Minimax Values and Metric Entropy Bounds for Portfolio Selection Problems
by
Nicolò Cesa-Bianchi
University of Milan, Italy
Coauthors: Gábor Lugosi (Pompeu Fabra University, Barcelona)

The game-theoretic version of the portfolio selection problem has been analyzed by Cover and Ordentlich who established the extent to which a nonanticipating investment strategy can achieve, in the worst-case market, the performance of the best constant rebalanced portfolio computed in hindsight.

For a given a class of investment strategies, the minimax wealth ratio is the the worst-case ratio, over all market sequences, between the wealth factor of the best strategy chosen in hindsight from the class and the wealth factor of the best possible nonanticipating investment strategy. Cover and Ordentlich derived the exact value and the asymptotics of the minimax wealth ratio for the class of all constant rebalanced portfolios.

Their proof reveals an intriguing connection between the portfolio selection problem for m assets and the problem of predicting an m-ary sequence under the logarithmic loss (as originally defined by Shtarkov). By generalizing their approach, we prove the equivalence between the minimax values of the investment and the prediction problems for any class of static investment strategies. This equivalence allows us to apply a series of recent results developed in the prediction context and prove a general upper bound on the minimax wealth ratio in terms of the metric entropy (under a suitable metric) of the given class of strategies. In the paper we show applications of this result to several classes of static investment strategies.

Nicolò Cesa-Bianchi's homepage

Date received: May 8, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafc-12.