# American Institute of Mathematical Sciences

January  2022, 18(1): 511-540. doi: 10.3934/jimo.2020166

## Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk

 1 School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China 2 Actuarial Studies, Department of Economics, University of Melbourne, Australia 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

* Corresponding author: Xun Li

Received  April 2020 Revised  September 2020 Published  January 2022 Early access  November 2020

Fund Project: This research is partially supported by the National Natural Science Foundation of China (Nos. 71871071, 72071051, 71471045), the Innovative Research Group Project of National Natural Science Foundation of China (No. 71721001), the Natural Science Foundation of Guangdong Province of China (Nos. 2018B030311004, 2017A030313399), and Research Grants Council of Hong Kong under grants 15213218 and 15215319

This paper investigates a multi-period asset allocation problem for a defined contribution (DC) pension fund facing stochastic inflation under the Markowitz mean-variance criterion. The stochastic inflation rate is described by a discrete-time version of the Ornstein-Uhlenbeck process. To the best of our knowledge, the literature along the line of dynamic portfolio selection under inflation is dominated by continuous-time models. This paper is the first work to investigate the problem in a discrete-time setting. Using the techniques of state variable transformation, matrix theory, and dynamic programming, we derive the analytical expressions for the efficient investment strategy and the efficient frontier. Moreover, our model's exceptional cases are discussed, indicating that our theoretical results are consistent with the existing literature. Finally, the results established are tested through empirical studies based on Australia's data, where there is a typical DC pension system. The impacts of inflation, investment horizon, estimation error, and superannuation guarantee rate on the efficient frontier are illustrated.

Citation: Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial and Management Optimization, 2022, 18 (1) : 511-540. doi: 10.3934/jimo.2020166
##### References:
 [1] A. Albert, Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM Journal on Applied Mathematics, 17 (1969), 434–440. doi: 10.1137/0117041. [2] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, The Review of Financial Studies, 23 (2010), 2970-3016. [3] D. Blake, D. Wright and Y. Zhang, Target-driven investing: Optimal investment strategies in defined contribution pension plans under loss aversion, Journal of Economic Dynamics and Control, 37 (2013), 195-209.  doi: 10.1016/j.jedc.2012.08.001. [4] D. Blake, D. Wright and Y. M. Zhang, Age-dependent investing: Optimal funding and investment strategies in defined contribution pension plans when members are rational life cycle financial planners, Journal of Economic Dynamics and Control, 38 (2014), 105-124.  doi: 10.1016/j.jedc.2013.11.001. [5] M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, The Journal of Finance, 57 (2002), 1201-1238. [6] A. Chen and L. Delong, Optimal investment for a defined-contribution pension scheme under a regime switching model, Astin Bulletin, 45 (2015), 397-419.  doi: 10.1017/asb.2014.33. [7] Z. Chen, Z. F. Li, Y. Zeng and J. Y. Sun, Asset allocation under loss aversion and minimum performance constraint in a DC pension plan with inflation risk, Insurance: Mathematics and Economics, 75 (2017), 137-150.  doi: 10.1016/j.insmatheco.2017.05.009. [8] X. Y. Cui, J. J. Gao, X. Li and D. Li, Optimal multi-period mean-variance policy under no-shorting constraint, European Journal of Operational Research, 234 (2014), 459-468.  doi: 10.1016/j.ejor.2013.02.040. [9] X. Y. Cui, X. Li and D. Li, Mean-variance policy for discrete-time cone constrained markets: The consistency in efficiency and minimum-variance signed supermartingale measure, Mathematical Finance, 27 (2017), 471-504.  doi: 10.1111/mafi.12093. [10] P. Devolder, M. Bosch Princep and I. Dominguez Fabian, Stochastic optimal control of annuity contracts, Insurance: Mathematics and Economics, 33 (2003), 227-238.  doi: 10.1016/S0167-6687(03)00136-7. [11] Y. Dong and H. Zheng, Optimal investment of DC pension plan under short-selling constraints and portfolio insurance, Insurance: Mathematics and Economics, 85 (2019), 47-59.  doi: 10.1016/j.insmatheco.2018.12.005. [12] Y. Dong and H. Zheng, Optimal investment with S-shaped utility and trading and Value at Risk constraints: An application to defined contribution pension plan, European Journal of Operational Research, 281 (2020), 341-356.  doi: 10.1016/j.ejor.2019.08.034. [13] P. Emms, Lifetime 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. [14] R. Gerrard, B. Hogaard and E. Vigna, Choosing the optimal annuitization time post retirement, Quantitative Finance, 12 (2012), 1143-1159.  doi: 10.1080/14697680903358248. [15] N. W. Han and M. W. Hung, Optimal asset allocation for DC pension plans under inflation, Insurance: Mathematics and Economics, 51 (2012), 172-181.  doi: 10.1016/j.insmatheco.2012.03.003. [16] N.-W. Han and M.-W. Hung, Optimal consumption, portfolio, and life insurance policies under interest rate and in ation risks, Insurance: Mathematics and Economics, 73 (2017), 54-67.  doi: 10.1016/j.insmatheco.2017.01.004. [17] L. He and Z. X. Liang, Optimal assets 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. [18] A. K. Konicz and J. M. Mulvey, Optimal savings management for individuals with defined contribution pension plans, European Journal of Operational Research, 243 (2015), 233-247.  doi: 10.1016/j.ejor.2014.11.016. [19] M. Kwak and B. H. Lim, Optimal portfolio selection with life insurance under inflation risk, Journal of Banking and Finance, 46 (2014), 59-71. [20] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [21] D. P. Li, X. M. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for a mean-variance insurer under stochastic interest rate model and inflation risk, Insurance: Mathematics and Economics, 64 (2015), 28-44.  doi: 10.1016/j.insmatheco.2015.05.003. [22] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504. [23] D. G. Luenberger, Optimization by Vector Space Methods, John Wiley & Sons, Inc., New York-London-Sydney, 1969 [24] Q.-P. Ma, On optimal pension management in a stochastic framework with exponential utility, Insurance: Mathematics and Economics, 49 (2011), 61-69.  doi: 10.1016/j.insmatheco.2011.02.003. [25] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [26] C. Munk, C. Sørensen and T. N. Vinther, Dynamic asset allocation under mean-reverting returns, stochastic interest rates, and inflation uncertainty: Are popular recommendations consistent with rational behavior?, International Review of Economics and Finance, 13 (2004), 141-166. [27] M. Simutin, Cash holding and mutual fund performance, Review of Finance, 18 (2014), 1425-1464.  doi: 10.1093/rof/rft035. [28] J. Y. Sun, Z. F. Li and Y. Zeng, Precommitment and equilibrium investment strategies for defined contribution pension plans under a jump-diffusion model, Insurance: Mathematics and Economics, 67 (2016), 158-172.  doi: 10.1016/j.insmatheco.2016.01.005. [29] M.-L. Tang, S.-N. Chen, G. C. Lai and T. P. Wu, Asset allocation for a DC pension fund under stochastic interest rates and inflation-protected guarantee, Insurance: Mathematics and Economics, 78 (2018), 87-104.  doi: 10.1016/j.insmatheco.2017.11.004. [30] 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. [31] H. X. Yao, Y. Z. Lai, Q. H. Ma and M. J. Jian, Asset allocation for a DC pension fund with stochastic income and mortality risk: A multi-period mean-variance framework, Insurance: Mathematics and Economics, 54 (2014), 84-92.  doi: 10.1016/j.insmatheco.2013.10.016. [32] H. X. Yao, Z. F. Li and D. Li, Multi-period mean-variance portfolio selection with stochastic interest rate and uncontrollable liability, European Journal of Operational Research, 252 (2016), 837-851.  doi: 10.1016/j.ejor.2016.01.049. [33] H. X. Yao and Z. Yang adn P. Chen, Markowitz's mean-variance defined contribution pension fund management under inflation: A continuous-time model, Insurance: Mathematics and Economics, 53 (2013), 851-863.  doi: 10.1016/j.insmatheco.2013.10.002. [34] A. Zhang and C.-O. Ewald, Optimal investment for a pension fund under inflation risk, Mathematical Methods of Operations Research, 71 (2010), 353-369.  doi: 10.1007/s00186-009-0294-5. [35] F. Z. Zhang, Matrix Theory: Basic Results and Techniques, Second edition, Universitext. Springer, New York, 2011. doi: 10.1007/978-1-4614-1099-7. [36] L. Zhang, H. Zhang and H. X. Yao, Optimal investment management for a defined contribution pension fund under imperfect information, Insurance: Mathematics and Economics, 79 (2018), 210-224.  doi: 10.1016/j.insmatheco.2018.01.007.

show all references

##### References:
 [1] A. Albert, Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM Journal on Applied Mathematics, 17 (1969), 434–440. doi: 10.1137/0117041. [2] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, The Review of Financial Studies, 23 (2010), 2970-3016. [3] D. Blake, D. Wright and Y. Zhang, Target-driven investing: Optimal investment strategies in defined contribution pension plans under loss aversion, Journal of Economic Dynamics and Control, 37 (2013), 195-209.  doi: 10.1016/j.jedc.2012.08.001. [4] D. Blake, D. Wright and Y. M. Zhang, Age-dependent investing: Optimal funding and investment strategies in defined contribution pension plans when members are rational life cycle financial planners, Journal of Economic Dynamics and Control, 38 (2014), 105-124.  doi: 10.1016/j.jedc.2013.11.001. [5] M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, The Journal of Finance, 57 (2002), 1201-1238. [6] A. Chen and L. Delong, Optimal investment for a defined-contribution pension scheme under a regime switching model, Astin Bulletin, 45 (2015), 397-419.  doi: 10.1017/asb.2014.33. [7] Z. Chen, Z. F. Li, Y. Zeng and J. Y. Sun, Asset allocation under loss aversion and minimum performance constraint in a DC pension plan with inflation risk, Insurance: Mathematics and Economics, 75 (2017), 137-150.  doi: 10.1016/j.insmatheco.2017.05.009. [8] X. Y. Cui, J. J. Gao, X. Li and D. Li, Optimal multi-period mean-variance policy under no-shorting constraint, European Journal of Operational Research, 234 (2014), 459-468.  doi: 10.1016/j.ejor.2013.02.040. [9] X. Y. Cui, X. Li and D. Li, Mean-variance policy for discrete-time cone constrained markets: The consistency in efficiency and minimum-variance signed supermartingale measure, Mathematical Finance, 27 (2017), 471-504.  doi: 10.1111/mafi.12093. [10] P. Devolder, M. Bosch Princep and I. Dominguez Fabian, Stochastic optimal control of annuity contracts, Insurance: Mathematics and Economics, 33 (2003), 227-238.  doi: 10.1016/S0167-6687(03)00136-7. [11] Y. Dong and H. Zheng, Optimal investment of DC pension plan under short-selling constraints and portfolio insurance, Insurance: Mathematics and Economics, 85 (2019), 47-59.  doi: 10.1016/j.insmatheco.2018.12.005. [12] Y. Dong and H. Zheng, Optimal investment with S-shaped utility and trading and Value at Risk constraints: An application to defined contribution pension plan, European Journal of Operational Research, 281 (2020), 341-356.  doi: 10.1016/j.ejor.2019.08.034. [13] P. Emms, Lifetime 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. [14] R. Gerrard, B. Hogaard and E. Vigna, Choosing the optimal annuitization time post retirement, Quantitative Finance, 12 (2012), 1143-1159.  doi: 10.1080/14697680903358248. [15] N. W. Han and M. W. Hung, Optimal asset allocation for DC pension plans under inflation, Insurance: Mathematics and Economics, 51 (2012), 172-181.  doi: 10.1016/j.insmatheco.2012.03.003. [16] N.-W. Han and M.-W. Hung, Optimal consumption, portfolio, and life insurance policies under interest rate and in ation risks, Insurance: Mathematics and Economics, 73 (2017), 54-67.  doi: 10.1016/j.insmatheco.2017.01.004. [17] L. He and Z. X. Liang, Optimal assets 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. [18] A. K. Konicz and J. M. Mulvey, Optimal savings management for individuals with defined contribution pension plans, European Journal of Operational Research, 243 (2015), 233-247.  doi: 10.1016/j.ejor.2014.11.016. [19] M. Kwak and B. H. Lim, Optimal portfolio selection with life insurance under inflation risk, Journal of Banking and Finance, 46 (2014), 59-71. [20] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [21] D. P. Li, X. M. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for a mean-variance insurer under stochastic interest rate model and inflation risk, Insurance: Mathematics and Economics, 64 (2015), 28-44.  doi: 10.1016/j.insmatheco.2015.05.003. [22] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504. [23] D. G. Luenberger, Optimization by Vector Space Methods, John Wiley & Sons, Inc., New York-London-Sydney, 1969 [24] Q.-P. Ma, On optimal pension management in a stochastic framework with exponential utility, Insurance: Mathematics and Economics, 49 (2011), 61-69.  doi: 10.1016/j.insmatheco.2011.02.003. [25] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [26] C. Munk, C. Sørensen and T. N. Vinther, Dynamic asset allocation under mean-reverting returns, stochastic interest rates, and inflation uncertainty: Are popular recommendations consistent with rational behavior?, International Review of Economics and Finance, 13 (2004), 141-166. [27] M. Simutin, Cash holding and mutual fund performance, Review of Finance, 18 (2014), 1425-1464.  doi: 10.1093/rof/rft035. [28] J. Y. Sun, Z. F. Li and Y. Zeng, Precommitment and equilibrium investment strategies for defined contribution pension plans under a jump-diffusion model, Insurance: Mathematics and Economics, 67 (2016), 158-172.  doi: 10.1016/j.insmatheco.2016.01.005. [29] M.-L. Tang, S.-N. Chen, G. C. Lai and T. P. Wu, Asset allocation for a DC pension fund under stochastic interest rates and inflation-protected guarantee, Insurance: Mathematics and Economics, 78 (2018), 87-104.  doi: 10.1016/j.insmatheco.2017.11.004. [30] 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. [31] H. X. Yao, Y. Z. Lai, Q. H. Ma and M. J. Jian, Asset allocation for a DC pension fund with stochastic income and mortality risk: A multi-period mean-variance framework, Insurance: Mathematics and Economics, 54 (2014), 84-92.  doi: 10.1016/j.insmatheco.2013.10.016. [32] H. X. Yao, Z. F. Li and D. Li, Multi-period mean-variance portfolio selection with stochastic interest rate and uncontrollable liability, European Journal of Operational Research, 252 (2016), 837-851.  doi: 10.1016/j.ejor.2016.01.049. [33] H. X. Yao and Z. Yang adn P. Chen, Markowitz's mean-variance defined contribution pension fund management under inflation: A continuous-time model, Insurance: Mathematics and Economics, 53 (2013), 851-863.  doi: 10.1016/j.insmatheco.2013.10.002. [34] A. Zhang and C.-O. Ewald, Optimal investment for a pension fund under inflation risk, Mathematical Methods of Operations Research, 71 (2010), 353-369.  doi: 10.1007/s00186-009-0294-5. [35] F. Z. Zhang, Matrix Theory: Basic Results and Techniques, Second edition, Universitext. Springer, New York, 2011. doi: 10.1007/978-1-4614-1099-7. [36] L. Zhang, H. Zhang and H. X. Yao, Optimal investment management for a defined contribution pension fund under imperfect information, Insurance: Mathematics and Economics, 79 (2018), 210-224.  doi: 10.1016/j.insmatheco.2018.01.007.
Efficient frontier with various time horizons
Inflation Vs non-inflation
Efficient frontier's sensitivity to the variation of $\phi$
Impact of SG rate on efficient frontier
Impact of the ratio $a$ on efficient frontier
Salary growth rate
 Date Total earnings ($fanxiexian_myfh$) Growth rate May-2012 1053.20 N/A Nov-2012 1081.30 1.0267 May-2013 1105.00 1.0219 Nov-2013 1114.20 1.0083 May-2014 1123.00 1.0079 Nov-2014 1128.70 1.0051 May-2015 1136.90 1.0073 Nov-2015 1145.70 1.0077 May-2016 1160.90 1.0133 Nov-2016 1163.50 1.0022 $q_k$(half yearly) N/A 1.0112 $q_k$(quarterly) N/A 1.0056
 Date Total earnings ($fanxiexian_myfh$) Growth rate May-2012 1053.20 N/A Nov-2012 1081.30 1.0267 May-2013 1105.00 1.0219 Nov-2013 1114.20 1.0083 May-2014 1123.00 1.0079 Nov-2014 1128.70 1.0051 May-2015 1136.90 1.0073 Nov-2015 1145.70 1.0077 May-2016 1160.90 1.0133 Nov-2016 1163.50 1.0022 $q_k$(half yearly) N/A 1.0112 $q_k$(quarterly) N/A 1.0056
Log-inflation rate
 $\hat{\bar{I}}$ $\hat{\sigma}$ $\hat{\phi}$ Confidence interval for $\hat{\phi}$ (95$\%$) 0.54$\%$ 0.45$\%$ 0.5709 (0.4731, 0.6687)
 $\hat{\bar{I}}$ $\hat{\sigma}$ $\hat{\phi}$ Confidence interval for $\hat{\phi}$ (95$\%$) 0.54$\%$ 0.45$\%$ 0.5709 (0.4731, 0.6687)
TN yield
 Year Weighted Average Issue Yield ($\%$) 2009 3.1537 2010 4.4971 2011 4.5861 2012 3.4670 2013 2.6450 2014 2.5127 2015 2.0541 2016 1.8134 2017 1.5807 $r_k^0$(annually) 2.9233 $r_k^0$(quarterly) 0.7229
 Year Weighted Average Issue Yield ($\%$) 2009 3.1537 2010 4.4971 2011 4.5861 2012 3.4670 2013 2.6450 2014 2.5127 2015 2.0541 2016 1.8134 2017 1.5807 $r_k^0$(annually) 2.9233 $r_k^0$(quarterly) 0.7229
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