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Abstract
In recent years much work has been carried out to design and analyse online investment strategies based on constant rebalanced portfolios. A constant rebalanced portfolio is a sequential investment strategy that maintains fixed through time, trading period by trading period, the wealth distribution among a set of assets. In this framework, Cover proposed the universal portfolio, which is competitive with the best constant rebalanced portfolio determined in hindsight, i.e. the constant rebalanced portfolio obtained by assuming perfect knowledge of future stock prices. However, the constant rebalanced portfolio is designed to deal with the portfolio selection problem in the case where no additional information concerning the stock market, is available. To overcome this limitation, Cover and Ordentlich proposed the state constant rebalanced portfolio, which is capable of appropriately exploiting the available side-information concerning the stock market. In this paper we study and analyse the topic introduced by Cover and Ordentlich and focus our attention on the interplay between constant rebalanced portfolios and side-information. We introduce a mathematical framework to deal with the constant rebalanced portfolio in the case where side-information, concerning the stock market, is available. The mathematical framework defines and analyses the mixture best constant rebalanced portfolio, which we propose as the investment benchmark to be considered in the case where side-information, concerning the stock market, is available. The mixture best constant rebalanced portfolio outperforms the best constant rebalanced portfolio by an exponential factor in terms of the achieved wealth and therefore offers an interesting opportunity for side-information specialized online investment algorithms. We describe a new online investment algorithm that exploits the definition of the mixture best constant rebalanced portfolio and the available side-information. The performance of the proposed online investment algorithm is investigated through a set of numerical experiments concerning four major stock market data sets, namely DJIA, S&P500, TSE and NYSE. The results emphasize the relevance of the proposed online investment strategy and underline the central role of the quality of the side-information in outperforming the best constant rebalanced portfolio.
Acknowledgements
The authors are grateful to the anonymous referees whose insightful comments enabled us to make significant improvements to the paper. Special thanks go to Professor Georg Pflug for his comments and suggestions concerning the mathematical notation, which allowed us to significantly improve the clarity of the paper.