Universal Portfolios


  • Thomas M. Cover

    1. Departments of Statistics and Electrical Engineering, Stanford University, Stanford, CA
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      I wish to thank Hal Stern for his invaluable contributions in obtaining and analyzing the stock market data. I also wish to thank the referee for helpful comments. This work was partially supported by NSF Grant NCR 89–14538.


We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the market. Let xi= (xi, xi2,…, xim)t denote the performance of the stock market on day i, where xii is the factor by which the jth stock increases on day i. Let bi= (bi1 bi2, bim)t, b;ij≫ 0, bij= 1, denote the proportion bij of wealth invested in the j th stock on day i. Then Sn= IIin= bitxi is the factor by which wealth is increased in n trading days. Consider as a goal the wealth Sn*= maxb IIin=1 btxi 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, Sn* usually exceeds these quantities by an exponential factor. Let x1, x2, be an arbitrary sequence of market vectors. It will be shown that the nonanticipating sequence of portfolios inline image db yields wealth inline image such that inline image, for every bounded sequence x1, x2…, and, under mild conditions, achieve


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.