Universal portfolios with side information

We present a sequential investment algorithm, the -weighted universal portfolio with side-information, which achieves, to rst order in the exponent, the same wealth as the best side-information dependent investment strategy (the best state-constant rebalanced portfolio) determined in hindsight from observed market and side-information outcomes. This is an individual sequence result which shows that the di erence between the exponential growth rates of wealth of the best state-constant rebalanced portfolio and the universal portfolio with side-information is uniformly less than (d=(2n)) log(n + 1) + (k=n) log 2 for every stock market and side-information sequence and for all time n. Here d = k(m 1) is the number of degrees of freedom in the state-constant rebalanced portfolio with k states of side-information and m stocks. The proof of this result establishes a close connection between universal investment and universal data compression.