When Harry Met Kelly: An Overlooked Result in the Classical Theory of Optimal Capital Growth

A classic problem in finance is the question of how an investor should optimally share wealth over one risky and one risk-free asset. This article reconsiders this theory under the assumptions that (1) the investor’s objective is to maximize the intertemporal growth rate of their wealth, (2) that the investor can make portfolio adjustments only in discrete time, and (3) that the risky asset is defined in each period according to only its mean return and the standard deviation thereof. This problem was resolved many years ago, but only for continuous time, thereby ignoring the discrete time optimal strategy. Here, the discrete time strategy for optimal capital growth in the one risky one risk-free setting is resolved for two different methodologies, and it is shown that the continuous time model is recoverable as a limit version of each of the two discrete time models that are analysed. Finally, it is also shown that for both of the methodologies considered, the same first-order Taylor’s approximation formula eventuates, and that in both models it always gives a more accurate approximation to the true optimal discrete time strategy than does the continuous time strategy