Universal Portfolios With and Without Transaction Costs

A constant rebalanced portfolio is an investment strategy which keeps the same distribution of wealth among a set of stocks from period to period. Recently there has been work on on-line investment strategies that are competitive with the best constant rebalanced portfolio determined in hindsight (Cover, 1991, 1996; Helmbold et al., 1996; Cover & Ordentlich, 1996a, 1996b; Ordentlich & Cover, 1996). For the universal algorithm of Cover (Cover, 1991),we provide a simple analysis which naturallyextends to the case of a fixed percentage transaction cost (commission ), answering a question raised in (Cover, 1991; Helmbold et al., 1996; Cover & Ordentlich, 1996a, 1996b; Ordentlich & Cover, 1996; Cover, 1996). In addition, we present a simple randomized implementation that is significantly faster in practice. We conclude by explaining how these algorithms can be applied to other problems, such as combining the predictions of statistical language models, where the resulting guarantees are more striking.