A MATHEMATICAL APPROACH TO INVESTMENT WITH CHARGE ON BALANCE AND VOLUNTARY CONTRIBUTIONS UNDER WEIBULL MORTALITY FORCE FUNCTION

Abstract

One of the many challenges encountered by most pension fund administrators (PFAs) in the Defined Contribution (DC) pension plan is the determination of a sustainable and suitable investment plan for their members under mortality risk. To achieve this, there is need to develop an optimal portfolio which considers the volatility of the stock market price consisting of one risk-free asset and a risky asset which follows the Heston volatility model (HVM). Also, the portfolio considers additional voluntary contributions (AVC) by members, tax on the stock market price, charge on balance (CB), and the mortality risk of the pension scheme members (PSM) modeled by the Weibull mortality force function. Furthermore, an optimization problem is established from the extended Hamilton Jacobi Bellman (EHJB) equation using variational method. By applying the variable separation technique and mean variance utility, the optimal control strategy (OCS) and the efficient frontier are obtained. Finally, some numerical simulations are presented to study the behavior of the OCS with respect to some sensitive parameters. It was discovered that the composition of the OCS depends on the instantaneous volatility, tax on investment, AVC, risk aversion coefficient (RAC), CB and the correlation coefficient. Hence, the understanding of the behaviour these parameters are very crucial in the determination of OCS.