مرکز منطقه ای اطلاع رساني علوم و فناوري - 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.