## 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
Full Text:

### References:

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