Universal Portfolios

Abstract

We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the market. Let xi = (xi1; xi2; : : : ; xim) t denote the performance of the stock market on day i ; where xij is the factor by which the j-th stock increases on day i : Let bi = (bi1;bi2; : : : ;bim) t ;bij 0; P j bij = 1 ; denote the proportion bij of wealth invested in the j-th stock on day i : Then Sn = Qn i=1 b t ixi is the factor by which wealth is increased in n trading days. Consider as a goal the wealth S n = maxb Qn i=1 b t xi 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 x1;x2; : : : ; be an arbitrary sequence of market vectors. It will be shown that the nonanticipating sequence of portfolios b̂k = R b Qk 1 i=1 b t xidb= R Qk 1 i=1 b t xidb yields wealth Ŝn = Qn k=1 b̂ t kxk such that ( 1 n ) ln(S n=Ŝn) ! 0 ; for every bounded sequence x1;x2; : : : ; and, under mild conditions, achieves Ŝn S n(m 1)!(2 =n) (m 1)=2 = j Jn j 1=2 ; where Jn is an (m 1) (m 1) sensitivity matrix. Thus this portfolio strategy has the same exponential rate of growth as the apparently unachievable S n : Departments of Statistics and Electrical Engineering, Durand Room 121, Stanford, CA, 94305. This work was partially supported by NSF Grant NCR 89-14538.

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@inproceedings{Cover1996UniversalP, title={Universal Portfolios}, author={Thomas M Cover}, year={1996} }