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Key Takeaways
- The Kelly criterion sizes a bet to maximize the expected long-run geometric growth rate of capital, not the expected profit of any single trade.
- For a binary bet with net win R times the stake and loss of the stake, f* = W − (1 − W) ÷ R.
- Evaluate growth with g(f) = W ln(1 + Rf) + (1 − W) ln(1 − f). The optimum is the peak of this function within the permitted allocation range.
- Fractional Kelly multiplies the model’s full fraction by a chosen scale, such as one-half or one-quarter.
- Use the complete payoff distribution and portfolio constraints when the position has more than two outcomes.
What is the Kelly criterion?
The Kelly criterion is a position-sizing rule that chooses the fraction of capital to stake by maximizing expected logarithmic wealth under a specified payoff model. For repeated independent identical bets, that choice maximizes the long-run compound growth rate. For a binary bet with win probability W that gains R times the stake or loses the stake, the unconstrained optimal fraction is W − (1 − W) ÷ R.
Formula
f* = W − (1 − W) / R- f* is the fraction of capital staked in this binary model, where the losing outcome forfeits that entire stake.
- W is the probability of a winning outcome, expressed as a decimal.
- R is the fixed net gain per unit staked in the winning outcome. W is between 0 and 1, and R is positive.
- For a nonnegative stake, use zero when the unconstrained fraction is negative. For a general position, maximize expected log wealth using its actual returns and permitted allocations.
Kelly criterion example: full and half Kelly
Use a binary model with a 55% winning probability. A win earns twice the stake as net profit, while a loss forfeits the stake.
- Win probability W = 0.55; net winning payoff R = 2
- Full fraction: 0.55 − 0.45 ÷ 2 = 0.325, or 32.5%
- Half fraction: 0.325 ÷ 2 = 0.1625, or 16.25%
- Expected log growth: 0.0986 at full size and 0.0750 at half size
For this binary model, the 32.5% stake maximizes expected log growth. The 16.25% half stake retains 76.1% of that growth rate. Compare those model outputs with the selected portfolio allocation limits.
At a 65% stake, expected log growth is −0.0143. The positive zero-growth boundary is 63.1%, below twice the optimal fraction. Use the actual growth function to locate the boundary.
How the winning payoff changes the Kelly fraction
Holding W fixed, increasing R increases the Kelly fraction: (1 − W) ÷ R becomes smaller. At W = 0.55, R = 1 gives 10%, R = 2 gives 32.5% and R = 3 gives 40%.
For a nonnegative stake, constrain a negative unconstrained result to zero. If considering a reverse trade, calculate that trade’s own payoff distribution.
Kelly sizing with actual position returns
The binary stake fraction is not automatically the fraction of capital invested in a security. If the two position returns are +B and −A, the unconstrained invested fraction is W ÷ A − (1 − W) ÷ B, with A and B positive and terminal wealth kept positive.
For more than two outcomes, maximize the probability-weighted log of terminal wealth across the complete distribution. Average winning and losing amounts alone do not determine that objective. Options outcomes come from the Options Profit Calculator, which shows the payoff at every price on your date and the maximum loss; weight them with your probability model to set the Kelly fraction.
Kelly sizing for several positions at once
For simultaneous trades, use their joint returns and shared capital constraints. Apply the selected fractional scale and portfolio limits to the resulting allocation.
A log-wealth objective also applies to a single allocation decision. Repetition supplies the long-run growth interpretation, rather than being required for the objective to exist.
Record the chosen capital fraction, trade quantity and completed result. Compare the actual portfolio path with the selected allocation process.
Limitations of the Kelly criterion
- Size from a conservative estimate of the edge. Overstating the probability or payoff pushes the stake above the true optimum, where growth falls, and in the example growth turns negative past 63.1% of capital.
- Scale down with fractional Kelly. Full Kelly produces large swings in wealth; in the example half Kelly keeps 76.1% of the growth rate, but halving the stake does not mechanically halve compounded drawdown.
- The binary formula fits only two-outcome bets that lose the full stake. Real positions need their complete payoff distribution.
- Size correlated and overlapping trades from their joint returns. The long-run growth reading assumes repeated, independent, identical bets, and changing odds, correlated trades and shared capital limits all change the answer.
Bottom line
The Kelly criterion turns an edge and a payoff into the stake that maximizes expected log wealth and, over repeated independent identical bets, long-run compound growth. Treat the full fraction as a ceiling, not a target: model the complete payoff distribution, size overlapping trades from their joint returns, and scale down with fractional Kelly to allow for estimation error.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- Kelly, J.L. (1956): A New Interpretation of Information Rate (Bell System Technical Journal)
- CFA Institute: The Kelly Criterion, You Don’t Know the Half of It
- Thorp, E.O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market
How entries are sourced, calculated and updated: editorial methodology.