Mining Matrix Data with Bregman Matrix Divergences for Portfolio Selection

Abstract

15.1 Introduction 0 If only we always knew ahead of time…. The dream of any stock portfolio manager 1 is to allocate stocks in his portfolio in hindsight so as to always reach maximum 2 wealth. With hindsight, over a given time period, the best strategy is to invest into 3 the best performing stock over that period. However, even this appealing strategy is 4 not without regret. Reallocating everyday to the best stock in hindsight (that is with 5 a perfect sense for ups and downs timing) notwithstanding, Cover has shown that a 6 Constant Rebalancing Portfolio (CRP) strategy can deliver superior results [10]. 7 These superior portfolios have been named Universal Portfolios (UP). In other 8 words, if one follows Cover's advice, a non anticipating portfolio allocation per-9 forms (asymptotically) as well as the best constant rebalancing portfolio allocation 10 determined in hindsight. This UP allocation is however not costless as it replicates 11 the payoff, if it existed, of an exotic option, namely a hindsight allocation option. 12 Buying this option, if it were traded, would enable a fund manager to behave as if 13 he always knew everything in hindsight. 14 Finding useful portfolio allocations, like the CRP allocation, is not however always 15 related to the desire to outperform some pre-agreed benchmark. As Markowitz has 16 shown, investors know that they cannot achieve stock returns greater than the risk-17 free rate without having to carry some risk [17]. Markowitz designed a decision 18 criterion which, taking both risk and return into account, enables any investor to