The Kelly Criterion was first implemented by Edward O. Thorp in his casino Blackjack card-counting system to optimize players' bet sizes. He derives the formula in his paper "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market" for a coin toss experiment using a biased coin and "even money" (i.e., the amount a player stands to win is equal to the amount the player stands to lose, and is also equal to the amount the player chooses to bet). He initially models the basic process as X(n) = X(0)*(1+f)^S*(1-f)^F where X(0) is the player's initial bankroll, X(n) is his bankroll after n trials, S is the number of Successful bets, F is the number of Failed bets such that S+F=n and S>F, and f is the fraction bet of the player's bankroll at each toss. (Since the probabilities of the bias of the coin don't change, neither does f throughout the experiment.) He then modifies this model to reflect just the average incremental gain to the player at each trial with the equation G=(1+f)^p*(1-f)^q, where p=S/n, q=F/n. and G=[X(n)/X(0)]^(1/n). To solve for "f", the logarithm of the equation is used since both p and q are fractions of 1 and a standard analytical solution for polynomial equations won't eliminate the exponents. This results in the equation g(f)=p*ln(1+f)+q*ln(1-f) where g(f)= ln[G(f)]. With the function of "f" in this form and the values of p and q are known, the equation can now be solved for "f*" as the maximum value of "f", which is a concave function over the domain of 0 to 1 with a maximum value somewhere in between. The solution for f* is f*=p-q. To solve for f* where the outcome is not even money the initial model can be expressed as X(n) = X(0)*(1+b*f)^S*(1-a*f)*F, where "b" and "a" are the fraction (or multiple) of the amount bet (f) that the player stands to win or lose, respectively. Solving once again for "f*", the result for this more general application is:

f* = p/a - q/b.

Outside of determining the optimal fraction of one's bankroll (or "available resources") to bet (or "invest") at each trial, where the amount the player stands to win or lose is not necessarily equal to the amount they have bet, something I find fascinating about the Kelly Criterion is how it relates to what could be the longest running piece of advice in trading; namely, "cut your losses quickly and let your winners run". Minimizing the term "a" can cause "f*" to go to values greater than 1, as can maximizing "b" to a lesser extent. Just about every trading guru quotes that rule, and it seems to me that even Jesse Livermore might have said something along that line. What the Kelly Criterion does, however, is quantify that advice in a working equation to give us an insight into just how powerful those words of advice are. Not only does the reward for reducing the variable "a" to its smallest practical value stand out all the more since "a" is in the denominator, but taking that measure also allows us to increase the fraction of our bankroll in a completely quantifiable way because "f*" just got bigger. I can see why Thorp dubbed the Kelly Criterion as "Fortune's Formula."

I'll continue this in another post. :)