.1985 $[F_T-6000, F_T-4000)$ 0.0997 0.0885 0.0791 $[F_T-4000, F_T-2000)$ 0.1811 0.1801 0.1724 $[F_T-2000, F_T)$ 0.4168 0.4842 0.5500 $[F_T, +\infty)$ 0 0 0
The total number of bankruptcies-10000 simulations
 Events Deter. withdrawal Number of simulations Comb. withdrawal Number of simulations Prop. withdrawal Number of simulations $S=0$ 9692 9726 9729 $S\in(0, 50]$ 265 247 246 $S\in(50,100]$ 39 23 22 $S\in(100,150]$ 4 4 3 $S\in(150, +\infty)$ 0 0 0 Deter. withdrawal Comb. withdrawal Prop. withdrawal Mean of $S$ 0.7341 0.5870 0.5060
 Events Deter. withdrawal Number of simulations Comb. withdrawal Number of simulations Prop. withdrawal Number of simulations $S=0$ 9692 9726 9729 $S\in(0, 50]$ 265 247 246 $S\in(50,100]$ 39 23 22 $S\in(100,150]$ 4 4 3 $S\in(150, +\infty)$ 0 0 0 Deter. withdrawal Comb. withdrawal Prop. withdrawal Mean of $S$ 0.7341 0.5870 0.5060
The first occurrence of bankruptcy
 Events Deter. withdrawal Number of simulations Comb. withdrawal Number of simulations Prop. withdrawal Number of simulations $\tau\in[0, 60)$ 100 117 153 $\tau\in[60,120)$ 142 118 92 $\tau\in[120 180]$ 66 39 26 Deter. withdrawal Comb. withdrawal Prop. withdrawal Mean of $\tau$ 86 76 65
 Events Deter. withdrawal Number of simulations Comb. withdrawal Number of simulations Prop. withdrawal Number of simulations $\tau\in[0, 60)$ 100 117 153 $\tau\in[60,120)$ 142 118 92 $\tau\in[120 180]$ 66 39 26 Deter. withdrawal Comb. withdrawal Prop. withdrawal Mean of $\tau$ 86 76 65
The mean of $\tilde \tau$ and $\tilde S$-10000 simulations
 Deter. withdrawal Comb. withdrawal Prop. withdrawal Mean of $\tilde \tau$ 64.4634 51.1655 41.9057 Mean of $\tilde S$ 24.1194 21.6262 18.9368
 Deter. withdrawal Comb. withdrawal Prop. withdrawal Mean of $\tilde \tau$ 64.4634 51.1655 41.9057 Mean of $\tilde S$ 24.1194 21.6262 18.9368
Accumulated withdrawal amounts and their relative changes
 Time Deter. withdrawal Comb. withdrawal Prop. withdrawal $n=36$ 027,922 028,338 029,112 $n=72$ 055,088 (27166) 056,223 (27885) 058,294 (29182) $n=108$ 082,255 (27167) 083,865 (27642) 086,875 (28581) $n=144$ 109,422 (27167) 110,911 (27046) 114,109 (27234) $n=180$ 136,589 (27167) 137,134 (26223) 139,640 (25531)
 Time Deter. withdrawal Comb. withdrawal Prop. withdrawal $n=36$ 027,922 028,338 029,112 $n=72$ 055,088 (27166) 056,223 (27885) 058,294 (29182) $n=108$ 082,255 (27167) 083,865 (27642) 086,875 (28581) $n=144$ 109,422 (27167) 110,911 (27046) 114,109 (27234) $n=180$ 136,589 (27167) 137,134 (26223) 139,640 (25531)
The total number of bankruptcy-10000 simulations
 Events Deter. withdrawal Comb. withdrawal Prop. withdrawal $r^f=1.0020$ $S=0$ 9672 9697 9701 Mean of $S$ 0.8268 0.6933 0.6123 $r^f=1.0025$ $S=0$ 9692 9726 9729 Mean of $S$ 0.7341 0.5870 0.5060 $r^f=1.0030$ $S=0$ 9690 9724 9718 Mean of $S$ 0.6565 0.5304 0.5001
 Events Deter. withdrawal Comb. withdrawal Prop. withdrawal $r^f=1.0020$ $S=0$ 9672 9697 9701 Mean of $S$ 0.8268 0.6933 0.6123 $r^f=1.0025$ $S=0$ 9692 9726 9729 Mean of $S$ 0.7341 0.5870 0.5060 $r^f=1.0030$ $S=0$ 9690 9724 9718 Mean of $S$ 0.6565 0.5304 0.5001
The total number of bankruptcies-10000 simulations
 Events Deter. withdrawal Comb. withdrawal Prop. withdrawal $F_T=1.4 (x_0/ä_{60})_{75}$ $S=0$ 9771 simulations 9820 simulations 9850 simulations Mean of $S$ 0.4555 0.3011 0.2060 $F_T=1.5 (x_0/ ä_{60})ä_{75}$ $S=0$ 9692 simulations 9726 simulations 9729 simulations Mean of $S$ 0.7341 0.5870 0.5060 $F_T=1.6$(x_0/ä_{60})ä_{75} S=0 9608 simulations 9611 simulations 9540 simulations Mean of S 0.9601 0.8639 0.8636  Events Deter. withdrawal Comb. withdrawal Prop. withdrawal F_T=1.4 (x_0/ä_{60})_{75} S=0 9771 simulations 9820 simulations 9850 simulations Mean of S 0.4555 0.3011 0.2060 F_T=1.5 (x_0/ ä_{60})ä_{75} S=0 9692 simulations 9726 simulations 9729 simulations Mean of S 0.7341 0.5870 0.5060 F_T=1.6$(x_0/ä_{60})ä_{75}$ $S=0$ 9608 simulations 9611 simulations 9540 simulations Mean of $S$ 0.9601 0.8639 0.8636
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