# Optimization of investment strategies for a Defined Contribution (DC) plan member with Couple Risky Assets, Tax and Proportional Administrative Fee

## Keywords:

Ornstein-Uhlenbeck, process, optimal investment plan, proportional administrative fee, Legendre transforms, logarithm utility.## Abstract

**Communication in Physical Sciences, 2020, 7(1):31-39**

**Authors: Edikan E. Akpanibah and Udeme O. Ini**

**Received 11 February 2021/Accepted 23 March 2021**

**T**he aim of this paper is to study the optimal investment plans of a member in a defined contribution (DC) pension scheme with proportional administrative fee and tax on invested funds under logarithm utility function and Ornstein-Uhlenbeck (O-U) model. This is done by considering a portfolio consisting of a risk-free asset (bank security) and two risky assets (stocks) where the stock market prices are driven by the Ornstein-Uhlenbeck (O-U) process. An optimization problem known as the Hamilton Jacobi Bellman (HJB) equation is obtained by maximizing the expected utility of the member’s terminal wealth. Since the HJB equation is a nonlinear partial differential equation (PDE) and could be complex to solve, we use the Legendre transformation method and dual theory to reduce it to a linear PDE. By method of variable change and separation of variable, closed form solutions of the optimal investment plans are obtained using logarithm utility function. More so, sensitivity analysis of some parameters are carried out theoretically on the optimal investment plans with observations that apart from the changes experienced in the stock market prices caused by the O-U process, the optimal investment plans for the risky assets are inversely proportional to the contribution rate, tax rate imposed on the invested fund, proportional administration fee, investment time, but directly proportional to the appreciation rate of the risky assets

### Downloads

## References

Akpanibah, E. E. & Ogheneoro O. (2018). Optimal Portfolio Selection in a DC Pension with Multiple Contributors and the Impact of Stochastic Additional Voluntary Contribution on the Optimal Investment Strategy.international Journal of Mathematical and Computational Sciences, 12, 1, pp. 14-19.

Akpanibah, E. E., Osu, B. O., Njoku, K.N. C. & Akak E. O. (2017). Optimization of Wealth Investment Strategies for a DC Pension Fund with Stochastic Salary and Extra Contributions. International Journal of Partial Diff. Equations and Application, 5,1, pp. 33-41.

Akpanibah, E. E. & Ini, U. O. (2020). An Investor’s Investment Plan with Stochastic Interest Rate under the CEV Model and the Ornstein-Uhlenbeck Process. Preprint submitted to Journal of Non-linear Science and Application.

Battocchio, P. & Menoncin F. (2004). Optimal pension management in a stochastic framework. Insurance. 34, 1, pp. 79–95.

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

Cairns, A. J. G. Blake, D. & Dowd K. (2006). Stochastic lifestyling: optimal dynamic asset allocation for defined contribution pension plans, Journal of Economic Dynamics &Control 30, 5, (2006) pp. 843–877.

Deelstra, G. Grasselli, M. & Koehl P. F. (2003). Optimal investment strategies in the presence of a minimum guarantee. Insurance, 33, 1, pp. 189–207.

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

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

Gu, M. Yang, Y. Li, S. & Zhang, J. (2010). Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance: Mathematics and Economics. 46, 3, pp. 580-587.

Ihedioha, S.A. Danat, N. T. & Buba. A. (2020). Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models via Exponential Utility Maximization. Pure and Applied Mathematics Journal, 9, 3, pp. 55-63.

Ihedioha, S. A., Oruh, B. I. & Osu, B. O. (2017). Effects of Correlation of Brownian Motions on an Investor’s Optimal Investment and Consumption Decision under Ornstein Uhlenbeck Model. Academic Journal of Applied Mathematical Sciences. 3, 6, pp. 52-61.

Ini, U. O. & Akpanibah, E. E. (2020). Strategic Portfolio Management for a Pension Plan Member with Couple Risky Assets and Transaction Cost under the O-U Process. Preprint submitted to Matlab Journal.

Jose, Luis, Menaldi. (2006). Controlled Markov processes and viscosity solutions 25.

Jonsson, M. & Sircar, R. (2002). Optimal investment problems and volatility homogenization approximations. in Modern. Methods in Scientific Computing and Applications, 75, 2, pp 255–281

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

Li D., Rong, X. & Zhao H., (2013). Optimal investment problem with taxes, dividends and transaction costs under the constant elasticity of variance model. Transaction on Mathematics, 12, 3, pp. 243-255.

Njoku, K.N. C, Osu, B. O. Akpanibah, E. E. & Ujumadu, R. N. (2017). Effect of Extra Contribution on Stochastic Optimal Investment Strategies for DC Pension with Stochastic Salary under the Affine Interest Rate Model. Journal of Mathematical Finance, 7, pp. 821-833.

Osu, B. O., Akpanibah, E. E. & Oruh, B I. (2017). Optimal investment strategies for defined contribution (DC) pension fund with multiple contributors via Legendre transform and dual theory, International journal of pure and applied researches, 2, 2, pp. 97-105.

Sheng, D. L. & Rong, X. M. (2014). Optimal time consistent investment strategy for a DC pension with the return of premiums clauses and annuity contracts, Discrete Dynamics. Natural Science,, pp. 1-13

Xiao, X. & Yonggui, K, K. (2020). The optimal investment strategy of a DC pension plan under deposit loan spread and the O-U process. Preprint submitted to Elsevier.

Xiao, J., Hong, Z. & Qin, C. (2007). The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance, 40, 2, pp. 302–310.

Zhang, C. & Rong, X. (2013). Optimal investment strategies for DC pension with a stochastic salary under affine interest rate model. Hindawi Publishing Corporation, vol 2013 http://dx.doi.org/10.1155/2013/297875.

Zhao, H. & Rong, X. (2012). Portfolio selection problem with multiple risky assets under the constant elasticity of variance model, Mathematics and Economics, 50, pp. 179-190.

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2010 The Journal and the author

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.