Generalizing the Kelly Criterion
Formulated by John L. Kelly and popularized by the practical success of Ed Thorp, the Kelly Criterion is a formula used to determine the optimal bet size for a given set of probabilities and payoffs. While the formula can be stated in several ways, one format is an expanded version of the formula that appeared in Thorp’s interview in the book Hedge Fund Market Wizards:
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If one knows the odds and payouts of a given bet with precision, the Kelly Criterion bet size will maximize one’s capital over the long run. To give a common example, assume that someone decides to make you a generous offer in a coin-tossing game. The coin is fair, and if it comes up heads, you will win $2 for every $1 you bet; if it comes up tails, you will lose each dollar you bet. Moreover, you are permitted to keep playing the game under these terms for as long as you’d like, as long as you don’t run out of money. You, rightly, conclude that this is a game worth playing and gather all your money together to play.
You also don’t want to make tiny bets because while you will be profitable, you know that the bet is so favorable to you, that you want to make a big enough bet to adequately grow your capital. Given the terms of the bet, the amount you should bet on each toss to maximize your winnings over the long term is 25% of your bankroll, as calculated with the Kelly Criterion in the exhibit to the right.
When it comes to applying the Kelly Criterion to investing, one hurdle is that one doesn’t get precise odds and only in the rare special situation or arbitrage does one get a decent picture of the payout. Another hurdle is that when utilizing Kelly, the long run is based on the number of events, not a time frame. An investor, especially a value investor, will have trouble making enough investments to get the full long-term benefits of applying Kelly. As Ed Thorp wrote in his 2007 article, “Understanding Fortune’s Formula,” “The caveat here is that an investor or bettor [may] not choose to make, or be able to make, enough Kelly bets for the probability to be ‘high enough’ for these asymptotic properties to prevail.”
A third hurdle is volatility. Kelly is designed for optimal long-term return while avoiding the risk of ruin. While we certainly don’t equate risk with volatility, the drawdown that can be experienced using the Kelly formula to size positions can leave even the most steadfast investors a bit squeamish, even if the odds and payouts were known with 100% certainty, which they of course never will be. And a final hurdle of note, also mentioned in Thorp’s article, is that humans tend to underestimate the role of infrequent, high-impact events: what Nassim Taleb refers to as Black Swans. The probability and downside of negative Black Swans may not be given enough consideration when investors look to apply the Kelly Criterion, and thus the formula may tend to overestimate F when applied by the human mind. And continual overestimation leads to ruin; anything above the optimal bet size will lead to total loss sooner or later.
See full Generalizing the Kelly Criterion in PDF format here.