ANALYSIS OF FINITE HORIZON INVESTMENT STRATEGIES FOR A LOGARITHMIC UTILITY FUNCTION

ANALYSIS OF FINITE HORIZON INVESTMENT STRATEGIES FOR A LOGARITHMIC UTILITY FUNCTION

ABSTRACT
We considered a simple optimization problem for an investor whose actions cannot affect the market prices and
have no other profit than the returns on the financial investments. Under general conditions on the nature of the
market model, the optimal consumption and the optimal investment were derived from our wealth process.
Risky asset price () obeys a geometric Brownian motion. The method used is dynamic programming principle
which is used to derive the Hamilton-Jacobi-Bellman equation (HJB) which is a second order non-linear
differential equation. In addition we analyzed the optimal consumption of an investor for 10years and our result
shows that the optimal consumption of the investor is to consume a fraction of his wealth which also coincides
with the wealth at terminal. We also focused on analyzing the optimal bond/stock mix for a single stock. Our
result shows that for a logarithmic investor, investing large part of their wealth on stock tends to yield higher
return. Illustrative example is given.

ANALYSIS OF FINITE HORIZON INVESTMENT STRATEGIES FOR A LOGARITHMIC UTILITY FUNCTION