AN INVESTOR’S OPTIMAL PLAN IN A DC SCHEME WITH REFUND OF CONTRIBUTIONS FOR MORTGAGE HOUSING SCHEME
Abstract
In this paper, we investigate an investor’s portfolios in a defined contributory (DC) Pension Scheme with return of contributions for a mortgage housing scheme and managerial fees for time-inconsistent utility. A portfolio with a fixed deposit (risk-free asset) and two stocks (risky assets) is taken into consideration, where the stock market prices of the risky assets follow the geometric Brownian motion (GBM) and the instantaneous volatilities form a positive definite matrix. To determine the number of scheme members (SM) interested in the mortgage housing, the Abraham De Moivre function is used. Furthermore, the dynamic programming and game techniques were used to obtain our optimization problem by maximizing the expected utility (mean-variance utility) subject to the SM’s wealth. Using the variable change technique, the optimal value function (OVF), investor’s optimal plan (IOP), and the efficient frontier were obtained under mean variance utility function. . Furthermore, some numerical results of some sensitive parameters such as risk-free interest rate (RIR), risk averse coefficient (RAC), entry age (EA) of SM, managerial charges (MC), optimal fund size (OFS), instantaneous volatilities (IV) and appreciation rates (AR) of the risky assets were presented to explain their impact on the IOP. It was observed that the IOP, which is the fraction of SM’s accumulations invested in the risky assets, is a decreasing function of RIR, RAC, IV, EA, OFS, and MC but an increasing function of the AR.
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References
P. Antolin, S. Payet, and J. Yermo, "ACCESSING DEFAULT INVESTMENT STRATEGIES IN DC PENSION PLANS," OECD Journal of Financial Market trend, vol. 2010, pp. 1-30, 2010.
R. Abade, "PENSION REFORMS ACT 2004: WHAT’S IN IT FOR YOU?," 2004.
P. Wang, Y, Shen, L. Zhang, and Y, Kang, "EQUILIBRIUM INVESTMENT STRATEGY FOR A DC PENSION PLAN WITH LEARNING ABOUT STOCK RETURN PREDICTABILITY," Insurance Math. Econom, vol. 100, pp. 384-407, 2021. doi: http://doi.org/10.1016/j.inmatheco.2021.07.001.
J. Alonso-Garcia and P. Devolder , "OPTIMAL MIX BETWEEN PAY-AS –YOU-GO AND FUNDING FOR DC PENSION SCHEMES IN AN OVERLAPPING GENERATIONS MODEL," Insurance Mathematics and Economics, vol. 70, pp. 224-236, 2016. doi: http://doi.org/10.1016/j.inmatheco.2016.06.011.
Z. Chen, Z, Li, L. Zhang, Y, Zeng and J. Sun, "ASSET ALLOCATION UNDER LOSS AVERSION AND MINIMUM PERFORMANCE CONSTRAINT IN DC PENSION PLAN WITH INFLATION RISK," Insurance Math. Econom, pp. 137-150, doi: http://doi.org/10.1016/j.inmatheco.2017.05.009 2017.
K, T, Ng and W. F. Chong, "OPTIMAL INVESTMENT IN DC PENSION SCHEMES WITH FORWARD UTILITIES PREFERENCES," Insurance Mathematics and Economics, vol. 114, pp. 192-211, doi: http://doi.org/10.1016/j.inmatheco.2023.12.001 2024.
B. O. Osu, E. E. Akpanibah and K. N. C. Njoku, "ON THE EFFECT OF STOCHASTIC EXTRA CONTRIBUTION ON OPTIMAL INVESTMENT STRATEGIES FOR STOCHASTIC SALARY UNDER THE AFFINE INTEREST RATE MODEL IN A DC PENSION FUND," General Letters in Mathematics, vol. 2, no. 3, pp. 138-149, doi: http://doi.org/10.31559/glm2016.2.3.15 2017.
X. Peng, and L. Luo, "OPTIMAL INVESTMENT STATEGY WITH MEAN WEIGHTED VARIANCE-CVAR CRITERION UNDER PARTIAL INFORMATION," Insurance: Mathematics and Economics, vol. 120, pp. 302-324, http://doi.org/10.1016/j.inmatheco.2024.12.006 2025.
G. E. Muller and P. J. Witbooi, "AN OPTIMAL PORTFOLIO AND CAPITAL MANAGEMENT STRATEGY FOR BASEL III COMPLIANT COMMERCIAL BANKS," Journal of Applied Mathematics, vol. 2014, pp. 1-11, doi: https://doi.org/10.1155/2014/723873 2014..
G. E. Muller, "AN OPTIMAL INVESTMENT STRATEGY AND MULTIPERIOD DEPOSIT INSURANCE PRICING MODEL FOR COMMERCIAL BANKS," Journal of Applied Mathematics, vol. 2018, pp. 1-10, doi: https://doi.org/10.1155/2018/8942050 2018.
L. He, and Z. Liang, "THE OPTIMAL INVESTMENT STRATEGY FOR THE DC PLAN WITH THE RETURN OF PREMIUMS CLAUSES IN A MEAN-VARIANCE FRAMEWORK," Insurance, vol. 53, pp. 643-649, http://doi.org/10.1016/j.inmatheco.2013.09.002. 2013.
D. Li, X. Rong, H. Zhao, and B. Yi, "EQUILIBRIUM INVESTMENT STRATEGY FOR DC PENSION PLAN WITH DEFAULT RISK AND RETURN OF PREMIUMS CLAUSES UNDER CEV MODEL," Insurance, vol. 72, pp. 6-20, doi: http://doi.org/10.1016/j.inmatheco.2016.10.007. 2017.
B. O. Osu, K. N. C. Njoku and B. I. Oruh, "ON THE INVESTMENT STRATEGY, EFFECT OF INFLATION AND IMPACT OF HEDGING ON PENSION WEALTH DURING ACCUMULATION AND DISTRIBUTION PHASE," Journal of the Nigerian Society of Physical Sciences, vol. 2, no. 3, pp. 170-179, doi: http://doi.org/10.46481/jnsps.2020.62 2020
E. E. Akpanibah and U. O. Ini, "PORTFOLIO STRATEGY FOR AN INVESTOR WITH LOGARITHM UTILITY AND STOCHASTIC INTEREST RATE UNDER CONSTANT ELASTICITY OF VARIANCE MODEL," Journal of the Nigerian Society of Physical Sciences, vol. 2, no. 3, pp. 186-196, doi: http://doi.org/10.46481/jnsps.2020.86 2020.
E. E. Akpanibah, B. O. Osu, B. I. Oruh, and C. N. Obi, "STRATEGIC OPTIMAL PORTFOLIO MANAGEMENT FOR A DC PENSION SCHEME WITH RETURN OF PREMIUM CLAUSES," Transaction of the Nigerian association of mathematical physics, vol. 8, no. 1, pp. 121-130, 2019.
M. Barr, N. T. Kashkari, A. Lehnert, and P. Swagel, CRISI-ERA HOUSING PROGRAMS, New Heaven CT, Yale University Press., 2020, pp. 320-358.
L. He and Z. Liang, "OPTIMAL FINANCING AND DIVIDEND CONTROL OF THE INSURANCE COMPANY WITH FIXED AND PROPORTIONAL TRANSACTION COSTS,” Insurance: Mathematics & Economics, vol. 44, pp. 88-94, 2009. doi: http://doi.org/10.1016/j.insmatheco.2008.10.001
D. Sheng and X. Rong, "OPTIMAL TIME CONSISTENT INVESTMENT STRATEGY FOR A DC PENSION WITH THE RETURN OF PREMIUMS CLAUSES AND ANNUITY CONTRACTS," vol. 2014, pp. 1-13, 2014. doi: http://dx.doi.org/10.1155/2014/862694
J. Pan, and X. Y. Zhuo, "OPTIMAL INVESTMENT STRATEGIES FOR DC PENSION PLAN WITH ADMINISTRATIVE FEES AND RETURN OF PREMIUM CLAUSES AND UNDER HESTON MODEL," Journal of Operation Research Society of China, vol. 7, no. 2, pp. 1-12, 2023. doi: http://doi.org/10.1007/s40305023005050
H. Chang, and J. Li, “ROBUST EQUILIBRIUM STRATEGY FOR DC PENSION PLAN WITH THE RETURN OF PREMIUMS CLAUSES IN A JUMP DIFFUSION MODEL, “ Optimization, vol. 72, no 2, pp. 463-492, 2021. doi: http://doi.org/10.1080/02331934.2021.1970754
Z. Hao, H. Chang and M. Kou, "PRE-COMMITMENT AND EQUILIBRIUM INVEST STRATEGIES OF DC PENSION PLANS WITH RETURN OF PREMIUM CLAUSES UNDER INFLATION," Communication in Statistics-Theory and Methods, vol. 54, no. 6, pp. 1569-1595, 2024. doi: http://doi.org/10.1080/03610926.2024.2347334
M. Yan, Z. Cao, T. Wang and S. Zhang, "ROBUST OPTIMAL INVESTMENT STRATEGY OF DC PENSION PLANS WITH STOCHASTIC SALARY AND A RETURN OF PREMIUM CLAUSE,” vol. 51, no 22, pp. 7980-8011," Communication in Statistics-Theory and Methods, vol. 51, no. 2, pp. 7980-8011, 2021. doi: http://doi.org/10.1080/03610926.2021.1887236
C. Lai, S. Liu, and Y. Wu, “OPTIMAL PORTFOLIO SELECTION FOR A DEFINED CONTRIBUTION PLAN UNDER TWO ADMINISTRATIVE FEES AND RETURN OF PREMIUM CLAUSES,”Journal of Computational and Applied Mathematicsvol. 398, pp. 1-20, 2021. http://doi.org/10.1016/j.cam.2021.113694
L. Bian, Z. Li, and H. Yao, "‘ROBUST TIME-CONSISTENT STRATEGIES OF DC PENSION PLANS WITH REGIME SWITCHING AND A RETURN OF PREMIUM CLAUSE," Insurance: Mathematics & Economics, vol. 81, no. 6, pp. 78-94, 2018 . doi: http://doi.org/10.1016/j.inmatheco.2018.05.005.
E. E. Akpanibah, and S. K. Samaila, "OPTIMAL INVESTMENT POLICY IN A PENSION FUND SYSTEM WITH RETURN CLAUSE AND MULTIPLE ASSETS UNDER VOLATILITY RISKS," General Letters in Mathematics, vol. 7, no. 2, pp. 1-12, 2019. doi: http://doi.org/10.31559/glm2019.7.1.1
X. Tang, W. Liu, W. Wu, and Y. Hu, "AN OPTIMAL INVESTMENT STRATEGY FOR DC PENSION PLANS WITH COST AND THE RETURN OF PREMIUM CLAUSES AND UNDER CEV MODEL," Aims Mathematics, vol. 9, no. 8, pp. 21731-21754, 2024. doi: http://doi.org/10.3934/math.20241057
X. J. Li, H. Chang, and X. J. Chen, "TIME-CONSISTENT STRATEGIES FOR DC PENSION PLAN WITH RETURN OF PREMIUM CLAUSES IN STOCHASTICS ENVIRONMENT," Journal of Operation Research Society of China, vol. 7, no. 2, pp. 1-12,2024. doi: http://doi.org/10.1007/s4030502400554
P. A., Azor, A.Igodo, and E. M. Ase, "EXPLICIT SOLUTION OF AN INVESTMENT PLAN FOR A DC PENSION SCHEME WITH VOLUNTARY CONTRIBUTIONS AND RETURN CLAUSE UNDER LOGARITHM UTILITY," International Journal of physical and mathematical sciences, vol. 17, no. 8, pp. 115-121, 2023.
A. M. Esebai, E. E. Akpanibah, and S. K. Samaila, "ON INVESTMENT MODEL FOR A CARA PENSION SCHEME WITH RETURN OF CONTRIBUTIONS CLAUSE FOR MORTGAGE HOUSING SCHEME," Communication in Physical Sciences, vol. 11, no. 3, pp. 548-558, 2024.
E. E. Akpanibah and B. O. Osu, "OPTIMAL PORTFOLIO SELECTION FOR A DEFINED CONTRIBUTION PENSION FUND WITH RETURN CLAUSES OF PREMIUM WITH PREDETERMINED INTEREST RATE UNDER MEAN VARIANCE UTILITY," Asian Journal of Mathematical Science, vol. 2, no. 2, pp. 19-29, 2018.
E. E. Akpanibah, B. O. Osu and S. A. Ihedioha, "ON THE OPTIMAL ASSET ALLOCATION STRATEGY FOR A DEFINED CONTRIBUTION PENSION SYSTEM WITH REFUND CLAUSE OF PREMIUM WITH PREDETERMINED INTEREST UNDER HESTON’S VOLATILITY MODEL," J. Nonlinear Sci. Appl, vol. 13, no. 1, pp. 53-64, 2020. doi: http://doi.org/10.22436/jnsa.013.01.05
H. Zhao and X. Rong, "PORTFOLIO SELECTION PROBLEM WITH MULTIPLE RISKY ASSETS UNDER THE CONSTANT ELASTICITY OF VARIANCE MODEL," Insurance: Mathematics and Economics, vol. 50, pp. 179-190, 2012. doi: http://doi.org/10.1016/j.inmatheco.2011.10.013
E. E. Akpanibah, P. Benneth, and A. M. Esebai, "A MATHEMATICAL APPROACH TO INVESTMENT WITH CHARGE ON BALANCE AND VOLUNTARY CONTRIBUTIONS UNDER WEIBULL MORTALITY FORCE FUNCTION," Barekeng: Journal of Mathematics and its Application, vol. 19, no. 1, pp. 427-440, 2025. doi: http://doi.org/10.30598/barekengvol19iss1pp0427-0440
P. H. Kohler and I. Kohler, "FRAILTY MODELING FOR ADULT AND OLD AGE MORTALITY: THE APPLICATION OF A MODIFIED DE MOIVRE HAZARD FUNCTION TO SEX DIFFERENTIALS IN MORTALITY," Demographic Research, vol. 3, no. 8, 2000. doi: http://www.demographic-research.org/Volumes/Vol3/8/
T. Björk, A. Murgoci, and Z. Xunyu, “MEAN-VARIANCE PORTFOLIO OPTIMIZATION WITH STATE DEPENDENT RISK AVERSION” Mathematical Finance, vol 24 no. 1, 1–24, 2012. doi: http://doi.org/10.1111/j.1467-9965.2011.00515.x
Z. Liang, and J. Huang, "OPTIMAL DIVIDEND AND INVESTING CONTROL OF AN INSURANCE COMPANY WITH HIGHER SOLVENCY CONSTRAINTS, ‘INSURANCE: MATHEMATICS & ECONOMICS," Insurance: Mathematics & Economics, vol. 49, pp. 501-511, 2011. doi: http://doi.org/10.1016/j.inmatheco.2011.08.008
Y. Zeng and Z. Li, "OPTIMAL TIME CONSISTENT INVESTMENT AND REINSURANCE POLICIES FOR MEAN-VARIANCE INSURERS," Insurance: Mathematics & Economic, vol. 49, pp. 145-154, 2011. doi: http://doi.org/10.1016/j.inmatheco.2018.01.001
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