I keep reading that the expected time for your capital to reach any predefined number is minimized by the strategy that sizes bets according to the Kelly Criterion. But it seems trivially easy to come up with a counterexample.
Suppose we are playing a game (the details of which are unimportant) for which Kelly dictates that you bet 99% of your capital on each round (i.e. it's a highly favorable game for us). Say our goal is to increase our capital by a mere 1%. If we bet 99% on round one and lose, then it will take many more rounds for us to be able to reach our goal. On the other hand, if on each round we only bet as much as will allow us to exactly reach our target capital and no more when we win (1% on round one in this example), we retain the ability to be able to bet enough to reach our target on round two (as well as on a few more rounds after successive losses). So clearly the second strategy will on average reach the target capital in fewer time periods.
So either I am missing something, or the way that claim about the Kelly Criterion is stated is not completely accurate. Which is it? If the latter, how can the "overshooting" of Kelly in such scenarios be addressed rigorously?