Modelling and Optimization of Portfolio in a DC Scheme with Return of Contributions and Tax using Weibull Force Function

One of the major challenges faced by most pension fund managers in the defined pension (DC) scheme is how best member’s contributions can be invested to yield maximum returns. To achieve this, there is need to model and developed a robust investment plan which takes into consideration the volatility of the stock market price, tax on investment on risky assets and the mortality risk of its members. Based on this, the optimal portfolio distribution of a DC pension scheme with return of premium clause is studied where the mortality force function is characterized by the Weibull model and the investment in risky asset is subject to a certain proportion of tax. A portfolio with a risk-free asset and a risky asset modeled by the geometric Brownian motion such that the remaining accumulations are equally distributed between the remaining members is considered. Furthermore, the game theoretic approach is used to establish an optimization problem from the extended Hamilton Jacobi Bellman (HJB) equation which is a non-linear partial differential equation (PDE). Using variable separation method, closed form solutions of the optimal portfolio distribution and the efficient frontier are obtained. Lastly, some numerical simulations are used to study the impact of some the parameters on the optimal portfolio distribution with observations that the optimal portfolio distribution developed by the fund manager is inversely proportional to the tax imposed on the risky asset, risk averse coefficient, initial fund size, and risk free interest rate but directly proportional to time.