Universal portfolio selection
A typical problem in portfolio selection in stock markets is that it is not clear which of the many available strategies should be used. We apply a general algorithm of prediction with expert advice (the Aggregating Algorithm) to two different idealizations of the stock market. One is the well-known game introduced by Cover in connection with his “universal portfolio” algorithm; the other is a more realistic modification of Cover’s game introduced in this paper, where market’s participants are allowed to take “short positions”, so that the algorithm may be applied to currency and futures markets. Besides applying the Aggregating Algorithm to a countable (or finite) family of arbitrary investment strategies, we also apply it, in the case of Cover’s game, to the uncountable family of “constant rebalanced portfolios” considered by Cover. We generalize Cover’s worst-case bounds for his “universal portfolio” algorithm (which can be regarded as a special case of the Aggregating Algorithm corresponding to learning rate 1) to the case of learning rates not exceeding 1. Finally, we discuss a general approach to designing investment strategies in which, instead of making statistical or other assumptions about the market, natural assumptions of computability are made about possible investment strategies; this approach leads to natural extensions of the notion of Kolmogorov complexity.