Kelly Criterion Calculator

The Kelly Criterion is the bet size that maximizes the expected logarithm of your bankroll, which is the same as maximizing long-run compound growth. Bell Labs scientist John Kelly derived it in 1956 by asking what fraction f of bankroll maximizes the expected value of log(1 + f * b) when you win with probability p and lose with probability q = 1 - p. Taking the derivative and setting it to zero gives the closed form f* = (b * p - q) / b, where b is the payoff per $1 staked (so European decimal odds minus one). Maximizing log-utility rather than raw expected dollars is what makes Kelly the unique fraction that wins out over any other fixed strategy across enough bets.

Full Kelly is provably growth-optimal only when you know p and b exactly. In real life your edge is an estimate, and Kelly is highly sensitive to that estimate: overstating your win rate by even a few percentage points can swing the recommended stake well past the true optimum. Full Kelly also produces stomach-churning drawdowns, with a roughly one-in-three chance of halving your bankroll at some point before reaching a new high. Half Kelly captures about 75% of the growth rate at one quarter of the volatility, which is why sports bettors, blackjack teams, and quantitative hedge funds nearly all stake a fraction of full Kelly. Quarter Kelly is even more conservative and is common when edges are noisy or correlated.

Kelly's formula returns a negative fraction whenever p * b is less than q, which is the mathematical definition of a negative-edge bet. A negative fraction means the growth-optimal action is to take the other side of the wager, not to bet a smaller amount on the losing side. Since you cannot short a sports bet or a casino game, the practical answer collapses to zero: do not bet. Casinos and sportsbooks make their living on negative-edge spots, which is why Kelly applied to a standard roulette wheel, baccarat, or a -110 sports bet without a real handicap edge always tells you to keep your money in your pocket.

Kelly assumes independent, repeated bets with a known edge and no constraints other than your bankroll. It breaks down when bets are correlated (a portfolio of long stocks all moving together is one trade, not many), when leverage is capped (you cannot stake more than your account allows even when Kelly says to), when your utility function is not logarithmic (most retirees care more about ruin avoidance than long-run growth), and when your edge estimate is unreliable. It also ignores frictions like commissions, slippage, and minimum bet sizes that erode small edges. For correlated portfolios use a multi-asset Kelly generalization or mean-variance optimization, and for retirement planning use a safe-withdrawal model rather than Kelly.

Ed Thorp learned the Kelly formula directly from Claude Shannon at MIT and used it as the bet-sizing engine behind his 1962 book Beat the Dealer. By counting cards Thorp could estimate his per-hand edge in real time, then size each bet as a fraction of his bankroll proportional to that edge. The result was the first mathematically documented winning strategy against the casino. Thorp later carried the same idea to his hedge fund Princeton Newport Partners, sizing convertible arbitrage and statistical arbitrage trades by a fractional Kelly rule. The fund returned roughly 19% annually for nearly two decades with almost no losing quarters, and Thorp's approach influenced a generation of quantitative investors including Bill Gross and Jim Simons.