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doi: 10.3934/jimo.2021138
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## Asset allocation for a DC pension plan with learning about stock return predictability

 a. Department of FinTech, School of Economics and Trade, Guangdong University of Finance, Guangzhou 510521, China b. Department of Finance, Southern University of Science and Technology, Shenzhen 518055, China c. School of Management, Guangzhou Xinhua College, Guangzhou 510520, China

* Corresponding author: Zhongfei Li

Received  February 2021 Revised  May 2021 Early access August 2021

Fund Project: This research was funded by the National Natural Science Foundation of China (Nos. 71721001, 71991474, 71971070, 71601055), the Humanity and Social Science Foundation of Ministry of Education of China (No. 20YJC790139), the Guangdong Basic and Applied Basic Research Foundation (Nos. 2020A1515010419, 2020A1515110606), and the Chinese Postdoctoral Science Foundation (No. 2018M641594)

This paper investigates an optimal investment problem for a defined contribution pension plan member who receives a stochastic salary, and considers inflation risk and stock return predictability. The member aims to maximize the expected power utility from her terminal real wealth by investing her pension account wealth in a financial market consisting of a risk-free asset, an inflation-indexed bond and a stock. The expected excess return on the stock can be predicted by both an observable predictor and an unobservable predictor, and the member has to estimate the unobservable predictor by learning the history information. By using the filtering techniques and dynamic programming approach, the closed-form optimal investment strategy and the corresponding value function are derived. Finally, with the help of numerical analysis, we explore the impact of model parameters on the optimal investment strategy, and analyze the welfare benefits from leaning and using inflation-indexed bond to hedge the stock return predictors.

Citation: Pei Wang, Ling Zhang, Zhongfei Li. Asset allocation for a DC pension plan with learning about stock return predictability. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021138
##### References:
 [1] J. Boudoukh, R. Michaely, M. Richardson and M. R. Roberts, On the importance of measuring payout yield: Implications for empirical asset pricing, The Journal of Finance, 62 (2007), 877-915.  doi: 10.1111/j.1540-6261.2007.01226.x.  Google Scholar [2] N. Branger, L. S. Larsen and C. Munk, Robust portfolio choice with ambiguity and learning about return predictability, Journal of Banking and Finance, 37 (2013), 1397-1411.  doi: 10.1016/j.jbankfin.2012.05.009.  Google Scholar [3] Z. Chen, Z. Li, Y. Zeng and J. 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 [4] J. B. Detemple, Asset pricing in a production economy with incomplete information, The Journal of Finance, 41 (1986), 383-391.  doi: 10.1111/j.1540-6261.1986.tb05043.x.  Google Scholar [5] M. U. Dothan and D. Feldman, Equilibrium interest rates and multiperiod bonds in a partially observable economy, The Journal of Finance, 41 (1986), 369-382.  doi: 10.1111/j.1540-6261.1986.tb05042.x.  Google Scholar [6] D. Duffie, Presidential address: Asset price dynamics with slow-moving capital, The Journal of Finance, 65 (2010), 1237-1267.  doi: 10.1111/j.1540-6261.2010.01569.x.  Google Scholar [7] E. F. Fama and K. R. French, Dividend yields and expected stock returns, Journal of Financial Economics, 22 (1988), 3-25.  doi: 10.1016/0304-405X(88)90020-7.  Google Scholar [8] G. Gennotte, Optimal portfolio choice under incomplete information, The Journal of Finance, 41 (1986), 733-746.  doi: 10.1111/j.1540-6261.1986.tb04538.x.  Google Scholar [9] 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 [10] T. S. Kim and E. Omberg, Dynamic nonmyopic portfolio behavior, Review of Financial Studies, 9 (1996), 141-161.  doi: 10.1093/rfs/9.1.141.  Google Scholar [11] R. Korn and H. Kraft, On the stability of continuous-time portfolio problems with stochastic opportunity set, Mathematical Finance, 14 (2004), 403-414.  doi: 10.1111/j.0960-1627.2004.00197.x.  Google Scholar [12] Y. W. Li, S. Y. Wang, Y. Zeng and H. Qiao, Equilibrium investment strategy for a DC plan with partial information and mean–variance criterion, IEEE Systems Journal, 11 (2017), 1492-1504.  doi: 10.1109/JSYST.2016.2533920.  Google Scholar [13] R. S. Liptser and A. N. Shiryayev, Statistics of Random Processes. Applications, vol. II., Springer-Verlag, 2001.  Google Scholar [14] J. Liu, Portfolio selection in stochastic environments, Review of Financial Studies, 20 (2007), 1-39.  doi: 10.1093/rfs/hhl001.  Google Scholar [15] J. H. van Binsbergen and R. S. J. Koijen, Predictive regressions: A present-value approach, The Journal of Finance, 65 (2010), 1439-1471.  doi: 10.1111/j.1540-6261.2010.01575.x.  Google Scholar [16] J. A. Wachter, Portfolio and consumption decisions under mean-reverting returns: An exact solution for complete markets, Journal of Financial and Quantitative Analysis, 37 (2002), 63-91.  doi: 10.2307/3594995.  Google Scholar [17] P. Wang, Z. Li and J. Sun, Robust portfolio choice for a DC pension plan with inflation risk and mean-reverting risk premium under ambiguity, Optimization, 70 (2021), 191-224.  doi: 10.1080/02331934.2019.1679812.  Google Scholar [18] H. Wu, L. Zhang and H. Chen, Nash equilibrium strategies for a defined contribution pension management, Insurance: Mathematics and Economics, 62 (2015), 202-214.  doi: 10.1016/j.insmatheco.2015.03.014.  Google Scholar [19] Y. Xia, Learning about predictability: The effects of parameter uncertainty on dynamic asset allocation, The Journal of Finance, 56 (2001), 205-246.  doi: 10.1111/0022-1082.00323.  Google Scholar [20] H. Yao, P. Chen, M. Zhang and X. Li, Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk, Journal of Industrial and Management Optimization. doi: 10.3934/jimo.2020166.  Google Scholar [21] 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 [22] L. Zhang, Z. Li, Y. Xu and Y. Li, Multi-period mean variance portfolio selection under incomplete information, Applied Stochastic Models in Business and Industry, 32 (2016), 753-774.  doi: 10.1002/asmb.2191.  Google Scholar [23] L. Zhang, H. Zhang and H. 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 [24] Q. Zhao, R. Wang and J. 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.  Google Scholar

show all references

##### References:
 [1] J. Boudoukh, R. Michaely, M. Richardson and M. R. Roberts, On the importance of measuring payout yield: Implications for empirical asset pricing, The Journal of Finance, 62 (2007), 877-915.  doi: 10.1111/j.1540-6261.2007.01226.x.  Google Scholar [2] N. Branger, L. S. Larsen and C. Munk, Robust portfolio choice with ambiguity and learning about return predictability, Journal of Banking and Finance, 37 (2013), 1397-1411.  doi: 10.1016/j.jbankfin.2012.05.009.  Google Scholar [3] Z. Chen, Z. Li, Y. Zeng and J. 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 [4] J. B. Detemple, Asset pricing in a production economy with incomplete information, The Journal of Finance, 41 (1986), 383-391.  doi: 10.1111/j.1540-6261.1986.tb05043.x.  Google Scholar [5] M. U. Dothan and D. Feldman, Equilibrium interest rates and multiperiod bonds in a partially observable economy, The Journal of Finance, 41 (1986), 369-382.  doi: 10.1111/j.1540-6261.1986.tb05042.x.  Google Scholar [6] D. Duffie, Presidential address: Asset price dynamics with slow-moving capital, The Journal of Finance, 65 (2010), 1237-1267.  doi: 10.1111/j.1540-6261.2010.01569.x.  Google Scholar [7] E. F. Fama and K. R. French, Dividend yields and expected stock returns, Journal of Financial Economics, 22 (1988), 3-25.  doi: 10.1016/0304-405X(88)90020-7.  Google Scholar [8] G. Gennotte, Optimal portfolio choice under incomplete information, The Journal of Finance, 41 (1986), 733-746.  doi: 10.1111/j.1540-6261.1986.tb04538.x.  Google Scholar [9] 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 [10] T. S. Kim and E. Omberg, Dynamic nonmyopic portfolio behavior, Review of Financial Studies, 9 (1996), 141-161.  doi: 10.1093/rfs/9.1.141.  Google Scholar [11] R. Korn and H. Kraft, On the stability of continuous-time portfolio problems with stochastic opportunity set, Mathematical Finance, 14 (2004), 403-414.  doi: 10.1111/j.0960-1627.2004.00197.x.  Google Scholar [12] Y. W. Li, S. Y. Wang, Y. Zeng and H. Qiao, Equilibrium investment strategy for a DC plan with partial information and mean–variance criterion, IEEE Systems Journal, 11 (2017), 1492-1504.  doi: 10.1109/JSYST.2016.2533920.  Google Scholar [13] R. S. Liptser and A. N. Shiryayev, Statistics of Random Processes. Applications, vol. II., Springer-Verlag, 2001.  Google Scholar [14] J. Liu, Portfolio selection in stochastic environments, Review of Financial Studies, 20 (2007), 1-39.  doi: 10.1093/rfs/hhl001.  Google Scholar [15] J. H. van Binsbergen and R. S. J. Koijen, Predictive regressions: A present-value approach, The Journal of Finance, 65 (2010), 1439-1471.  doi: 10.1111/j.1540-6261.2010.01575.x.  Google Scholar [16] J. A. Wachter, Portfolio and consumption decisions under mean-reverting returns: An exact solution for complete markets, Journal of Financial and Quantitative Analysis, 37 (2002), 63-91.  doi: 10.2307/3594995.  Google Scholar [17] P. Wang, Z. Li and J. Sun, Robust portfolio choice for a DC pension plan with inflation risk and mean-reverting risk premium under ambiguity, Optimization, 70 (2021), 191-224.  doi: 10.1080/02331934.2019.1679812.  Google Scholar [18] H. Wu, L. Zhang and H. Chen, Nash equilibrium strategies for a defined contribution pension management, Insurance: Mathematics and Economics, 62 (2015), 202-214.  doi: 10.1016/j.insmatheco.2015.03.014.  Google Scholar [19] Y. Xia, Learning about predictability: The effects of parameter uncertainty on dynamic asset allocation, The Journal of Finance, 56 (2001), 205-246.  doi: 10.1111/0022-1082.00323.  Google Scholar [20] H. Yao, P. Chen, M. Zhang and X. Li, Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk, Journal of Industrial and Management Optimization. doi: 10.3934/jimo.2020166.  Google Scholar [21] 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 [22] L. Zhang, Z. Li, Y. Xu and Y. Li, Multi-period mean variance portfolio selection under incomplete information, Applied Stochastic Models in Business and Industry, 32 (2016), 753-774.  doi: 10.1002/asmb.2191.  Google Scholar [23] L. Zhang, H. Zhang and H. 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 [24] Q. Zhao, R. Wang and J. 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.  Google Scholar
Effect of the investment horizon $T$ on the optimal investment strategy
Effect of $\kappa_{x}$ and $\sigma_x$ on the optimal investment strategy
Effect of $\kappa_{y}$ and $\sigma_y$ on the optimal investment strategy
Effect of $\mu_{I}$ and $\sigma_I$ on the optimal investment strategy
Effect of $\mu_{L}$ and $\sigma_L$ on the optimal investment strategy
Effect of $\rho_{yI}$ and $\rho_{xI}$ on the inflation-indexed bond hedge against the stock return predictors
Sample paths for $W^{*}(t)$, $x(t)$, $y(t)$ and $L(t)$ and the optimal investment strategy
Effect of risk aversion parameter $\gamma$ on the optimal investment strategy
Effect of $T$, $\rho_{xy}$ and $\rho_{SI}$ on the utility losses
Effect of $\rho_{yI}$ and $\rho_{xI}$ on the utility losses
Effect of $b_x$ and $b_y$ on the utility losses
Value of parameters
 $\mu_I$ $\sigma_I$ $\sigma_S$ $R$ $b_x$ $b_y$ $\kappa_x$ $\kappa_y$ 0.03 0.1 0.2 0.050 0.3263 2.6807 0.2935 4.0942 $\mu_x$ $\mu_y$ $\sigma_x$ $\sigma_y$ $\rho_{Sx}$ $\rho_{Sy}$ $\rho_{xy}$ $r$ -2.1493 0.0000 0.1467 0.2620 -0.2186 -0.2164 -0.4913 0.0366 $\mu_L$ $\sigma_L$ $\rho_{SI}$ $\rho_{xI}, \rho_{yI}$ 0.04 0.08 0.0000 0.3000
 $\mu_I$ $\sigma_I$ $\sigma_S$ $R$ $b_x$ $b_y$ $\kappa_x$ $\kappa_y$ 0.03 0.1 0.2 0.050 0.3263 2.6807 0.2935 4.0942 $\mu_x$ $\mu_y$ $\sigma_x$ $\sigma_y$ $\rho_{Sx}$ $\rho_{Sy}$ $\rho_{xy}$ $r$ -2.1493 0.0000 0.1467 0.2620 -0.2186 -0.2164 -0.4913 0.0366 $\mu_L$ $\sigma_L$ $\rho_{SI}$ $\rho_{xI}, \rho_{yI}$ 0.04 0.08 0.0000 0.3000
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