Expectation and Optimal f : Expected Growth with and without Reinvestment for Discretely-Distributed Outcomes of Finite Length

Abstract

Presented is the formulation for determining the exact, expected growth-optimal fraction of equity to risk for all conditions, rather than merely the asymptotic growth-optimal fraction. The formulation presented represents the surface in the leverage space manifold, wherein the loci at the peak of the surface are those fractions for maximizing expected growth. Other criteria can be solved for upon the surface in the leverage space manifold utilizing the equation specified here. Of equal importance, since solving for the expected growth-optimal fraction involves a set of inputs and the "expectation" of those inputs, we see that the notion of mathematical expectation, as ubiquitously employed in numerous disciplines is also an asymptotic proxy for what an individual participant "expects" shall occur over a finite sequence of propositions. Thus, we determine what this expectation is over a finite sequence, demonstrate that it approaches the classical expectation as the length of the sequence becomes ever-longer, and see how this determination of "actual" expectation, versus the asymptotic "classical" one, is innately part of the human decision-making process.

Keywords: Growth-optimal portfolio, expectation, mathematical expectation, finite trials, gambling, risk management, Kelly Criterion, finite investment horizon, drawdown

JEL Classification: C02, C13, C44, C65, C70, C79, G02, G11, G12