Applying the Kelly Criterion - Targeting Specific Capital Gains

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?

• You've changed the problem so the results will surely change. The criterion you're describing is just for long-term growth. There are many variations like "How to Gamble if you Must" (subfair games), "How to Gamble if You're in a Hurry" (finite horizon), etc. – Jared May 2 '18 at 15:52

Full Kelly bet criteria maximizes the expected logarithmic rate of return. In your example, you propose to reach a specific rate of return. If ever the target is to achieve a specific rate of return which is less than maximal, then the optimal bet size is said to be fractional Kelly. In other words, the fractional Kelly bet which achieves the target rate of return is said to be Kelly-optimal for that target rate. Likewise, if the target is to maximize the logarithmic rate of return, then the optimal bet size is said to be full Kelly optimal.

Generally, for asset returns which are approximately normally distributed, and if one has the ability to make many bets over any finite time horizon, the expected variance of return falls proportionately to bet size, while expected returns falls at about half that rate. This artifact is often used as to justify fractional Kelly betting as optimal in the real world to compensate for fat tailed results, estimation errors, and the real life consequences of Gambler's ruin. In the case of a single discrete bet, the relationship between return and variance is unity. In either case, if the optimal rate of return is sub-Kelly optimal, the optimal bet size will also be less than full Kelly.