Universal Semiconstant Rebalanced Portfolios

In this paper, we investigate investment strategies that can rebalance their target portfolio vectors at arbitrary investment periods. These strategies are called semiconstant rebalanced portfolios in Blum and Kalai and Helmbold et al. Unlike a constant rebalanced portfolio, which must rebalance at every investment interval, a semiconstant rebalanced portfolio rebalances its portfolio only on selected instants. Hence, a semiconstant rebalanced portfolio may avoid rebalancing if the transaction costs outweigh the benefits of rebalancing. In a competitive algorithm framework, we compete against all such semiconstant portfolios with an arbitrary number of rebalancings and corresponding rebalancing instants. We investigate this framework with and without transaction costs and demonstrate sequential portfolios that asymptotically achieve the wealth of the best semiconstant rebalanced portfolios whose number of rebalancings and instants of rebalancings are tuned to the individual sequence of price relatives.