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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. Assuming that the expected returns are known, 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). J. L. Kelly Jr, a researcher at Bell Labs, described the criterion in 1956. 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. It is also the standard replacement of statistical power in anytime-valid statistical tests and confidence intervals, based on e-values and e-processes. Kelly criterion for binary return rates In a system where the return on an investment or a bet is binary, so an interested party either wins or loses a fixed percentage of their bet, the expected growth rate coefficient yields a very specific solution for an optimal betting percentage. Gambling formula Where losing the bet involves losing the entire wager, the Kelly bet is: f ∗ = p − q b = p − 1 − p b {\displaystyle f^{*}=p-{\frac {q}{b}}=p-{\frac {1-p}{b}}} where: f ∗ {\displaystyle f^{*}} is the fraction of the current bankroll to wager. p {\displaystyle p} is the probability of a win. q {\displaystyle q} is the probability of a loss ( q = 1 − p {\displaystyle q=1-p} ). b {\displaystyle b} 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 b = $ 20 / $ 10 = 2.0 {\displaystyle b=\$20/\$10=2.0} . As an example, if a gamble has a 60% chance of winning ( p = 0.6 {\displaystyle p=0.6} , q = 0.4 {\displaystyle q=0.4} ), and the gambler receives 1-to-1 odds on a winning bet ( b = 1 {\displaystyle b=1} ), then to maximize the long-run growth rate of the bankroll, the gambler should bet 20% of the bankroll at each opportunity ( f ∗ = 0.6 − 0.4 1 = 0.2 {\textstyle f^{*}=0.6-{\frac {0.4}{1}}=0.2} ). If the gambler has zero edge (i.e., if b = q / p {\displaystyle b=q/p} ), then the criterion recommends the gambler bet nothing. If the edge is negative ( b < q / p {\displaystyle b<q/p} ), 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 ( b = 1 {\displaystyle b=1} ) on red, when there are 18 red numbers and 20 non-red numbers on the wheel ( p = 18 / 38 {\displaystyle p=18/38} ). The Kelly bet is − 1 / 19 {\displaystyle -1/19} , 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:: 7  f ∗ = p a − q b {\displaystyle f^{*}={\frac {p}{a}}-{\frac {q}{b}}} where: f ∗ {\displaystyle f^{*}} is the fraction of the assets to apply to the security. p {\displaystyle p} is the probability that the investment increases in value. q {\displaystyle q} is the probability that the investment decreases in value ( q = 1 − p {\displaystyle q=1-p} ). a {\displaystyle a} is the fraction that is lost in a negative outcome.: 7  If the security price falls 10%, then a = original value − final value original value = 1 − .9 1 = 0.1 {\displaystyle a={\frac {{\text{original value}}-{\text{final value}}}{\text{original value}}}={\frac {1-.9}{1}}=0.1} b {\displaystyle b} is the fraction that is gained in a positive outcome.: 7  If the security price rises 10%, then b = final value − original value original value = 1.1 − 1 .... Discover the Q Kelly popular books. Find the top 100 most popular Q Kelly books.