American Institute of Mathematical Sciences

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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,

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  January 2019 Early access  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.

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 and Management Optimization, 2019, 15 (1) : 401-427. doi: 10.3934/jimo.2018059
References:
[1]

P. Albrecht and R. Maurer, Self-annuitization, consumption shortfall in retirement and asset allocation: the annuity benchmark, Journal of Pension Economics and Finance, 1 (2002), 269-288.  doi: 10.1017/S1474747202001117.

[2]

D. Blake, A. J. G. Cairns and K. Dowd, Pensionmetrics 2: stochastic pension plan design during the distribution phase, Insurance: Mathematics and Economics, 33 (2003), 29-47.  doi: 10.1016/S0167-6687(03)00141-0.

[3]

J. F. Boulier, S. Michel and V. Wisnia, Optimizing investment and contribution policies of a defined benefit pension fund, Proceedings of the 6th AFIR International Colloquium, 1 (1996), 593-607. 

[4]

J. R. Brown, Rational and behavioral perspectives on the role of annuities in retirement planning, NBER Working Paper No. 13537, (2007), 1-36.  doi: 10.3386/w13537.

[5]

J. R. Brown, Private pensions, mortality risk, and the decision to annuitize, Journal of Public Economics, 82 (2001), 29-62.  doi: 10.3386/w7191.

[6]

A. Cairns, Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time, ASTIN Bulletin, 30 (2000), 19-55.  doi: 10.2143/AST.30.1.504625.

[7]

K. C. Cheung and H. L. Yang, Optimal investment-consumption strategy in discrete-time model with regime switching, Discrete and Continuous Dynamical Systems, 8 (2007), 315-332.  doi: 10.3934/dcdsb.2007.8.315.

[8]

M. Di Giacinto, S. Federico, F. Gozzi and E. Vigna, Constrained portfolio choices in the decumulation phase of a pension plan, Working Paper, Available at http://www.carloalberto.org/assets/working-papers/no.155.pdf. doi: 10.2139/ssrn.1600130.

[9]

M. Di Giacinto, S. Federico, F. Gozzi and E. Vigna, Income drawdown option with minimum guarantee, European Journal of Operational Research, 234 (2014), 610-624.  doi: 10.1016/j.ejor.2013.10.026.

[10]

P. Emms, Relative choice models for income drawdown in a defined contribution pension scheme, North American Actuarial Journal, 14 (2010), 176-197.  doi: 10.1080/10920277.2010.10597584.

[11]

P. Emms, Optimal investment and consumption using a defined-contribution pension scheme, Journal of Economic Dynamics and Control, 36 (2012), 1303-1321.  doi: 10.1016/j.jedc.2012.01.012.

[12]

P. Emms and S. Haberman, Income drawdown schemes for a defined-contribution pension plan, The Journal of Risk and Insurance, 75 (2008), 739-761. 

[13]

A. Finkelstein and J. Poterba, Selection effects in the united kingdom individual annuities market, Economic Journal, 112 (2002), 28-50.  doi: 10.1111/1468-0297.0j672.

[14]

R. Gerrard, S. Haberman, B. Hojgaard and E. Vigna, The income drawdown option: Quadratic loss, Actuarial Research Paper No. 155, Cass Business School, London, 2004.

[15]

R. Gerrard, S. Haberman and E. Vigna, Optimal investment choices post-retirement in a defined contribution pension scheme, Insurance: Mathematics and Economics, 35 (2004), 321-342.  doi: 10.1016/j.insmatheco.2004.06.002.

[16]

R. Gerrard, S. Haberman and E. Vigna, The management of decumulation risks in a defined contribution pension plan, North American Actuarial Journal, 10 (2006), 84-110.  doi: 10.1080/10920277.2006.10596241.

[17]

R. Gerrard, B. Hojgaard and E. Vigna, Choosing the optimal annuitization time post-retirement, Quantitative Finance, 12 (2012), 1143-1159.  doi: 10.1080/14697680903358248.

[18]

M. R. Hardy, A regime-switching model of long-term stock returns, North American Actuarial Journal, 5 (2001), 41-53.  doi: 10.1080/10920277.2001.10595984.

[19]

J. Inkmann, P. Lopes and A. Michaelides, How deep is the annuity market participation puzzle?, The Review of Financial Studies, 24 (2011), 279-319. 

[20]

L. M. Lockwood, Bequest motives and the annuity puzzle, Review of Economic Dynamics, 15 (2012), 226-243.  doi: 10.1016/j.red.2011.03.001.

[21]

M. A. Milevsky, Optimal asset allocation towards the end of the life cycle: To annuitize or not to annuitize?, The Journal of Risk and Insurance, 65 (1998), 401-426. 

[22]

M. A. Milevsky, Optimal annuitization policies: Analysis of the options, North American Actuarial Journal, 5 (2001), 57-69.  doi: 10.1080/10920277.2001.10595953.

[23]

M. A. Milevsky, K. S. Moore and V. R. Young, Optimal asset allocation and ruin minimization annuitization strategies, Mathematical Finance, 16 (2006), 647-671. 

[24]

M. A. Milevsky and C. Robinson, Self-annuitization and ruin in retirement, North American Actuarial Journal, 4 (2000), 112-129.  doi: 10.1080/10920277.2000.10595940.

[25]

M. A. Milevsky and V. R. Young, Optimal asset allocation and the real option to delay annuitization: It's not now-or-never, Schulich School of Business, Working Paper, (2002).

[26]

M. A. Milevsky and V. R. Young, Annuitization and asset allocation, Journal of Economic Dynamics and Control, 31 (2007), 3138-3177.  doi: 10.1016/j.jedc.2006.11.003.

[27]

O. S. Mitchell, J. M. Poterba, M. J. Warshawsky and J. R. Brown, New evidence on the money's worth of individual annuities, American Economic Review, 89 (1999), 1299-1318.  doi: 10.3386/w6002.

[28]

G. Stabile, Optimal timing of the annuity purchase: combined stochastic control and optimal stopping problem, International Journal of Theoretical and Applied Finance, 9 (2006), 151-170.  doi: 10.1142/S0219024906003524.

[29]

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.

[30]

H. L. Wu and Y. Zeng, Equilibrium investment strategy for defined-contribution pension schemes with generalized mean-variance criterion and mortality risk, Insurance: Mathematics and Economics, 64 (2015), 396-408.  doi: 10.1016/j.insmatheco.2015.07.007.

[31]

V. R. Young, Optimal investment strategy to minimize the probability of lifetime ruin, North American Actuarial Journal, 8 (2004), 106-126. 

[32]

Q. Zhao, R. M. Wang and J. Q. Wei, Time-consistent consumption-investment problem for a member in a defined contribution pension plan, Journal of Industrial and Management Optimization, 12 (2016), 1557-1585.  doi: 10.3934/jimo.2016.12.1557.

show all references

References:
[1]

P. Albrecht and R. Maurer, Self-annuitization, consumption shortfall in retirement and asset allocation: the annuity benchmark, Journal of Pension Economics and Finance, 1 (2002), 269-288.  doi: 10.1017/S1474747202001117.

[2]

D. Blake, A. J. G. Cairns and K. Dowd, Pensionmetrics 2: stochastic pension plan design during the distribution phase, Insurance: Mathematics and Economics, 33 (2003), 29-47.  doi: 10.1016/S0167-6687(03)00141-0.

[3]

J. F. Boulier, S. Michel and V. Wisnia, Optimizing investment and contribution policies of a defined benefit pension fund, Proceedings of the 6th AFIR International Colloquium, 1 (1996), 593-607. 

[4]

J. R. Brown, Rational and behavioral perspectives on the role of annuities in retirement planning, NBER Working Paper No. 13537, (2007), 1-36.  doi: 10.3386/w13537.

[5]

J. R. Brown, Private pensions, mortality risk, and the decision to annuitize, Journal of Public Economics, 82 (2001), 29-62.  doi: 10.3386/w7191.

[6]

A. Cairns, Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time, ASTIN Bulletin, 30 (2000), 19-55.  doi: 10.2143/AST.30.1.504625.

[7]

K. C. Cheung and H. L. Yang, Optimal investment-consumption strategy in discrete-time model with regime switching, Discrete and Continuous Dynamical Systems, 8 (2007), 315-332.  doi: 10.3934/dcdsb.2007.8.315.

[8]

M. Di Giacinto, S. Federico, F. Gozzi and E. Vigna, Constrained portfolio choices in the decumulation phase of a pension plan, Working Paper, Available at http://www.carloalberto.org/assets/working-papers/no.155.pdf. doi: 10.2139/ssrn.1600130.

[9]

M. Di Giacinto, S. Federico, F. Gozzi and E. Vigna, Income drawdown option with minimum guarantee, European Journal of Operational Research, 234 (2014), 610-624.  doi: 10.1016/j.ejor.2013.10.026.

[10]

P. Emms, Relative choice models for income drawdown in a defined contribution pension scheme, North American Actuarial Journal, 14 (2010), 176-197.  doi: 10.1080/10920277.2010.10597584.

[11]

P. Emms, Optimal investment and consumption using a defined-contribution pension scheme, Journal of Economic Dynamics and Control, 36 (2012), 1303-1321.  doi: 10.1016/j.jedc.2012.01.012.

[12]

P. Emms and S. Haberman, Income drawdown schemes for a defined-contribution pension plan, The Journal of Risk and Insurance, 75 (2008), 739-761. 

[13]

A. Finkelstein and J. Poterba, Selection effects in the united kingdom individual annuities market, Economic Journal, 112 (2002), 28-50.  doi: 10.1111/1468-0297.0j672.

[14]

R. Gerrard, S. Haberman, B. Hojgaard and E. Vigna, The income drawdown option: Quadratic loss, Actuarial Research Paper No. 155, Cass Business School, London, 2004.

[15]

R. Gerrard, S. Haberman and E. Vigna, Optimal investment choices post-retirement in a defined contribution pension scheme, Insurance: Mathematics and Economics, 35 (2004), 321-342.  doi: 10.1016/j.insmatheco.2004.06.002.

[16]

R. Gerrard, S. Haberman and E. Vigna, The management of decumulation risks in a defined contribution pension plan, North American Actuarial Journal, 10 (2006), 84-110.  doi: 10.1080/10920277.2006.10596241.

[17]

R. Gerrard, B. Hojgaard and E. Vigna, Choosing the optimal annuitization time post-retirement, Quantitative Finance, 12 (2012), 1143-1159.  doi: 10.1080/14697680903358248.

[18]

M. R. Hardy, A regime-switching model of long-term stock returns, North American Actuarial Journal, 5 (2001), 41-53.  doi: 10.1080/10920277.2001.10595984.

[19]

J. Inkmann, P. Lopes and A. Michaelides, How deep is the annuity market participation puzzle?, The Review of Financial Studies, 24 (2011), 279-319. 

[20]

L. M. Lockwood, Bequest motives and the annuity puzzle, Review of Economic Dynamics, 15 (2012), 226-243.  doi: 10.1016/j.red.2011.03.001.

[21]

M. A. Milevsky, Optimal asset allocation towards the end of the life cycle: To annuitize or not to annuitize?, The Journal of Risk and Insurance, 65 (1998), 401-426. 

[22]

M. A. Milevsky, Optimal annuitization policies: Analysis of the options, North American Actuarial Journal, 5 (2001), 57-69.  doi: 10.1080/10920277.2001.10595953.

[23]

M. A. Milevsky, K. S. Moore and V. R. Young, Optimal asset allocation and ruin minimization annuitization strategies, Mathematical Finance, 16 (2006), 647-671. 

[24]

M. A. Milevsky and C. Robinson, Self-annuitization and ruin in retirement, North American Actuarial Journal, 4 (2000), 112-129.  doi: 10.1080/10920277.2000.10595940.

[25]

M. A. Milevsky and V. R. Young, Optimal asset allocation and the real option to delay annuitization: It's not now-or-never, Schulich School of Business, Working Paper, (2002).

[26]

M. A. Milevsky and V. R. Young, Annuitization and asset allocation, Journal of Economic Dynamics and Control, 31 (2007), 3138-3177.  doi: 10.1016/j.jedc.2006.11.003.

[27]

O. S. Mitchell, J. M. Poterba, M. J. Warshawsky and J. R. Brown, New evidence on the money's worth of individual annuities, American Economic Review, 89 (1999), 1299-1318.  doi: 10.3386/w6002.

[28]

G. Stabile, Optimal timing of the annuity purchase: combined stochastic control and optimal stopping problem, International Journal of Theoretical and Applied Finance, 9 (2006), 151-170.  doi: 10.1142/S0219024906003524.

[29]

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.

[30]

H. L. Wu and Y. Zeng, Equilibrium investment strategy for defined-contribution pension schemes with generalized mean-variance criterion and mortality risk, Insurance: Mathematics and Economics, 64 (2015), 396-408.  doi: 10.1016/j.insmatheco.2015.07.007.

[31]

V. R. Young, Optimal investment strategy to minimize the probability of lifetime ruin, North American Actuarial Journal, 8 (2004), 106-126. 

[32]

Q. Zhao, R. M. Wang and J. Q. Wei, Time-consistent consumption-investment problem for a member in a defined contribution pension plan, Journal of Industrial and Management Optimization, 12 (2016), 1557-1585.  doi: 10.3934/jimo.2016.12.1557.

Figure 1.  Fn over time
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Figure 2.  The targets and the expected wealths over time
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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
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Figure 4.  Accumulated withdrawal amounts over time
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Figure 5.  Expected withdrawal amounts at each time
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Figure 6.  $F_n-{\rm{E}}_{0, x_0}(X_n), ~n = 0, 1, \ldots, T$
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Figure 7.  Expectation of the accumulated withdrawal amount over time
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Figure 8.  Growth ranges of the accumulated withdrawal amounts
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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
$[F_T-4000, F_T-2000)$ 0.1811 0.1801 0.1724
$[F_T-2000, F_T)$ 0.4168 0.4842 0.5500
$[F_T, +\infty)$ 0 0 0
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$X_T\in[a, b)$ 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
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
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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
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
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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
Table 4.  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
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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
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)
$n=144$ 109,422 (27167) 110,911 (27046) 114,109 (27234)
$n=180$ 136,589 (27167) 137,134 (26223) 139,640 (25531)
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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)
Table 6.  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
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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
Table 7.  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$$(x_0/ä_{60})ä_{75}$ $S=0$ 9608 simulations 9611 simulations 9540 simulations
Mean of $S$ 0.9601 0.8639 0.8636
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Events Deter. withdrawal Comb. withdrawal Prop. withdrawal
$F_T=1.4 (x_0/ä_{60})_{75}$ $S=0$ 9771 simulations 9820 simulations 9850 simulations
Mean of $S$ 0.4555 0.3011 0.2060
$F_T=1.5 (x_0/ ä_{60})ä_{75}$ $S=0$ 9692 simulations 9726 simulations 9729 simulations
Mean of $S$ 0.7341 0.5870 0.5060
$F_T=1.6$$(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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