Statistical properties of estimators for the log-optimal portfolio

The best constant re-balanced portfolio represents the standard estimator for the log-optimal portfolio. It is shown that a quadratic approximation of log-returns works very well on a daily basis and a mean-variance estimator is proposed as an alternative to the best constant re-balanced portfolio. It can easily be computed and the numerical algorithm is very fast even if the number of dimensions is high. Some small-sample and the basic large-sample properties of the estimators are derived. The asymptotic results can be used for constructing hypothesis tests and for computing confidence regions. For this purpose, one should apply a finite-sample correction, which substantially improves the large-sample approximation. However, it is shown that the impact of estimation errors concerning the expected asset returns is serious. The given results confirm a general rule, which has become folklore during the last decades, namely that portfolio optimization typically fails on estimating expected asset returns.