Universal investment and universal data compression

Investors typically experience some regret about their investment decisions, especially when faced with hindsight knowledge of actual market behavior. Is it possible to find an implementable investment strategy that in some sense eliminates or at least minimizes this regret? In this thesis we study the extent to which an investor can minimize his or her regret (measured in terms of relative rate of return) relative to the performance of the best constant rebalanced portfolio determined in hindsight. Constant rebalanced portfolios buy and sell assets to ensure that the fraction of wealth invested in each of the m available assets is constant over time. These strategies are therefore parameterized by an m dimensional vector of fractions summing to one. More specifically, we solve the following problem. For any implementable investment strategy, consider the minimum over all possible market behavior up through time n of the ratio of the strategy's wealth to the wealth obtained by the best constant rebalanced portfolio computed for that market behavior. We find the maximum value of this ratio over all implementable investment strategies. It turns out that the max-min ratio is also the solution to a known problem in universal data compression and can be shown to decrease polynomially in n. Therefore, the regret of the max-min optimal strategy, measured in terms of relative rate of return, tends to zero as desired; and does so in a strong pointwise uniform sense. We also present an infinite horizon investment strategy that achieves the max-min ratio to within a constant factor for any time n. We show that the worst-case markets for both strategies are "data compression" or gambling markets, and that in these markets the strategies reduce to well known universal data compression schemes. Additional results include an analysis of the infinite horizon universal investment strategy in a side-information setting; a new derivative security motivated by the max-min regret problem called the hindsight allocation option; and an efficient method for computing the various universal investment strategies presented. A final result provides new motivation for targeting the performance of the best constant rebalanced portfolio.