UNIVERSAL PORTFOLIOS WITH SIDE INFORMATION

We present a sequential investment algorithm, the mu-weighted universa l portfolio with side information, which achieves, to first order in t he exponent, the same wealth as the best side-information dependent in vestment strategy (the best state-constant rebalanced portfolio) deter mined in hindsight from observed market and side-information outcomes. This is an individual sequence result which shows that the difference between the exponential growth rates of wealth of the best state-cons tant rebalanced portfolio and the universal portfolio with side inform ation is uniformly less than (d/(2n))log(n + 1) + (k/n) log 2 for ever y 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 univer sal investment and universal data compression.