# American Institute of Mathematical Sciences

May  2021, 17(3): 1147-1171. doi: 10.3934/jimo.2020015

## Mean-variance investment and contribution decisions for defined benefit pension plans in a stochastic framework

 1 School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai 201620, China 2 School of Risk and Actuarial Studies, University of New South Wales, Sydney NSW 2052, Australia 3 Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, Shanghai 200241, China

* Corresponding author: Jiaqin Wei

Received  January 2019 Revised  September 2019 Published  May 2021 Early access  January 2020

Fund Project: This work was supported by the 111 Project (B14019); the National Natural Science Foundation of China (11601157, 11571113, 11601320, 11971301); and the Natural Sciences and Engineering Research Council of Canada (RGPIN-2016-05677)

In this paper we investigate the management of a defined benefit pension plan under a model with random coefficients. The objective of the pension sponsor is to minimize the solvency risk, contribution risk and the expected terminal value of the unfunded actuarial liability. By measuring the solvency risk in terms of the variance of the terminal unfunded actuarial liability, we formulate the problem as a mean-variance problem with an additional running cost. With the help of a system of backward stochastic differential equations, we derive a time-consistent equilibrium strategy towards investment and contribution rate. The obtained equilibrium strategy turns out to be a good candidate for a stable contribution plan. When the interest rate is given by the Vasicek model and all other coefficients are deterministic, we obtain closed-form solutions of the equilibrium strategy and efficient frontier.

Citation: Qian Zhao, Yang Shen, Jiaqin Wei. Mean-variance investment and contribution decisions for defined benefit pension plans in a stochastic framework. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1147-1171. doi: 10.3934/jimo.2020015
##### References:
 [1] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016. [2] P. Battocchio 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] T. Björk, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance and Stochastics, 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5. [4] T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Mathematical Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x. [5] P. Briand and F. Confortola, BSDEs with stochastic lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Processes and their Applications, 118 (2008), 818-838.  doi: 10.1016/j.spa.2007.06.006. [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] S. C. Chang, L. Y. Tzeng and J. C. Y. Miao, Pension funding incorporating downside risks, Insurance: Mathematics and Economics, 32 (2003), 217-228.  doi: 10.1016/S0167-6687(02)00211-1. [8] L. Colombo and S. Haberman, Optimal contributions in a defined benefit pension scheme with stochastic new entrants, Insurance: Mathematics and Economics, 37 (2005), 335-354.  doi: 10.1016/j.insmatheco.2005.02.011. [9] C. Czichowsky, Time-consistent mean-variance portfolio selection in discrete and continuous time, Finance and Stochastics, 17 (2013), 227-271.  doi: 10.1007/s00780-012-0189-9. [10] L. Delong, R. Gerrard and S. Haberman, Mean-variance optimization problems for an accumulation phase in a defined benefit plan, Insurance: Mathematics and Economics, 42 (2008), 107-118.  doi: 10.1016/j.insmatheco.2007.01.005. [11] I. Ekeland, O. Mbodji and T. A. Pirvu, Time-consistent portfolio management, SIAM Journal on Financial Mathematics, 3 (2012), 1-32.  doi: 10.1137/100810034. [12] I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Mathematics and Financial Economics, 2 (2008), 57-86.  doi: 10.1007/s11579-008-0014-6. [13] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022. [14] S. Haberman, Stochastic investment returns and contribution rate risk in a defined benefit pension scheme, Insurance: Mathematics and Economics, 19 (1997), 127-139.  doi: 10.1016/S0167-6687(96)00019-4. [15] S. Haberman, Z. Butt and C. Megaloudi, Contribution and solvency risk in a defined benefit pension scheme, Insurance: Mathematics and Economics, 27 (2000), 237-259.  doi: 10.1016/S0167-6687(00)00051-2. [16] S. Haberman and J.-H. Sung, Dynamic approaches to pension funding, Insurance: Mathematics and Economics, 15 (1994), 151-162. [17] Y. Hu, H. Q. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM Journal on Control and Optimization, 50 (2012), 1548-1572.  doi: 10.1137/110853960. [18] H.-C. Huang and A. J. G. Cairns, On the control of defined-benefit pension plans, Insurance: Mathematics and Economics, 38 (2006), 113-131.  doi: 10.1016/j.insmatheco.2005.08.005. [19] R. Josa-Fombellida and J. Rincón-Zapatero, Minimization of risks in pension funding by means of contributions and portfolio selection, Insurance: Mathematics and Economics, 29 (2001), 35-45.  doi: 10.1016/S0167-6687(01)00070-1. [20] 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. [21] R. Josa-Fombellida and J. P. Rincón-Zapatero, Mean-variance portfolio and contribution selection in stochastic pension funding, European Journal of Operational Research, 187 (2008), 120-137.  doi: 10.1016/j.ejor.2007.03.002. [22] R. Josa-Fombellida and J. P. Rincón-Zapatero, Optimal asset allocation for aggregated defined benefit pension funds with stochastic interest rates, European Journal of Operational Research, 201 (2010), 211-221.  doi: 10.1016/j.ejor.2009.02.021. [23] E. Lee, An Introduction to Pension Schemes, Institute and Faculty of Actuaries, London, UK, 1986. [24] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [25] A. E. B. Lim, Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Mathematics of Operations Research, 29 (2004), 132-161.  doi: 10.1287/moor.1030.0065. [26] J. Marín-Solano and J. Navas, Consumption and portfolio rules for time-inconsistent investors, European Journal of Operational Research, 201 (2010), 860-872.  doi: 10.1016/j.ejor.2009.04.005. [27] D. Marshall and J. Reeve, Defined Benefit Pension Schemes: Funding for Ongoing Security, Presented to Staple Inn Actuarial Society, London, UK, 1993. [28] M.-A. Morlais, Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem, Finance and Stochastics, 13 (2009), 121-150.  doi: 10.1007/s00780-008-0079-3. [29] 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. [30] R. H. Strotz, Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, (1973), 128–143. doi: 10.1007/978-1-349-15492-0_10. [31] J. Q. Wei and T. X. Wang, Time-consistent mean-variance asset-liability management with random coefficients, Insurance: Mathematics and Economics, 77 (2017), 84-96.  doi: 10.1016/j.insmatheco.2017.08.011. [32] J. M. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations: Time-consistent solutions, Transactions of the American Mathematical Society, 369 (2017), 5467-5523.  doi: 10.1090/tran/6502. [33] Q. Zhao, Y. Shen and J. Q. Wei, Consumption-investment strategies with non-exponential discounting and logarithmic utility, European Journal of Operational Research, 238 (2014), 824-835.  doi: 10.1016/j.ejor.2014.04.034. [34] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.

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##### References:
 [1] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016. [2] P. Battocchio 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] T. Björk, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance and Stochastics, 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5. [4] T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Mathematical Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x. [5] P. Briand and F. Confortola, BSDEs with stochastic lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Processes and their Applications, 118 (2008), 818-838.  doi: 10.1016/j.spa.2007.06.006. [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] S. C. Chang, L. Y. Tzeng and J. C. Y. Miao, Pension funding incorporating downside risks, Insurance: Mathematics and Economics, 32 (2003), 217-228.  doi: 10.1016/S0167-6687(02)00211-1. [8] L. Colombo and S. Haberman, Optimal contributions in a defined benefit pension scheme with stochastic new entrants, Insurance: Mathematics and Economics, 37 (2005), 335-354.  doi: 10.1016/j.insmatheco.2005.02.011. [9] C. Czichowsky, Time-consistent mean-variance portfolio selection in discrete and continuous time, Finance and Stochastics, 17 (2013), 227-271.  doi: 10.1007/s00780-012-0189-9. [10] L. Delong, R. Gerrard and S. Haberman, Mean-variance optimization problems for an accumulation phase in a defined benefit plan, Insurance: Mathematics and Economics, 42 (2008), 107-118.  doi: 10.1016/j.insmatheco.2007.01.005. [11] I. Ekeland, O. Mbodji and T. A. Pirvu, Time-consistent portfolio management, SIAM Journal on Financial Mathematics, 3 (2012), 1-32.  doi: 10.1137/100810034. [12] I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Mathematics and Financial Economics, 2 (2008), 57-86.  doi: 10.1007/s11579-008-0014-6. [13] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022. [14] S. Haberman, Stochastic investment returns and contribution rate risk in a defined benefit pension scheme, Insurance: Mathematics and Economics, 19 (1997), 127-139.  doi: 10.1016/S0167-6687(96)00019-4. [15] S. Haberman, Z. Butt and C. Megaloudi, Contribution and solvency risk in a defined benefit pension scheme, Insurance: Mathematics and Economics, 27 (2000), 237-259.  doi: 10.1016/S0167-6687(00)00051-2. [16] S. Haberman and J.-H. Sung, Dynamic approaches to pension funding, Insurance: Mathematics and Economics, 15 (1994), 151-162. [17] Y. Hu, H. Q. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM Journal on Control and Optimization, 50 (2012), 1548-1572.  doi: 10.1137/110853960. [18] H.-C. Huang and A. J. G. Cairns, On the control of defined-benefit pension plans, Insurance: Mathematics and Economics, 38 (2006), 113-131.  doi: 10.1016/j.insmatheco.2005.08.005. [19] R. Josa-Fombellida and J. Rincón-Zapatero, Minimization of risks in pension funding by means of contributions and portfolio selection, Insurance: Mathematics and Economics, 29 (2001), 35-45.  doi: 10.1016/S0167-6687(01)00070-1. [20] 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. [21] R. Josa-Fombellida and J. P. Rincón-Zapatero, Mean-variance portfolio and contribution selection in stochastic pension funding, European Journal of Operational Research, 187 (2008), 120-137.  doi: 10.1016/j.ejor.2007.03.002. [22] R. Josa-Fombellida and J. P. Rincón-Zapatero, Optimal asset allocation for aggregated defined benefit pension funds with stochastic interest rates, European Journal of Operational Research, 201 (2010), 211-221.  doi: 10.1016/j.ejor.2009.02.021. [23] E. Lee, An Introduction to Pension Schemes, Institute and Faculty of Actuaries, London, UK, 1986. [24] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [25] A. E. B. Lim, Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Mathematics of Operations Research, 29 (2004), 132-161.  doi: 10.1287/moor.1030.0065. [26] J. Marín-Solano and J. Navas, Consumption and portfolio rules for time-inconsistent investors, European Journal of Operational Research, 201 (2010), 860-872.  doi: 10.1016/j.ejor.2009.04.005. [27] D. Marshall and J. Reeve, Defined Benefit Pension Schemes: Funding for Ongoing Security, Presented to Staple Inn Actuarial Society, London, UK, 1993. [28] M.-A. Morlais, Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem, Finance and Stochastics, 13 (2009), 121-150.  doi: 10.1007/s00780-008-0079-3. [29] 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. [30] R. H. Strotz, Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, (1973), 128–143. doi: 10.1007/978-1-349-15492-0_10. [31] J. Q. Wei and T. X. Wang, Time-consistent mean-variance asset-liability management with random coefficients, Insurance: Mathematics and Economics, 77 (2017), 84-96.  doi: 10.1016/j.insmatheco.2017.08.011. [32] J. M. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations: Time-consistent solutions, Transactions of the American Mathematical Society, 369 (2017), 5467-5523.  doi: 10.1090/tran/6502. [33] Q. Zhao, Y. Shen and J. Q. Wei, Consumption-investment strategies with non-exponential discounting and logarithmic utility, European Journal of Operational Research, 238 (2014), 824-835.  doi: 10.1016/j.ejor.2014.04.034. [34] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.
The variance and total expected contribution versus $\rho$
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