American Institute of Mathematical Sciences

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doi: 10.3934/jimo.2020166

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

Haixiang Yao 1, , Ping Chen 2, , Miao Zhang 2, and  Xun Li 3,,

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.

Keywords: Stochastic inflation rate, multi-period mean-variance formulation, portfolio selection, defined contribution pension fund, efficient frontier.
Mathematics Subject Classification: Primary: 90C26; Secondary: 91B28, 49N15.
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
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.  Google Scholar

[2]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, The Review of Financial Studies, 23 (2010), 2970-3016.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, The Journal of Finance, 57 (2002), 1201-1238.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. G. Luenberger, Optimization by Vector Space Methods, John Wiley & Sons, Inc., New York-London-Sydney, 1969  Google Scholar

[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.  Google Scholar

[25]

H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.   Google Scholar

[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.   Google Scholar

[27]

M. Simutin, Cash holding and mutual fund performance, Review of Finance, 18 (2014), 1425-1464.  doi: 10.1093/rof/rft035.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, The Review of Financial Studies, 23 (2010), 2970-3016.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, The Journal of Finance, 57 (2002), 1201-1238.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. G. Luenberger, Optimization by Vector Space Methods, John Wiley & Sons, Inc., New York-London-Sydney, 1969  Google Scholar

[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.  Google Scholar

[25]

H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.   Google Scholar

[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.   Google Scholar

[27]

M. Simutin, Cash holding and mutual fund performance, Review of Finance, 18 (2014), 1425-1464.  doi: 10.1093/rof/rft035.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Efficient frontier with various time horizons
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Figure 2.  Inflation Vs non-inflation
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Figure 3.  Efficient frontier's sensitivity to the variation of $ \phi $
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Figure 4.  Impact of SG rate on efficient frontier
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Figure 5.  Impact of the ratio $ a $ on efficient frontier
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Table 1.  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
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Download as excel

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
Table 2.  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)
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$ \hat{\bar{I}} $ $ \hat{\sigma} $ $ \hat{\phi} $ Confidence interval for $ \hat{\phi} $ (95$ \% $)
0.54$ \% $ 0.45$ \% $ 0.5709 (0.4731, 0.6687)
Table 3.  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 $(quarterly) 0.7229
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Year Weighted Average Issue Yield ($ \% $)
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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