Universal Portfolios

This is the authors' abstract. We don't add key points for this paper.

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 .