How the Kelly Formula Works

## The Formula The Kelly Criterion is a staking formula derived in 1956 by John L. Kelly Jr. at Bell Labs. Its purpose: to find the stake size that maximises the long-run growth rate of a bankroll. **Kelly stake (as a fraction of bankroll) = (bp – q) / b** Where: - b = the decimal odds minus 1 (i.e. the net return per unit staked) - p = your estimated probability that the bet wins - q = 1 – p (the probability it loses) ## A Worked Example You estimate a team has a 50% chance of winning. The bookmaker offers 2.20 (decimal). - b = 2.20 – 1 = 1.20 - p = 0.50 - q = 0.50 Kelly stake = (1.20 × 0.50 – 0.50) / 1.20 = (0.60 – 0.50) / 1.20 = 0.10 / 1.20 ≈ 8.3% The formula says to stake 8.3% of your bankroll on this bet. ## Why This Maximises Growth Kelly was mathematically proven to maximise the expected logarithm of wealth. Over a long sequence of bets, no strategy produces a higher terminal bankroll in expectation. At the same time, the Kelly fraction is the threshold above which overbetting destroys expected long-run wealth. ## The Critical Input: Your Probability Estimate The formula is only as good as your estimate of p. Overestimate your edge and Kelly tells you to overbet — producing catastrophic draw-downs. Underestimate and you leave growth on the table. The quality of p is everything.