Utility maximization and portfolio management with administrative fee and dividend

Abstract

The study of utility maximization cannot be over emphasized in the study of portfolio management
as it plays crucial role in the determination of investor’s strategy over any investment. In this
research work, we investigate some utility functions and their applications to portfolio optimization
and risk management. Also, a portfolio consisting of one risk free asset and a risky asset which
follows the geometric Brownian motion was considered. We took into consideration transaction
cost, dividend and tax on invested funds. By applying Ito’s lemma and maximum principle, we
obtained an optimization problem which comprise of a nonlinear PDE (Hamilton Jacobi Bellman
equation) and a control problem (optimal investment strategy). These problems are functions of the value function which is dependent on the utility of the investor at the expiration of such investment. Furthermore, the exponential and logarithm utility were used in solving for the optimal investment strategies and some numerical results also presented. It was observed for both cases that the optimal investment strategies (OIS) were directly proportional to appreciation rate, dividend and inversely proportional to the administrative charges, instantaneous volatility, tax and risk averse coefficient. Also, the OIS under logarithmic utility does not depend on the risk averse coefficient and tax while the optimal investment strategy under exponential utility does not depend on wealth.