Relative utility bounds for empirically optimal portfolios

Abstract

We consider a single-period portfolio selection problem for an investor, maximizing the expected ratio of the portfolio utility to the utility of a best asset taken in hindsight. The decision rules are based on the history of stock returns with unknown distribution. Assuming that the utility function is Lipschitz or Holder continuous (the concavity is not required), we obtain high probability utility bounds under the sole assumption that the returns are independent and identically distributed. These bounds depend only on the utility function, the number of assets and the number of observations. For concave utilities similar bounds are obtained for the portfolios produced by the exponentiated gradient algorithm. Also we use statistical experiments to study risk and generalization properties of empirically optimal portfolios. Herein we consider a model with one risky asset and several datasets, containing real stock prices.