In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet) is a formula for sizing a bet. The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate. It assumes that the expected returns are known and is optimal for a bettor who values their wealth logarithmically. J. L. Kelly Jr, a researcher at Bell Labs, described the criterion in 1956. Under the stated assumptions, the Kelly criterion leads to higher wealth than any other strategy in the long run (i.e., the theoretical maximum return as the number of bets goes to infinity).
The practical use of the formula has been demonstrated for gambling, and the same idea was used to explain diversification in investment management. In the 2000s, Kelly-style analysis became a part of mainstream investment theory and the claim has been made that well-known successful investors including Warren Buffett and Bill Gross use Kelly methods. Also see Intertemporal portfolio choice.
Gambling formula
Where losing the bet involves losing the entire wager, the Kelly bet is:
where: is the fraction of the current bankroll to wager. is the probability of a win. is the probability of a loss (). is the proportion of the bet gained with a win. E.g., if betting $10 on a 2-to-1 odds bet (upon win you are returned $30, winning you $20), then . As an example, if a gamble has a 60% chance of winning (, ), and the gambler receives 1-to-1 odds on a winning bet (), then to maximize the long-run growth rate of the bankroll, the gambler should bet 20% of the bankroll at each opportunity ().
If the gambler has zero edge, i.e., if , then the criterion recommends for the gambler to bet nothing.
If the edge is negative () the formula gives a negative result, indicating that the gambler should take the other side of the bet. For example, in American roulette, the bettor is offered an even money payoff () on red, when there are 18 red numbers and 20 non-red numbers on the wheel (). The Kelly bet is , meaning the gambler should bet one-nineteenth of their bankroll that red will not come up. There is no explicit anti-red bet offered with comparable odds in roulette, so the best a Kelly gambler can do is bet nothing.
Investment formula
A more general form of the Kelly formula allows for partial losses, which is relevant for investments:
where: is the fraction of the assets to apply to the security. is the probability that the investment increases in value. is the probability that the investment decreases in value (). is the fraction that is lost in a negative outcome. If the security price falls 10%, then is the fraction that is gained in a positive outcome. If the security price rises 10%, then . Note that the Kelly criterion is valid only for known outcome probabilities, which is not the case with investments. Risk averse investors should not invest the full Kelly fraction.
The general form can rewritten as follows
where: is the win-loss probability (WLP) ratio, which is the ratio of winning to losing bets. is the win-loss ratio (WLR) of bet outcomes, which is the wining skew. It is clear that, at least, one of the factors or needs to be larger than 1 for having an edge (so ). It is even possible that the win-loss probability ratio is unfavorable , but one has an edge as long as .
Kelly formula can easily result in the fractions higher than 1, for instance, when the losing size (see the above expression with factors of and ). This happens somewhat counterintuitively, because Kelly fraction formula compensates by larger size of the bet. But, in most of real situations, there is large uncertainty about all parameters entering Kelly formula. In the case of Kelly fraction higher than 1, it is theoretically advantageous to use leverage to purchase additional securities on margin.
Betting example - behavioural experiment
In a study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at $250. But the behavior of the test subjects was far from optimal: { { blockquote|Remarkably, 28% of the participants went bust, and the average payout was just $91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment.<ref>"Buttonwood", "Irrational tossers", The Economist, 1 November 2016.</ref>}}