The Kelly Criterion is just a
postulate. Kelly (and Graham as well as Shannon, both of who signed off on the paper) solved for this supposed asymptotic growth-optimal fraction in the 1956 Bell Labs paper, despite the fact that they thought they did solve for it. Actually, they solved for something else. They deluded themselves as they were solving for a subset of problems, the answer to which equaled the asymptotic growth optimal fraction in the narrow cases they were concerned with! Yet people accepted it as fact and still mistakenly do.
The first one to solve for the asymptotic growth optimal fraction was Thorp with his "Kelly Formulas," closed-end formulas for solving for the asymptotic growth-optimal fraction for
binomially-distributed outcomes.
The Optimal
f formula I put forth in the late 1980s does solve for the asymptotic growth-optimal fraction, for one or more simultaneous propositions (portfolio components/systems/markets). But it too is asymptotic, that is, it is a fraction approached (albeit very quickly) as the number of trials/trades/holding periods increases.
Consider however a game with a positive expectation where you wat to determine the expected growth optimal fraction, but you are quitting after only one play? In such cases, the expected growth optimal fraction, the Optimal f value is 1, or risk 100% to maximize your expected growth after one play.
It's actually a little more complicated than this. Assume a p=.1 to win 10 units and q=.9 to lose 1 unit. Though there is a positive expectation, it is a game you should not play if you are going to quit after one play. Similarly, if we have a negative expectation game with p=.9 to win 1 unit and q=.9 to lose 10 units, and you wish to maximize your expected growth, quitting after one play, your Optimal
f = 1, or risk it all on the one play. The Kelly Criterion, even Thorp's Kelly formulas for these binomial outcomes give you entirely different (and incorrect) results.
I refer you to my paper on this at
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2577782
This will give you what the expected growth optimal fraction(s) are for one or more simultaneous propositions, for 1 to infinitely many trials.Further, there are other considerations, such as the fact the there is a cost of funds involved (trades don't transpire instantaneously in most cases), etc.
Anyone who has used "Kelly" with success has been using something that is mathematically incorrect, and the success may in many cases be attributable to luck (you cannot determine an expected growth optimal fraction for most capital market situations with any of what has been referred to as "Kelly," nor can you even apply it, correctly mathematically to the game of blackjack, and favorable results garnered from such were done so with grossly inaccurate calculations).
Finally, all of this assumes your criterion in trading is to be expected growth-optimal, all else (including drawdown or other risk measures) be damned. This is ok provided that really is your criterion.
Since the actual Optimal
f value (or value sets) exit between the bookends of 0 and 1, there are other points of geometrical interest within that manifold. Further, everyone in every assumed proposition or set of propositions exists within that manifold,unwittingly in most cases, and are covering paths through it, unwittingly again in most cases.They are paying the consequence and reaping the benefits in terms of performance characteristics that these various points of geometric consequence (aside form the peak, the Optimal
f points) imbue. In fact, various points and paths in this manifold can be constructed to satisfy ant trading criteria. It really is not very complicated -- I'm not trying to sell anything, razzle-dazzle anyone, but rather to edify this area that is always misunderstood.
