Universal Portfolios

We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the market. Let x i = (x i , x i2 ,…, x im ) t denote the performance of the stock market on day i, where x ii is the factor by which the jth stock increases on day i. Let b i = ( bi1 b i2 , b im ) t , b; ij ≫ 0, b ij = 1, denote the proportion b ij of wealth invested in the j th stock on day i. Then S n = II i n = bi t x i is the factor by which wealth is increased in n trading days. Consider as a goal the wealth S n *= max b II i n = 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 Sn * 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 db yields wealth such that , for every bounded sequence x 1 , x 2 …, and, under mild conditions, achieve image where J, is an (m ‐ 1) x (m ‐ I) sensitivity matrix. Thus this portfolio strategy has the same exponential rate of growth as the apparently unachievable S* n .