An Investment Strategy with Overshoot Rebates which Minimizes the Time to Attain a Specified Goal

A strategy is developed which minimizes the expected time (or plays) to reach a given financial goal when “time rebates” are given if the goal is more than attained in the last investment period. It is shown that the optimal strategy is the one which maximizes the expected logarithm of the investment relative (for discrete distributions this is equivalent to maximizing the geometric mean). This strategy is optimal for all goals and levels of capital. When no time rebates are given, however, the proposed strategy will generally not be optimal. In this case the theory shows the strategy is uniformly good. Further, at least in a limited sense, the true optimal will generally differ significantly only in endgame play.