# American Institute of Mathematical Sciences

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  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 & Management Optimization, doi: 10.3934/jimo.2020166
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##### References:
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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