A Numerical Solution of Optimal Portfolio Selection Problem with General Utility Functions
In this paper, we adopt a monotone numerical scheme to solve the Hamilton-Jacobi-Bellman equation arising from the optimal portfolio selection problem with general utility functions. To explore the relationship between the relative risk aversion and optimal investment strategy, three different examples are presented with a constant relative risk aversion (CRRA) utility, an increasing relative risk aversion (IRRA) utility and a decreasing relative risk aversion (DRRA) utility, respectively. Our numerical results suggest that non-CRRA investors should adjust the portfolio according to both wealth and time positions. More specifically, both IRRA and DRRA investors should reduce their allocation to the risky asset as time approaches the investment horizon, which is consistent with the life-cycle investment advice. On the other hand, as the wealth increases, IRRA investors should decrease their allocation to the risky asset, while DRRA investors should behave the other way around.