Mathematical Modelling of Stochastic Supplementary Contributions on DC’s Investment Plan
Saheed K. Olarewaju *
Department of Mathematics and Statistics, Federal University Otuoke, Bayelsa, Nigeria.
Yusuff Abdul Salam
Department of Electrical Electronics Engineering, African Institute of Science Administration and Commercial Studies, IAEC LoMe, Togo.
*Author to whom correspondence should be addressed.
Abstract
In this research work, we consider investments in a risk-free asset and a risky asset with the sole objective of increasing the accumulated wealth and returns from investment of the contributing members and assumed that the supplementary contributions are stochastic. Also, a stochastic differential equation which consist of the DC plan members’(DCPM) monthly contributions and the invested funds are obtained. Furthermore, an optimization problem is obtained in the form of Hamilton Jacobi Bellman (HJB) equation by maximizing the value function of the DCPM subject to her wealth. We determined the DCPM's investment strategy with stochastic voluntary contribution using the Legendre transformation and dual theory approach. The matching ideal fund size was also determined. With theoretical arguments, we examined how the stock market appreciation rate, risk-free interest rate, risk aversion coefficient, and voluntary contribution affected the MIP. We found that voluntary contributions help PFAs cut back on investments in riskier assets, which lowers the MIP on those assets. If the ideal fund size is chosen at the outset, the MIP also gradually drops.
Keywords: Geometric Brownian motion, stock market prices, Ito’s lemma, defined contribution pension scheme, supplementary contributions