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.
Formula
Copy the exact expression or work through it step by step below.
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
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Sources & References
- 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)
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