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Universal portfolios. (English) Zbl 0900.90052
Summary: We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the market. Let \({\mathbf x}_i=(x_{i1},x_{i2},\dots,x_{im})^t\) 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 \({\mathbf b}_i=(b_{i1},b_{i2},\dots,b_{im})^t\), \(b_{ij}\geq 0\), \(\sum_j b_{ij}=1\), denote the proportion \(b_{ij}\) of wealth invested in the \(j\)th stock on day \(i\). Then \(S_n=\prod^n_{i=1} {\mathbf b}^t_i {\mathbf x}_i\) is the factor by which wealth is increased in \(n\) trading days. Consider as a goal the wealth \(S^*_n=\max_{\mathbf b} \prod^n_{i=1}{\mathbf b}^t{\mathbf x}_i\) 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 \({\mathbf x}_1, {\mathbf x}_2,\dots,\) be an arbitrary sequence of market vectors. It will be shown that the nonanticipating sequence of portfolios \(\widehat{\mathbf b}_k=\int{\mathbf b}\prod^{k-1}_{i=1} {\mathbf b}^t{\mathbf x}_i d{\mathbf b}/\int\prod^{k-1}_{i=1}{\mathbf b}^t{\mathbf x}_i d{\mathbf b}\) yields wealth \(\widehat{S}_n=\prod^n_{k=1}\widehat{\mathbf b}^t_k {\mathbf x}_k\) such that \((1/n)\ln(S^*_n/\widehat{S}_n)\to 0\), for every bounded sequence \({\mathbf x}_1{\mathbf x}_2,\dots,\) and, under mild conditions, achieves \[ \widehat{S}_n\sim\;\frac{S^*_n(m-1)!(2\pi/n)^{(m-1)/2}}{|J_n|^{1/2}} , \] where \(J_n\) is an \((m-1)\times (m-1)\) sensitivity matrix. Thus this portfolio strategy has the same exponential rate of growth as the apparently unachievable \(S^*_n\).

MSC:
91G10 Portfolio theory
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