On the Competitive Theory and Practice of Portfolio Selection (Extended Abstract)

The portfolio selection problem is clearly one of the most fundamental problems in the eld of computational nance. Given a set of say m stocks (one of which may be \cash"), the natural online problem is to determine a portfolio for the i th trading period based on the sequence of prices (or equivalently relative prices) for the preceding i ? 1 trading periods. There has been both a growing interest and a growing skepticism concerning the value of a competitive theory of online portfolio selection algorithms. Competitive analysis is based on a worst case perspective and such a perspective is inconsistent with the more widely accepted analyses and theories based on statistical assumptions. The competitive framework does (perhaps surprisingly) permit non trivial upper bounds on relative performance against CBAL-OPT, an optimal ooine constant rebalancing portfolio. Perhaps more impressive are some preliminary experimental results showing that certain algorithms that enjoy \respectable" competitive (i.e. worst case) performance also seem to perform quite well on historical sequences of data. These algorithms and the emerging competitive theory are directly related to studies in information theory and computational learning theory and indeed some of these algorithms have been pioneered within the information theory and computational learning communities. One goal of this paper is to try to better understand the extent to which competitive portfolio algorithms are indeed \learning". In doing so we discuss some simple strategies which can adapt to the data sequence. We present a mixture of both theoretical and experimental results. We also present a more inclusive study of the performance of existing and new algorithms with respect to a standard sequence of historical data cited in many studies. Furthermore, we present experiments from three other historical data sequences. We conclude that there is great potential for portfolio selection algorithms that are motivated by both competitive considerations as well as by an attempt to learn statistical properties of the data.