Relative utility bounds for empirically optimal portfolios

TL;DR

This paper derives high-probability utility bounds for empirical portfolio optimization under i.i.d. returns, applicable to Lipschitz or Hölder utilities, and evaluates risk and generalization using real NYSE data.

Contribution

It introduces utility bounds for empirical portfolios with Lipschitz/Hölder utilities and analyzes their risk and generalization properties using real stock data.

Findings

High-probability utility bounds depend only on utility, assets, and observations.

Bounds are applicable to portfolios optimized via exponentiated gradient for concave utilities.

Empirical analysis on NYSE data demonstrates practical relevance.

Abstract

We consider a single-period portfolio selection problem for an investor, maximizing the expected ratio of the portfolio utility and 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 H\"{o}lder 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 method. 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 a dataset, containing the stock prices from NYSE.