The log-optimal or Kelly portfolio forms the basis of a theoretically appealing investment strategy. However, it is difficult to compute, and this hinders its adoption in practice. In this paper we consider an approximate Kelly portfolio based on maximizing the expected value of a quadratic approximation to log utility. We show that this semi-log approximation gives an information-theoretic justification for portfolio selection based on either the mean-variance efficient frontier or the Sharpe ratio. We further show that there is a strong connection between estimated approximate fractional Kelly portfolios and shrinkage estimators, which leads to an optimal choice of a fractional Kelly parameter. We conclude by showing that the fractional Kelly portfolio succeeds not because of reduced risk, but because of reduced estimation error. We simulate to show that this property is largely responsible for the good empirical performance of fractional Kelly strategies.