An algorithm for maximizing expected log investment return

Abstract

Let the random (stock market) vector X \geq 0 be drawn according to a known distribution function F(x), x \in R^{m} . A log-optimal portfolio b^{\ast} is any portfolio b achieving maximal expected \log return W^{\ast}=\sup_{b} E \ln b^{t}X , where the supremum is over the simplex b \geq 0, \sum_{i=1}^{m} b_{i} = 1 . An algorithm is presented for finding b^{\ast} . The algorithm consists of replacing the portfolio b by the expected portfolio b^{'}, 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 in W(b) after each iteration is lower-bounded by the Kullback-Leibler information number D(b^{'}\|b) between the current and updated portfolios. Thus the algorithm monotonically improves the return W . An upper bound on W^{\ast} is given in terms of the current portfolio and the gradient, and the convergence of the algorithm is established.