Universal Portfolios

We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the market. Let x i = x i 1 ; x i 2 ; : : : ; x im t denote the performance of the stock market on day i ; where x ij is the factor by which the j-th stock increases on day i : Let b i = b i 1 ; b i 2 ; : : : ; b im t ; b ij 0; P j b ij = 1 ; denote the proportion b ij of wealth invested in the j-th stock o n d a y i : Then S n = Q n i=1 b t i x i is the factor by which w ealth is increased in n trading days. Consider as a goal the wealth S n = max b Q n i=1 b t x i that can be achieved by the best constant rebalanced portfolio chosen after the stock outcomes are revealed. It can be shown that S n exceeds the best stock, the Dow Jones average, and the value line index at time n: In fact, S n usually exceeds these quantities by an exponential factor. Let x 1 ; x 2 ; : : : ; be an arbitrary sequence of market vectors. It will be shown that the nonanticipating sequence of portfolios ^ b k = R b Q k,1 i=1 b t x i db= R Q k,1 i=1 b t x i db yields wealth ^