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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
Chuangwei Lin 1, , Li Zeng 2,, and Huiling Wu 3,
Research Center for International Trade and Economic, Guangdong University of Foreign Studies, Guangzhou 510006, China
School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China
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)
Figure(8) / Table(7)
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.
Keywords: Defined-contribution pension scheme, decumulation phase, income drawdown option, quadratic loss function, portfolio selection optimization.
Mathematics Subject Classification: Primary: 91G10, 91G80; Secondary: 90C90, 91B02.
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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Figure 1. Fn over time
Figure 2. The targets and the expected wealths over time
Figure 3. ${\rm{E}}_{0, x_0}\left((1-\rho_n)X_n-c_n\right)$ and ${\rm{E}}_{0, x_0}(\pi_n^*)$ over time
Figure 4. Accumulated withdrawal amounts over time
Figure 5. Expected withdrawal amounts at each time
Figure 6. $F_n-{\rm{E}}_{0, x_0}(X_n), ~n = 0, 1, \ldots, T$
Figure 7. Expectation of the accumulated withdrawal amount over time
Figure 8. Growth ranges of the accumulated withdrawal amounts
Table 1. 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
$[F_T-2000, F_T)$ 0.4168 0.4842 0.5500
$[F_T, +\infty)$ 0 0 0
Table 2. 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
Table 3. The first occurrence of bankruptcy
$\tau\in[0, 60)$ 100 117 153
$\tau\in[60,120)$ 142 118 92
$\tau\in[120 180]$ 66 39 26
Mean of $\tau$ 86 76 65
Table 4. The mean of $\tilde \tau$ and $\tilde S$-10000 simulations
Mean of $\tilde \tau$ 64.4634 51.1655 41.9057
Mean of $\tilde S$ 24.1194 21.6262 18.9368
Table 5. 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)
Table 6. The total number of bankruptcy-10000 simulations
Events Deter. withdrawal Comb. withdrawal Prop. withdrawal
$r^f=1.0020$ $S=0$
$r^f=1.0025$ $S=0$ 9692 9726 9729
$F_T=1.4 (x_0/ä_{60})_{75}$ $S=0$ 9771 simulations 9820 simulations 9850 simulations
$F_T=1.5 (x_0/ ä_{60})ä_{75}$ $S=0$ 9692 simulations 9726 simulations 9729 simulations
$F_T=1.6$$(x_0/ä_{60})ä_{75}$ $S=0$ 9608 simulations 9611 simulations 9540 simulations
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Chuangwei Lin Li Zeng Huiling Wu