Formula, in Risk & Portfolio Construction
Kelly Criterion Formula
The Kelly criterion gives the fraction of capital to stake that maximizes the expected logarithm of wealth, which maximizes long-run compound growth. For a bet with win probability p and net odds b to 1, the optimal fraction is f = (bp - q) / b, where q is the loss probability. Betting more than Kelly raises growth volatility while lowering growth; betting less is safer but slower.
4 variables. Published May 26, 2026. By AI Fin Hub Research.
f* = (b x p - q) / b = p - q/b
where q = 1 - p Variables
f*
Optimal fraction
The share of current capital to wager. It maximizes the expected log growth rate of wealth. A negative f* means the bet has negative edge and should not be taken.
p
Win probability
Probability the bet wins. In trading this is the estimated hit rate of the setup. Kelly is exquisitely sensitive to errors in p, which is the main practical caution.
q
Loss probability
Probability the bet loses, equal to 1 - p for a binary outcome.
b
Net odds (payoff ratio)
Profit per unit staked if the bet wins, the win/loss payoff ratio. Odds of b to 1 mean a winning 1-unit bet returns b in profit. In trading b is the average win divided by the average loss.
Step by step
- 1
Estimate the win probability p and the loss probability q = 1 - p.
A setup wins 55% of the time: p = 0.55, q = 0.45.
- 2
Estimate the net payoff odds b, the ratio of average win to average loss.
Average win is 1.5x the average loss, so b = 1.5.
- 3
Apply the formula f* = p - q/b.
0.55 - 0.45/1.5 = 0.55 - 0.30 = 0.25.
- 4
Stake that fraction of capital, recognizing full Kelly is aggressive and most practitioners use a fraction of it.
Full Kelly says stake 25% of capital; many traders would deploy a quarter or half of that.
Worked example
Sizing a trade with a 55% hit rate and 1.5:1 reward-to-risk
Win probability p
0.55
Loss probability q
0.45
Net odds b
1.5
f* = p - q/b = 0.55 - 0.45/1.5 = 0.55 - 0.30 = 0.25. Equivalently f* = (b x p - q)/b = (1.5 x 0.55 - 0.45)/1.5 = (0.825 - 0.45)/1.5 = 0.375/1.5 = 0.25.
Full Kelly fraction of 25%. Staking a quarter of capital per trade maximizes long-run log growth under these exact inputs, but it tolerates brutal drawdowns: a string of losses near full Kelly can cut capital in half. Because p and b are estimated, the real edge is uncertain, so most practitioners apply fractional Kelly to protect against overbetting a mismeasured edge.
Common variations
Run the numbers next
- Calculators Position Sizing under Edge Variance
Bayesian-Kelly bet sizing when your edge is itself uncertain. Compare deterministic Kelly, Bayesian-adjusted, and conservative lower-bound versions.
- Calculators Risk-Adjusted Returns Calculator
Paste a returns CSV. Sharpe, Sortino, Calmar, Omega, alpha, beta, tracking error, information ratio, max drawdown, and tail moments — plus.
Sources
- A New Interpretation of Information Rate , John L. Kelly Jr., Bell System Technical Journal (1956)
- The Kelly Capital Growth Investment Criterion , MacLean, Thorp, Ziemba (eds.), World Scientific (2011)
Keep the topic connected
- Formula Fractional Kelly Formula
The fractional Kelly formula stakes a fixed multiple of full Kelly to cut volatility for a small growth cost. Half-Kelly is the textbook benchmark; quarter-Kelly is the more common default when edge estimates are uncertain.
- Formula Expectancy Formula
The expectancy formula: win rate times average win minus loss rate times average loss. The expected profit per trade in R-multiples, with an example.
- Formula Profit Factor Formula
The profit factor formula: gross profit divided by gross loss. A single number summarizing a strategy's edge, what values mean, with a worked example.
- Learn Kelly Criterion
What the Kelly criterion is, when full Kelly blows up, and why most working quants size at half- or quarter-Kelly.