A Maximum Wealth Asset Index and Mixture Strategies for Universal Portfolios on Subsets of Stocks

Abstract In this paper we first show how to compute a combination of assets producing an appropriate index of past performance. The desired index is equal to Smax T = maxb ST (b) which is the maximum of T period investment return ST (b) = ∏T t=1 b · xt, where xi is the vector of returns for the tth investment period, and b is the portfolio vector specifying the fraction of wealth allocated to each asset. We provide an iterative algorithm to approximate this index, where at step k the algorithm produces a portfolio with at most k assets selected among M available assets. We show that the multi-period wealth factor ST (bk) converges to the maximum Smax T as k increases. Furthermore, in the exponent the wealth factor is within c2/k of the maximum, where c is determined by the empirical volatility of the stock returns, and we compare this computation to what is achieved by general procedures for convex optimization. This Smax T provides an index of historical asset performance which corresponds to the best constant rebalanced portfolio with hindsight. Surprisingly, we find empirically that a small handful of stocks among hundreds of candidate stocks are sufficient to have come close to Smax T .