An algorithm for maximizing expected log investment return

Abstract

Let the random (stock market) vectorX \geq 0be drawn according to a known distribution functionF(x), x \in R^{m}. A log-optimal portfoliob^{\ast}is any portfoliobachieving maximal expected\logreturnW^{\ast}=\sup_{b} E \ln b^{t}X, where the supremum is over the simplexb \geq 0, \sum_{i=1}^{m} b_{i} = 1. An algorithm is presented for findingb^{\ast}. The algorithm consists of replacing the portfoliobby the expected portfoliob^{'}, b_{i}^{'} = E(b_{i}X_{i}/b^{t}X), corresponding to the expected proportion of holdings in each stock after one market period. The improvement inW(b)after each iteration is lower-bounded by the Kullback-Leibler information numberD(b^{'}\|b)between the current and updated portfolios. Thus the algorithm monotonically improves the returnW. An upper bound onW^{\ast}is given in terms of the current portfolio and the gradient, and the convergence of the algorithm is established.

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