What it is
The Kelly criterion, developed by John Kelly at Bell Labs in 1956, is a formula for determining the optimal size of a bet or investment to maximize the long-run growth rate of capital. Betting the Kelly fraction maximizes expected logarithmic utility, which is equivalent to maximizing the geometric mean return over many repetitions.
The criterion solves a fundamental problem: expected value maximization suggests betting your entire bankroll on any positive expected value bet, but doing so leads to ruin with probability one in the long run (because even a single loss is catastrophic). The Kelly criterion finds the sizing that avoids ruin while maximizing growth.
The formula
For a simple binary bet with probability p of winning and probability q = 1-p of losing, where a win returns b times the stake:
f* = (bp - q) / b
Where f* is the fraction of capital to bet.
Example: a coin flip with 60% probability of winning and 40% of losing, paying 2:1 (win $2, lose $1):
f* = (2 × 0.60 - 0.40) / 2 = (1.20 - 0.40) / 2 = 0.80 / 2 = 0.40
Optimal bet: 40% of capital per flip.
At this fraction, the geometric growth rate is maximized. At a higher fraction (say, 60%), growth is slower despite higher expected value per bet — because large losses compound poorly. At a lower fraction (say, 20%), growth is also slower because the edge is underutilized.
Why overbetting destroys compounding
The Kelly criterion maximizes the geometric mean, not the arithmetic mean. The geometric mean is what determines actual wealth after many repetitions — it accounts for the compounding of gains and losses.
Losing 50% requires a 100% gain to recover. Losing 25% requires a 33% gain. The asymmetry of percentage gains and losses means that large losses disproportionately harm long-run growth. A strategy with positive expected value can destroy wealth in the long run if it bets too large: the occasional catastrophic loss offsets many smaller wins.
At exactly 2× Kelly, the expected long-run return is zero — a strategy that seems clearly profitable on a per-bet basis generates zero growth in the long run because the variance is too high. Beyond 2× Kelly, the expected return is negative and ruin is certain.
Kelly in practice: concentration vs. diversification
Warren Buffett's concentrated portfolio of high-conviction investments is often cited as Kelly-inspired. The logic: if you have genuine edge — a reason to believe your expected return from a given investment is higher than the market's expected return — Kelly says you should size the bet proportionally to that edge. Small edge → small bet. Large, high-confidence edge → large bet.
Index fund investors implicitly bet at roughly equal weights, regardless of any view on individual expected returns. Kelly would say this is suboptimal if you have genuine informational advantages — but most investors do not have such advantages, which is why index investing is rational for them.
One thing most people get wrong
Kelly assumes you know your edge — the precise probability of winning and the payoff ratio — with certainty. In practice, you almost never do. Your estimate of the probability is uncertain. Your payoff estimate is uncertain. Your model of the situation may be wrong.
When your edge estimate is uncertain, the mathematically correct response is to bet less than Kelly — sometimes called fractional Kelly (e.g., half-Kelly). If your true edge is 0.10 but your estimated edge is 0.20 due to model error, you will bet at the Kelly fraction for 0.20 but receive returns as if the true edge were 0.10 — which means significant overbetting. Practitioners routinely use half-Kelly or quarter-Kelly precisely because the downside of overestimating edge is catastrophic, while the cost of underestimating edge is merely suboptimal growth. The asymmetry of consequences, under uncertainty, recommends conservative fractional Kelly.