Abstract:

It is proved that if and only if the stock market return vector $X = (X_1,\ldots,X_m)$ satisfies the condition $E \vert\log \sum_{j=1}^m X_j\vert < \infty $ a log-optimal portfolio exists in a reasonable sense. Its uniqueness is guaranteed under the assumption that the underlying distribution of $ X $ is not concentrated on a hyperplane in $R^m$ containing the diagonal $D = \{(d,\ldots,d)\in R^m: d\in R\}$. Under these assumptions, approximations of log-optimal portfolio by means of more easily evaluated portfolios are considered. In particular, a strongly consistent estimate of log-optimal portfolio based on independent observations $ X_1,\ldots,X_n$ of $ X$ is obtained.