The Kelly Criterion is the mathematically optimal bet sizing formula. It maximizes long-run wealth growth while avoiding ruin. Most traders either ignore it or misapply it. Here is how to use it correctly.
John Kelly Jr. published his criterion in 1956 while working at Bell Labs. The formula answers a specific question: given a bet with known probability and payoff, what fraction of your capital should you wager to maximize the expected logarithm of wealth?
Maximizing expected log wealth is equivalent to maximizing long-run geometric growth rate. This is the right objective for any investor with a long time horizon — not maximizing expected value (which ignores risk of ruin) and not minimizing variance (which sacrifices too much return).
Kelly Formula (simple binary case)
f* = (bp - q) / b
f* = fraction of capital to bet
b = net odds (profit per unit wagered)
p = probability of winning
q = probability of losing (1 - p)
For a coin flip with 60% win probability and 1:1 payoff: f* = (1 × 0.6 - 0.4) / 1 = 0.20. Bet 20% of your capital each flip. This is the fraction that maximizes long-run growth. Bet more and you grow slower (or go broke). Bet less and you also grow slower.
In theory, full Kelly is optimal. In practice, it requires perfectly accurate probability estimates. Real trading strategies have estimation error — your win rate is not exactly 60%, your edge is not exactly what you think it is. When you overestimate your edge and bet full Kelly, the drawdowns are severe.
The standard practitioner approach is to use fractional Kelly — typically half Kelly (f*/2) or quarter Kelly (f*/4). Half Kelly gives you about 75% of the geometric growth rate of full Kelly with dramatically reduced variance. The drawdowns are smaller, the ride is smoother, and the penalty for estimation error is much lower.
Warren Buffett, Ed Thorp, and Renaissance Technologies all use Kelly-based position sizing, but none of them use full Kelly. Thorp, who literally wrote the book on blackjack card counting and later applied Kelly to markets, used quarter to half Kelly in practice.
The single-asset Kelly formula does not generalize directly to portfolios. For multiple simultaneous bets, you need the continuous Kelly criterion, which involves solving a quadratic optimization problem using the expected returns vector and the covariance matrix of returns.
The result is a vector of optimal position sizes — the fraction of capital to allocate to each asset — that maximizes the expected log return of the portfolio. This is closely related to mean-variance optimization (Markowitz) but with a specific utility function (log wealth) rather than an arbitrary risk aversion parameter.
The Kelly Criterion endpoint in the MainState Labs Finance API handles both the single-asset and multi-asset cases. For multi-asset portfolios, it accepts expected returns and a covariance matrix and returns optimal allocations, the expected geometric growth rate, and the maximum drawdown estimate.
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