Maximizing an Investment Portfolio for a DC Pension with a Return Clause and Proportional Administrative Charges under Weilbull Force Function


  • Njoku, K. N. C. Imo State University, Owerri, Imo State, Nigeria


Extended Hamilton Jacobi Bellman equation, Investment strategy, Return clause of premium, Administrative charges, Weibull mortality function


Communication in Physical Sciences, 2023, 10(1): 14-30

Author: Njoku, K. N. C.

Received 26 August 2023/Accepted 09 September 2023

In this paper, investment in a defined contributory (DC) pension fund system with a return clause of premium and proportional administrative charges is studied under geometric Brownian motion (GBM) and Weilbull mortality force function. To actualize this, an investment portfolio with a risk-free asset and a risky asset which follows the GBM model is considered such that the returned premium is with interest from an investment in risk-free asset and the Weilbull force function is used to determine the mortality rate of members during accumulation phase. Furthermore, the game-theoretic technique is applied to obtain an optimization problem from the extended Hamilton Jacobi Bellman equation. By using the mean-variance utility and variable separation technique, an investment strategy (IS) is obtained for the risky asset comprising of the risk-free interest rate, instantaneous volatility, administrative charges, the appreciation rate of the risky asset and the mortality force function was obtained together with the efficient frontier which gives the relationship between the investment expectation and the risk involvement in the investment. Furthermore, some numerical simulations were obtained to study the impact of some sensitive parameters of the IS. It was observed that the administrative charges and the mortality rate affect the IS to be adopted. Therefore, an insight into how these parameters behave is very essential in the development of an IS


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Author Biography

Njoku, K. N. C., Imo State University, Owerri, Imo State, Nigeria

Department of Mathematics


Li, D., Rong, X. and Zhao, H. (2013). Optimal investment problem with taxes, dividends and transaction costs under the constant elasticity of variance model, Transaction on Mathematics: 12, 243-255.

Witbooi, P. J., van Schalkwyk, G. J. and Muller, G. E. (2011). An optimal investment strategy in bank management. Mathematical Methods in the Applied Sciences, 34(13): 1606-1617.

Njoku, K. N. C., Bright, O. Osu, Edikan, E. Akpanibah and Rosemary, N. Ujumadu, (2017). Effect of Extra Contribution on Stochastic Optimal Investment Strategies for DC Pension with Stochastic Salary under the Affine Interest Rate Mode, Journal of Mathematical, 821-833.

Bright, O. Osu, Edikan, E. Akpanibah and Njoku, K. N. C., (2017). “On the Effect of Stochastic Extra Contribution on Optimal Investment Strategies with Stochastic Salaries under the Affine Interest Rate Model in a DC Pension Fund”, General Letters in Mathematics, Vol. 2, No. 3, Pp. 138-149.

Edikan, E. Akpanibah, Bright O. Osu, Njoku K. N. C., and Eyo, O. Akak, (2017). “Optimization of Wealth Investment Strategies for a DC Pension Fund with Stochastic Salary and Extra Contributions”, International Journal of Partial Differential Equations and Applications, Vol. 5, No. 1, Pp. 33-41.

Njoku, K. N. C. and Osu, B. O., (2019). “Effect of Inflation on Stochastic Optimal Investment strategies for DC Pension under the Affine Interest Rate Model”, Fundamental Journal of Mathematics and Application, pp. 91-100.

Osu, B. O., Njoku, K. N. C. and Basimanebotlhe, O. S., (2019). Fund Management Strategies for a Defined Contribution (DC) Pension Scheme under the Default Fund Phase IV, Commun. Math. Finance, 8:169-185.

Njoku, K. N. C. and Osu, B. O., (2019). On the Modified Optimal Investment Strategy for Annuity Contracts under the Constant elasticity of Variance (CEV) model. Earthline journal of Mathematical Sciences, 1:169:90.

Bright, O. Osu, Kevin, N. C. Njoku and Ben, I. Oruh, (2019). “On the Effect of Inflation and Impact of Hedging on the Pension Wealth Generation Strategies under the Geometric Brownian Motion model, “Earthline journal of Mathematical Sciences”, 1(2), pp., 119-142.

Njoku, K. N. C., Bright, O. Osu and Philip, U. Uzoma, (2019). “On the Investment Approach in a DC Pension Scheme for Default Fund Phase IV under the Constant Elasticity of Variance (CEV) model”, International Journal of Advances in Mathematics, vol. 2018, no. 3, pp. 65-76.

Osu, B. O., Njoku, K. N. C. and Oruh, B. I., (2020). ”On the Investment Strategies, Effect of Inflation and Impact of Hedging on Pension Wealth, during Accumulation and Distribution Phases”, Journal of Nigerian Society of Physical Sciences, 2, pp. 170-179.

Udeme, O. Ini, Ndipmong, A. Udoh, Njoku, K. N. C. and Edikan E. Akpanibah, (2021). ”Mathematical Modeling of an Insurer’s Portfolio and Reinsurance Strategy under the CEV Model and CRRA Utility “, Nigerian Journal of Mathematics and Applications, 31, 38-56.

Njoku, K. N. C. and Akpanibah, E. E., (2022). “Modeling and Optimization of Portfolio in a DC scheme with Return of Contributions and Tax using Weibull Force Function”, Asia Journal of Probability and Statistics, 16(3): 1-12.

Njoku, K. N. C., Chinwendu, Best Eleje, and Christian, Chukwuemeka Nwandu, (2022). “An Insurer’s Investment model with Reinsurance Strategy under the Modified Constant Elasticity of Variance Process”, World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences, Vol. 16, No. 12.

Akpanibah, E. E. and Osu, B. O. (2018). 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 Sciences. 2(2), 19 –29.

Osu, B. O., Okonkwo, C. U., Uzoma, P. U., and Akpanibah E. E. (2020). Wavelent analysis of the international markets: a look at the next eleven (N11), Scientific African 7, 1-16.

Wang, Y., Rong, X., and Zhao, H., (2018). Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math. 328, 414–431.

Akpanibah, E. E. and Ini U. O. (2021). An investor's investment plan with stochastic interest rate under the CEV model and the Ornstein-Uhlenbeck process, Journal of the Nigerian Society of Physical Sciences, 3 (3), 186-196.

Antolin, P. Payet S. and Yermo J. (2010). Accessing default investment strategies in DC pension plans. OECD Journal of Financial Market trend, 2010: 1-30.

Haberman, S., and Sung, J.H., (1994). Dynamics approaches to pension funding. Insurance: Mathematics and Economics 15, 151-162.

Josa{Fombellida, R., Rinc_on{Zapatero, J.P., 2001. Minimization of risks in pension funding by means of contribution and portfolio selection. Insurance: Mathematics and Economics 29, 35-45.

Josa{Fombellida, R., Rinc_on{Zapatero, J.P., 2004. Optimal risk management in defined benefit stochastic pension funds. Insurance: Mathematics and Economics 34, 489-503.

Wu, H and Zeng, Y., (2015). Equilibrium investment strategy for defined-contribution pension schemes with generalized mean–variance criterion and mortality risk. , Insurance: Mathematics and Economics. 64 : 396–408

Zhang, C. and Rong, X. (2013). Optimal investment strategies for DC pension with stochastic salary under affine interest rate model. Hindawi Publishing Corporation.

Akpanibah, E. E., Osu, B. O. Oruh, B. I. and Obi, C. N. (2019). Strategic optimal portfolio management for a DC pension scheme with return of premium clauses. Transaction of the Nigerian association of mathematical physics, 8(1): 121-130.

Sun, J., Li, Z., and Zeng, Y., (2016). Pre commitment and equilibrium investment strategies for defined contribution pension plans under a jump–diffusion model, Insurance Math. Econom. 67, 158–172.

Lai, C., Liu, S. and Wu, Y. (2021). Optimal portfolio selection for a defined contribution plan under two administrative fees and return of premium clauses. Journal of Computational and Applied Mathematics 398, 1-20.

Cairns, A. J. G., Blake, D. and Dowd, K., (2006). “Stochastic lifestyling: optimal dynamic asset allocation for defined contribution pension plans”, Journal of Economic Dynamics & Control 30, 843.

Chang, S.C., Tzeng, L.Y., Miao, J.C.Y., (2003). Pension funding incorporating downside risks. Insurance: Mathematics and Economics 32, 217-228.

Deelstra, G., Grasselli, M., Koehl, P.-F., (2003). Optimal investment strategies in the presence of a minimum guarantee. Insurance, Mathematics and Economics 33, 189-207.

Xu, J., Kannan, D., Zhang, B., (2007). Optimal dynamic control for the de_ned benefit pension plans with stochastic bene_t outgo. Stochastic Analysis and Applications 25, 201-236.

Akpanibah, E. E., Osu, B. O. and Ihedioha, S. A. (2020). 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. A ppl. 13(1),:53–64.

Delong, L., Gerrard, R., Haberman, S., (2008). Mean-variance optimization problems for an accumulation phase in a de_ned bene_t plan. Insurance: Mathematics and Economics 42, 107-118.

Le Cortois, O., and Menoncin, F., (2015). Portfolio optimisation with jumps: Illustration with a pension accumulation scheme. Journal of Banking and Finance 60, 127-137.

Gao, J. (2009). Optimal portfolios for DC pension plan under a CEV model. Insurance Mathematics and Economics 44(3): 479-490

Xiao, J., Hong, Z. and Qin, C. (2007). The Constant Elasticity of Variance (CEV) Model and the Legendre Transform-Dual Solution for Annuity Contracts. Insurance, 40, 302-310.

Gao, J. (2008). Stochastic optimal control of DC pension funds. Insurance, 42(3): 1159–1164.

Battocchio, P. and .Menoncin, F. (2004). Optimal pension management in a stochastic Framework. Insurance, 34(1): 79–95.

Boulier, J. F., Huang, S., and Taillard, G. (2001). Optimal management under stochastic interest rate: the case of a protected defined contribution pension fund. Insurance, 28(2): 173-189.

Sheng, D. and Rong, X. (2014). Optimal time consistent investment strategy for a DC pension with the return of premiums clauses and annuity contracts, Hindawi Publishing Corporation vol 2014 1-13.

He, L. and Liang, Z. (2013). The optimal investment strategy for the DC plan with the return of premiums clauses in a mean-variance framework, Insurance, 53, 643-649.

Li, D. Rong, X. Zhao, H. and Yi, B. (2017). Equilibrium investment strategy for DC pension plan with default risk and return of premiums clauses under CEV model, Insurance: 72, 6-20.

L. Chávez-Bedoya, (2016). Determining equivalent charges on flow and balance in individual account pension systems, J. Econ. Finance Adm. Sci. 21 (40), 2–7.

Bjork, T., Murgoci, A., (2010). A general theory of Markovian time inconsistent stochastic control problems, Working Paper, Stockholm School of Economics,

He, L. and Liang, Z. (2009). Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs. Insurance: Mathematics & Economics 44, 88–94.

Liang, Z. and Huang, J. (2011). Optimal dividend and investing control of an insurance company with higher solvency constraints. Insurance: Mathematics & Economics 49, 501–511.

Zeng, Y. and Li, Z. (2011). Optimal time consistent investment and reinsurance policies for mean-variance insurers. Insurance: Mathematics & Economics 49, 145–154.