The cost of achieving the best portfolio in hindsight

For a market with m assets consider the minimum, over all possible sequence s of asset prices through time n, of the ratio of the final wealth of a non anticipating investment strategy to the wealth obtained by the best constan t rebalanced portfolio computed in hindsight for that price sequence, we sh ow that the maximum value of this ratio over all nonanticipating investment strategies is V-n = [Sigma 2(m)(-nH(n1/n,...n)(/n))(n!/(n(1)!... n(m)!))]( -1), where H(.) is the Shannon entropy, and we specify a strategy achieving it. The optimal ratio V-n is shown to decrease only polynomially in rt, in dicating that the rate of return of the optimal strategy converges uniforml y to that of the best constant rebalanced portfolio determined with full hi ndsight. We also relate this result to the pricing of a new derivative secu rity which might be called the hindsight allocation option.