Universal Portfolios with Side Information 6.975

The general investment problem concerns the allocation of wealth among m assets (stocks) to generate high returns with low risk or uncertainty. Cover and Ordentlich consider this problem from an information theoretic perspective in the case where a side information sequence aids the investment decisions but no assumptions are made on the relative likelihoods (probabilities) of stock returns sequences or side information sequences nor on the relationship between the stock returns and side information. The authors present a method of constructing a sequential portfolio that is universal with respect to the class of so-called state-constant rebalanced (state-CRB) portfolios in the sense that for every stock returns and side information sequence, this universal portfolio asymptotically achieves the same growth rate of wealth as the best state-CRB portfolio for that particular stock returns and side information sequence. Thus, a universal portfolio without hindsight matches the performance of the best state-CRB portfolio chosen with hindsight. A key insight in the development of these universal portfolios is that the average of a set of exponentials grows exponentially as fast as the largest exponential. Thus, one can construct portfolios that are universal with respect to a class of portfolios by constructing a portfolio that is an average over that class. In their paper, Cover and Ordentlich demonstrate how to apply this intuition — which applies straightforwardly in the finite portfolio class case — to the state-CRB case, which involves a continuum of portfolios. Only the results of their derivations are presented in the talk.