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A class of two-stage distributionally robust games
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 |
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.
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. |








Deter. withdrawal Frequencies | Comb. withdrawal Frequencies | Prop. withdrawal Frequencies | |
|
0.3024 | 0.2472 | 0.1985 |
|
0.0997 | 0.0885 | 0.0791 |
|
0.1811 | 0.1801 | 0.1724 |
|
0.4168 | 0.4842 | 0.5500 |
|
0 | 0 | 0 |
Deter. withdrawal Frequencies | Comb. withdrawal Frequencies | Prop. withdrawal Frequencies | |
|
0.3024 | 0.2472 | 0.1985 |
|
0.0997 | 0.0885 | 0.0791 |
|
0.1811 | 0.1801 | 0.1724 |
|
0.4168 | 0.4842 | 0.5500 |
|
0 | 0 | 0 |
Events | Deter. withdrawal Number of simulations | Comb. withdrawal Number of simulations | Prop. withdrawal Number of simulations |
|
9692 | 9726 | 9729 |
|
265 | 247 | 246 |
|
39 | 23 | 22 |
|
4 | 4 | 3 |
|
0 | 0 | 0 |
Deter. withdrawal | Comb. withdrawal | Prop. withdrawal | |
Mean of |
0.7341 | 0.5870 | 0.5060 |
Events | Deter. withdrawal Number of simulations | Comb. withdrawal Number of simulations | Prop. withdrawal Number of simulations |
|
9692 | 9726 | 9729 |
|
265 | 247 | 246 |
|
39 | 23 | 22 |
|
4 | 4 | 3 |
|
0 | 0 | 0 |
Deter. withdrawal | Comb. withdrawal | Prop. withdrawal | |
Mean of |
0.7341 | 0.5870 | 0.5060 |
Events | Deter. withdrawal Number of simulations | Comb. withdrawal Number of simulations | Prop. withdrawal Number of simulations |
|
100 | 117 | 153 |
|
142 | 118 | 92 |
|
66 | 39 | 26 |
Deter. withdrawal | Comb. withdrawal | Prop. withdrawal | |
Mean of |
86 | 76 | 65 |
Events | Deter. withdrawal Number of simulations | Comb. withdrawal Number of simulations | Prop. withdrawal Number of simulations |
|
100 | 117 | 153 |
|
142 | 118 | 92 |
|
66 | 39 | 26 |
Deter. withdrawal | Comb. withdrawal | Prop. withdrawal | |
Mean of |
86 | 76 | 65 |
Deter. withdrawal | Comb. withdrawal | Prop. withdrawal | |
Mean of |
64.4634 | 51.1655 | 41.9057 |
Mean of |
24.1194 | 21.6262 | 18.9368 |
Deter. withdrawal | Comb. withdrawal | Prop. withdrawal | |
Mean of |
64.4634 | 51.1655 | 41.9057 |
Mean of |
24.1194 | 21.6262 | 18.9368 |
Time | Deter. withdrawal | Comb. withdrawal | Prop. withdrawal |
|
027,922 | 028,338 | 029,112 |
055,088 (27166) | 056,223 (27885) | 058,294 (29182) | |
082,255 (27167) | 083,865 (27642) | 086,875 (28581) | |
109,422 (27167) | 110,911 (27046) | 114,109 (27234) | |
136,589 (27167) | 137,134 (26223) | 139,640 (25531) |
Time | Deter. withdrawal | Comb. withdrawal | Prop. withdrawal |
|
027,922 | 028,338 | 029,112 |
055,088 (27166) | 056,223 (27885) | 058,294 (29182) | |
082,255 (27167) | 083,865 (27642) | 086,875 (28581) | |
109,422 (27167) | 110,911 (27046) | 114,109 (27234) | |
136,589 (27167) | 137,134 (26223) | 139,640 (25531) |
Events | Deter. withdrawal | Comb. withdrawal | Prop. withdrawal | |
|
9672 | 9697 | 9701 | |
Mean of |
0.8268 | 0.6933 | 0.6123 | |
|
9692 | 9726 | 9729 | |
Mean of |
0.7341 | 0.5870 | 0.5060 | |
|
9690 | 9724 | 9718 | |
Mean of |
0.6565 | 0.5304 | 0.5001 |
Events | Deter. withdrawal | Comb. withdrawal | Prop. withdrawal | |
|
9672 | 9697 | 9701 | |
Mean of |
0.8268 | 0.6933 | 0.6123 | |
|
9692 | 9726 | 9729 | |
Mean of |
0.7341 | 0.5870 | 0.5060 | |
|
9690 | 9724 | 9718 | |
Mean of |
0.6565 | 0.5304 | 0.5001 |
Events | Deter. withdrawal | Comb. withdrawal | Prop. withdrawal | |
|
9771 simulations | 9820 simulations | 9850 simulations | |
Mean of |
0.4555 | 0.3011 | 0.2060 | |
9692 simulations | 9726 simulations | 9729 simulations | ||
Mean of |
0.7341 | 0.5870 | 0.5060 | |
9608 simulations | 9611 simulations | 9540 simulations | ||
Mean of |
0.9601 | 0.8639 | 0.8636 |
Events | Deter. withdrawal | Comb. withdrawal | Prop. withdrawal | |
|
9771 simulations | 9820 simulations | 9850 simulations | |
Mean of |
0.4555 | 0.3011 | 0.2060 | |
9692 simulations | 9726 simulations | 9729 simulations | ||
Mean of |
0.7341 | 0.5870 | 0.5060 | |
9608 simulations | 9611 simulations | 9540 simulations | ||
Mean of |
0.9601 | 0.8639 | 0.8636 |
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