Portfolio selection with hyperexponential utility functions

Abstract

We analyze a single-period portfolio selection problem where the investor maximizes the expected utility of the terminal wealth. The utility function is hyperexponential. This is due to the fact that the risk tolerance of the investor at the end of the period when the terminal wealth is realized depends on the random state of the market at that time. This setting is also applicable in cases where an investment consultant is not sure about the risk profile of a client. It is well-known that an investor is memoryless in wealth for exponential utility functions with some known risk tolerance. In other words, the investment portfolio consisting of risky stocks does not depend on the level of wealth. However, we show that this is no longer true if the utility function is hyperexponential. We also obtain a number of interesting characterizations on the structure of the optimal policy.