Universal Portfolios

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 jth stock increases on day i. Let bi = (bi1 , bi2, …, bim)t, bij ≥ 0, Σj bij = 1, denote the proportion bij of wealth invested in the jth stock on day i. Then $S_{n} = \prod\nolimits_{i = 1}^{n} {\bold{b}_{i}^{t} \bold{x}_{i}}$ is the factor by which wealth is increased in n trading days. Consider as a goal the wealth $S_{n}^{*} = \max_{\bold{b}} \prod\nolimits_{i = 1}^{n} {\bold{b}^{\textrm{t}} {\bold{x}}_{i}}$ 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 x1, x2, …, be an arbitrary sequence of market vectors. It will be shown that the nonanticipating sequence of portfolios $\widehat{b}_{k} = \int \bold{b} \prod\nolimits_{i = 1}^{k - 1} \bold{b}^{\textrm{t}} \bold{x}_{i} \; d \bold{b} / \int \prod\nolimits_{i = 1}^{k - 1} \bold{b}^{\textrm{t}} \bold{x}_{i} \; d \bold{b}$ yields wealth $\widehat{S}_{n} = \prod\nolimits_{k = 1}^{n} \widehat{\bold{b}}_{k}^{\textrm{t}} \bold{x}_{k}$ such that $( 1/n ) \ln {(S_{n}^{*} / \widehat{S}_{n})} \to 0$, for every bounded sequence x1, x2, …, and, under mild conditions, achieves $$\widehat{S}_{n} \sim {{{S}_{n}^{*} (m - 1) \, ! \, (2\pi / n)^{(m - 1)/2}} \over {|J_{n}|^{1/2}}}$$ 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 $S_{n}^{*}$.