A bankroll-allocation formula that maximizes long-run log growth of capital.
John Kelly Jr. derived the formula in 1956 while working on signal noise at Bell Labs. The original paper was about gambling on horse races over a noisy telegraph wire, but the math generalizes to any positive-edge bet: how much of your bankroll should you stake to maximize the long-run growth rate of your capital?
The answer is the gap between your true edge and the price, divided by the price's payout odds. In its simplest form: stake fraction = (probability of winning × decimal odds − 1) / (decimal odds − 1). A 55% probability on a +100 line gives (0.55 × 2.0 − 1) / 1.0 = 10% of bankroll.
Kelly's strength is that it provably maximizes the geometric growth rate of bankroll over many bets. Bet less and you grow slower; bet more and you grow slower because the larger drawdowns punish compounding. The math is exact.
The catch is that Kelly assumes your probability estimate is correct. If you think you're 55% but you're really 52%, full Kelly will eat your bankroll. The standard professional response is fractional Kelly — quarter or half Kelly — which sacrifices some long-run growth for dramatically lower variance and lower exposure to estimation error. Most analytics-driven services that publish Kelly-style staking use fractional Kelly, both for safety and because no model's probabilities are perfect.
Our diamond unit system is a fractional-Kelly approximation tuned to our long-run hit rate. A 3-diamond pick is roughly quarter-Kelly on a 4% edge; 5-diamond is half-Kelly on a 7-plus percent edge. We don't ship full-Kelly stakes — the variance is unmanageable.