Kelly Criterion Calculator (Optimal Position Sizing)

The Kelly criterion answers a question expected value cannot: how much to stake, given an edge.

where b is the payoff ratio, p the win probability and q = 1 − p.

What makes Kelly different from naive expected-value maximisation is what it optimises. Maximising expected wealth on a repeated bet leads to staking everything, which produces the highest average outcome and near-certain ruin. Kelly maximises the expected logarithm of wealth, which is the compound growth rate of a bankroll bet repeatedly. That is the quantity a long-run investor actually experiences.

Two practical warnings follow, and both are more important than the formula.

Kelly is exquisitely sensitive to the win probability. Overestimating your edge overstakes you superlinearly. Most people who lose money using Kelly do so because their p was wrong, not because the formula was.

Full Kelly is very volatile. Drawdowns of 50% are routine at the optimal fraction. Half Kelly gives roughly three quarters of the growth with half the volatility, which is why practitioners overwhelmingly use fractional Kelly.

Edge is the expected profit per unit staked:

Full Kelly fraction is the edge divided by the payoff ratio:

Fractional Kelly simply scales that by your chosen multiple.

Expected log growth at fraction f is:

A negative f\* means the bet has negative expected value and should not be taken.

55% win rate, even money (b = 1), $100,000 bankroll:

• Edge: 1 × 0.55 − 0.45 = 10% per unit staked
• Full Kelly: 0.10 ÷ 1 = 10% of bankroll, or $10,000
• At half Kelly: $5,000
• Expected log growth at half Kelly: 0.38% per bet

50% win rate with a 2:1 payoff: edge is 2(0.5) − 0.5 = 0.5, so full Kelly is 25%. A losing win rate can still justify a large stake when the payoff is asymmetric.

45% win rate at even money: the edge is −10% and the Kelly fraction is negative. The correct stake is zero. A negative fraction is not an instruction to bet the other side unless you can actually take that side.

• Whether your probability estimate is right. This is the entire risk. Kelly assumes p is known, and in investing it never is.
• Estimation error. Because misestimating p leads to overstaking, many practitioners deliberately use a quarter or a half of Kelly as an error margin, not merely for comfort.
• Multiple simultaneous bets. Kelly for correlated positions requires solving jointly, and the single-bet formula overstates each position's size.
• Continuous rather than binary outcomes. Real investments have distributions of returns, not two branches. The continuous analogue is mean over variance.
• Transaction costs, spreads and slippage, all of which reduce the true edge.
• Bankroll that is not fully at risk. If part of your capital is committed elsewhere, the effective bankroll is smaller.
• Drawdown tolerance. Kelly is indifferent to path. A human investor is not.
• Ruin from a single catastrophic outcome, where losses can exceed the stake.

• Overestimating the edge. Kelly's output rises faster than the error in p, so optimism is punished harshly. A believed 55% that is truly 52% turns a 10% stake into an overstake.
• Using full Kelly. It maximises growth, and the drawdowns are brutal. Half Kelly captures around three quarters of the growth at roughly half the volatility.
• Applying it to correlated positions independently. Ten positions each sized at Kelly, all correlated, is effectively one enormous position.
• Treating a negative fraction as a short signal. It means do not take this bet. Reversing it only works if the opposite bet is genuinely available at those odds.
• Forgetting costs. An edge of 10% before a 2% round-trip cost is an 8% edge, and the correct stake falls accordingly.
• Ignoring that Kelly assumes you can bet repeatedly. For a one-off decision, maximising log wealth is not obviously the right objective.

• Kelly, J. L. (1956), "A New Interpretation of Information Rate", the original derivation.
• Standard formulation of the criterion as maximising expected log wealth, with the fractional Kelly adjustment widely used in practice to manage estimation error and volatility.