  In this paper, we consider a simple discrete-time optimal betting problem
using the celebrated Kelly criterion, which calls for maximization of the
expected logarithmic growth of wealth. While the classical Kelly betting
problem can be solved via standard concave programming technique, an
alternative but attractive approach is to invoke a Taylor-based approximation,
which recasts the problem into quadratic programming and obtain the closed-form
approximate solution. The focal point of this paper is to fill some voids in
the existing results by providing some interesting properties when such an
approximate solution is used. Specifically, the best achievable betting
performance, positivity of expected cumulative gain or loss and its associated
variance, expected growth property, variance of logarithmic growth, and results
related to the so-called survivability (no bankruptcy) are provided.
