# American Institute of Mathematical Sciences

October  2017, 13(4): 1641-1659. doi: 10.3934/jimo.2017011

## Optimal pension decision under heterogeneous health statuses and bequest motives

 a. China Financial Policy Research Center, School of Finance, Renmin University of China, Beijing, 100872, China b. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

* Corresponding author

Received  February 2016 Revised  October 02, 2016 Published  December 2016

In this paper, we study the optimal decision between ELA(Equity Linked Annuity) and ELID(Equity Linked Income Drawdown) pension plans under heterogeneous personal health statuses and bequest motives. In the ELA pension plan, the survival member receives the mortality credit, and leaves no bequest at the time of death, while the member receives no mortality credit and receives the fund wealth as bequest at the time of death in the ELID pension plan. The pension member controls the asset allocation and benefit outgo policies to achieve the objectives. We explore the square deviations between the actual benefit outgo and the pre-set target, and the square and negative linear deviations between the actual bequest and the pre-set target as the disutility function. The minimization of the disutility function is the objective of the stochastic optimal control problem. Using HJB (Hamilton-Jacobi-Bellman) equations and variational inequality methods, the closed-form optimal policies of the ELA and ELID pension plans are derived. Furthermore, the optimal decision boundary between the ELA and ELID plans is established. It is the first time to study the impacts of heterogeneous personal health status and bequest motive on the optimal choice between the ELA and ELID pension plans under the original performance criterions. The worse health status and higher bequest motive result in the higher utility of the ELID pension plan, and vice versa. The worse heath status increases the proportion allocated in the risky asset and increases the benefit outgo in both pension plans. The bequest motive has positive impacts on the proportion in the risky asset and negative impacts on the benefit outgo in the ELID pension plan.

Citation: Lin He, Zongxia Liang. Optimal pension decision under heterogeneous health statuses and bequest motives. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1641-1659. doi: 10.3934/jimo.2017011
##### References:
 [1] P. Albrecht and R. Maurer, Self-annuitization, ruin risk in retirment and asset allocation: The annuity benchmark, In Proceedings of the 11th International AFIR Colloquium, Toronto, 1 (2001), 19–37. [2] P. Battocchion and F. Menoncin, Optimal pension management in a stochastic framework, Insurance: Mathematics and Economics, 34 (2004), 79-95.  doi: 10.1016/j.insmatheco.2003.11.001. [3] B. D. Bernheim, How strong are bequest motives: Evidence based on estimates of the demand for life insurance and annuities, Journal of Political Economy, 99 (1991), 899-927. [4] D. Blake, A. J. G. Cairns and D. Blake, PensionmetricsII: stochastic pension plan design during the distribution phase, Insurance: Mathematics and Economics, 33 (2003), 29-47. [5] R. Bordley and M. Li Calzi, Decision analysis using targets instead of utility functions, Decision in Economics and Finance, 23 (2000), 53-74.  doi: 10.1007/s102030050005. [6] A. Brugiavini, Uncertainty resolution and the timing of annuity purchases, Journal of Public Economics, 50 (1993), 31-62.  doi: 10.1016/0047-2727(93)90059-3. [7] J. R. Brown, Private pensions, mortality risk, and the decision to annuitize, Journal of Public Economics, 82 (2001), 29-62. [8] A. J. G. Cairns, D. Blake and K. Dowd, Optimal dynamic asset allocation for defined contribution pension plans, Proceedings of the 10th AFIR Colloquium, Troms$φ$, 13 (2000), 131-154. [9] S. Chang, L. Tzeng and J. Miao, Pension funding incorporating downside risks, Insurance: Mathematics and Economics, 32 (2003), 217-228.  doi: 10.1016/S0167-6687(02)00211-1. [10] 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. [11] L. He and Z. X. Liang, Optimal dynamic asset allocation strategy for ELA scheme of DC pension plan during the distribution phase, Insurance: Mathematics and Economics, 52 (2013), 404-410.  doi: 10.1016/j.insmatheco.2013.02.005. [12] L. He and Z. X. Liang, Optimal asset allocation and benefit outgo policies of DC pension plan with compulsory conversion claims, Insurance: Mathematics and Economics, 61 (2015), 227-234.  doi: 10.1016/j.insmatheco.2015.01.006. [13] R. Josa-Fombellida and J. P. Rincón-Zapatero, Optimal risk management in defined benefit stochastic pension funds, Insurance: Mathematics and Economics, 34 (2004), 489-503.  doi: 10.1016/j.insmatheco.2004.03.002. [14] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1991. doi: 10.1007/978-1-4612-0949-2. [15] P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408. [16] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [17] M. A. Milevsky and C. Robinson, Self-annuitization and ruin in retirement, North American Acturial Journal, 4 (2000), 112-129.  doi: 10.1080/10920277.2000.10595940. [18] M. A. Milevsky and V. R. Young, Optimal asset allocation and the real option to defer annuitization: It's not now or never, Working paper, York University, Toronto, and University of Wisconsin-Madison, 2002. [19] 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. [20] B. Ngwira and R. Gerrard, Stochastic pension fund control in the presence of Poisson jumps, Insurance: Mathematics and Economics, 40 (2007), 283-292.  doi: 10.1016/j.insmatheco.2006.05.002. [21] E. Vigna and S. Haberman, Optimal investment strategy for defined contribution pension scheme, Insurance: Mathematics and Economics, 28 (2001), 233-262.  doi: 10.1016/S0167-6687(00)00077-9. [22] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, 1999. doi: 10.1007/978-1-4612-1466-3. [23] Q. Zhao, R. M. Wang and J. Q. Wei, Time-inconsistent 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. [24] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Springer, 2007. doi: 10.1007/978-3-540-69826-5.

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##### References:
 [1] P. Albrecht and R. Maurer, Self-annuitization, ruin risk in retirment and asset allocation: The annuity benchmark, In Proceedings of the 11th International AFIR Colloquium, Toronto, 1 (2001), 19–37. [2] P. Battocchion and F. Menoncin, Optimal pension management in a stochastic framework, Insurance: Mathematics and Economics, 34 (2004), 79-95.  doi: 10.1016/j.insmatheco.2003.11.001. [3] B. D. Bernheim, How strong are bequest motives: Evidence based on estimates of the demand for life insurance and annuities, Journal of Political Economy, 99 (1991), 899-927. [4] D. Blake, A. J. G. Cairns and D. Blake, PensionmetricsII: stochastic pension plan design during the distribution phase, Insurance: Mathematics and Economics, 33 (2003), 29-47. [5] R. Bordley and M. Li Calzi, Decision analysis using targets instead of utility functions, Decision in Economics and Finance, 23 (2000), 53-74.  doi: 10.1007/s102030050005. [6] A. Brugiavini, Uncertainty resolution and the timing of annuity purchases, Journal of Public Economics, 50 (1993), 31-62.  doi: 10.1016/0047-2727(93)90059-3. [7] J. R. Brown, Private pensions, mortality risk, and the decision to annuitize, Journal of Public Economics, 82 (2001), 29-62. [8] A. J. G. Cairns, D. Blake and K. Dowd, Optimal dynamic asset allocation for defined contribution pension plans, Proceedings of the 10th AFIR Colloquium, Troms$φ$, 13 (2000), 131-154. [9] S. Chang, L. Tzeng and J. Miao, Pension funding incorporating downside risks, Insurance: Mathematics and Economics, 32 (2003), 217-228.  doi: 10.1016/S0167-6687(02)00211-1. [10] 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. [11] L. He and Z. X. Liang, Optimal dynamic asset allocation strategy for ELA scheme of DC pension plan during the distribution phase, Insurance: Mathematics and Economics, 52 (2013), 404-410.  doi: 10.1016/j.insmatheco.2013.02.005. [12] L. He and Z. X. Liang, Optimal asset allocation and benefit outgo policies of DC pension plan with compulsory conversion claims, Insurance: Mathematics and Economics, 61 (2015), 227-234.  doi: 10.1016/j.insmatheco.2015.01.006. [13] R. Josa-Fombellida and J. P. Rincón-Zapatero, Optimal risk management in defined benefit stochastic pension funds, Insurance: Mathematics and Economics, 34 (2004), 489-503.  doi: 10.1016/j.insmatheco.2004.03.002. [14] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1991. doi: 10.1007/978-1-4612-0949-2. [15] P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408. [16] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [17] M. A. Milevsky and C. Robinson, Self-annuitization and ruin in retirement, North American Acturial Journal, 4 (2000), 112-129.  doi: 10.1080/10920277.2000.10595940. [18] M. A. Milevsky and V. R. Young, Optimal asset allocation and the real option to defer annuitization: It's not now or never, Working paper, York University, Toronto, and University of Wisconsin-Madison, 2002. [19] 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. [20] B. Ngwira and R. Gerrard, Stochastic pension fund control in the presence of Poisson jumps, Insurance: Mathematics and Economics, 40 (2007), 283-292.  doi: 10.1016/j.insmatheco.2006.05.002. [21] E. Vigna and S. Haberman, Optimal investment strategy for defined contribution pension scheme, Insurance: Mathematics and Economics, 28 (2001), 233-262.  doi: 10.1016/S0167-6687(00)00077-9. [22] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, 1999. doi: 10.1007/978-1-4612-1466-3. [23] Q. Zhao, R. M. Wang and J. Q. Wei, Time-inconsistent 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. [24] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Springer, 2007. doi: 10.1007/978-3-540-69826-5.
Optimal proportion in the risky asset $\pi_{1}^{*}$ and $\pi_{2}^{*}$ in ELA and ELID plans
Optimal benefit outgo $p_{1}^{*}$ and $p_{2}^{*}$, and bequest in ELA and ELID plans
The impacts of $\mu^{S}$ on optimal proportion in the risky asset $\pi_1^{*}$ in ELA plan
The impacts of $\mu^{S}$ on optimal proportion in the risky asset $\pi_2^{*}$ in ELID plan
The impacts of $\mu^{S}$ on optimal benefit outgo $p_1^{*}$ in ELA plan
The impacts of $\mu^{S}$ on optimal benefit outgo $p_2^{*}$ in ELID plan
The impacts of β on optimal proportion in the risky asset $\pi_2^{*}$ in ELID plan
The impacts of β on optimal benefit outgo $p_2^{*}$ in ELID plan
The impacts of $\mu^S$ on the objective functions V of ELA and ELID plans
The impacts of β on the objective functions V of ELA and ELID plans
Optimal choice between ELA and ELID plans under heterogeneous health status $\mu^{S}$s and bequest motive $\beta$s
VaR(95%) of the benefits outgo under the optimal and sub-optimal policies
 VaR(95%) ELA $\beta$ ELID $\beta=0.$ ELID $\beta=0.25.$ ELID $\beta=0.5.$ ELID $\beta=0.75.$ optimal sub-opt optimal sub-opt optimal sub-opt optimal sub-opt optimal sub-opt $\mu^S=0.025$ 2.2219 2.213 1.0604 0.3212 1.0604 0.331 1.1183 0.3473 1.117 0.3424 $\mu^S=0.05$ 2.3046 2.2859 1.2873 0.0789 1.3082 0.0719 1.309 0.0743 1.3166 0.0668 $\mu^S=0.075$ 2.3086 2.2663 1.2149 0.0085 1.2401 0.0042 1.3618 0.0037 1.384 0.0028 $\mu^S=0.01$ 2.3233 2.2675 1.0077 0.0072 1.1672 0.0033 1.3596 0.0017 1.4648 0.0011
 VaR(95%) ELA $\beta$ ELID $\beta=0.$ ELID $\beta=0.25.$ ELID $\beta=0.5.$ ELID $\beta=0.75.$ optimal sub-opt optimal sub-opt optimal sub-opt optimal sub-opt optimal sub-opt $\mu^S=0.025$ 2.2219 2.213 1.0604 0.3212 1.0604 0.331 1.1183 0.3473 1.117 0.3424 $\mu^S=0.05$ 2.3046 2.2859 1.2873 0.0789 1.3082 0.0719 1.309 0.0743 1.3166 0.0668 $\mu^S=0.075$ 2.3086 2.2663 1.2149 0.0085 1.2401 0.0042 1.3618 0.0037 1.384 0.0028 $\mu^S=0.01$ 2.3233 2.2675 1.0077 0.0072 1.1672 0.0033 1.3596 0.0017 1.4648 0.0011
ES of the benefits outgo under the optimal and sub-optimal policies
 ES ELA $\beta$ ElID $\beta=0.$ ELID $\beta=0.25.$ ELID $\beta=0.5.$ ELID $\beta=0.75.$ optimal sub-opt optimal sub-opt optimal sub-opt optimal sub-opt optimal sub-opt $\mu^S=0.025$ 0.299 0.3024 0.734 0.8892 0.7278 0.888 0.7096 0.8822 0.7065 0.876 $\mu^S=0.05$ 0.2644 0.2704 0.635 0.8238 0.6183 0.82 0.6039 0.8165 0.5871 0.814 $\mu^S=0.075$ 0.2405 0.2515 0.6034 0.8049 0.5865 0.8002 0.5418 0.7923 0.5128 0.785 $\mu^S=0.01$ 0.2196 0.2328 0.6125 0.8071 0.5645 0.7859 0.5066 0.7719 0.4559 0.7668
 ES ELA $\beta$ ElID $\beta=0.$ ELID $\beta=0.25.$ ELID $\beta=0.5.$ ELID $\beta=0.75.$ optimal sub-opt optimal sub-opt optimal sub-opt optimal sub-opt optimal sub-opt $\mu^S=0.025$ 0.299 0.3024 0.734 0.8892 0.7278 0.888 0.7096 0.8822 0.7065 0.876 $\mu^S=0.05$ 0.2644 0.2704 0.635 0.8238 0.6183 0.82 0.6039 0.8165 0.5871 0.814 $\mu^S=0.075$ 0.2405 0.2515 0.6034 0.8049 0.5865 0.8002 0.5418 0.7923 0.5128 0.785 $\mu^S=0.01$ 0.2196 0.2328 0.6125 0.8071 0.5645 0.7859 0.5066 0.7719 0.4559 0.7668
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