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

## Keywords:

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

**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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