The Cost of Achieving the Best Portfolio in Hindsight

For a market with m assets consider the minimum, over all possible sequences of asset prices through time n, of the ratio of the final wealth of a nonanticipating investment strategy to the wealth obtained by the best constant rebalanced portfolio computed in hindsight for that price sequence. We show that the maximum value of this ratio over all nonanticipating investment strategies is V_n = [∑ 2^{−nH(n_1/n,…,n_m/n)}(n!/(n₁! … 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 n, indicating that the rate of return of the optimal strategy converges uniformly to that of the best constant rebalanced portfolio determined with full hindsight. We also relate this result to the pricing of a new derivative security which might be called the hindsight allocation option.