Kelly criterion: a sizing formula with demanding assumptions
Kelly sizing maximizes expected logarithmic growth in a specified model. Its output is sensitive to probability errors and is not a personal spending recommendation or a guarantee against loss.
Define the binary model
Let p be the estimated probability of winning and d the decimal odds. With a fraction f of bankroll risked, a win multiplies bankroll by 1 + f(d − 1), and a loss multiplies it by 1 − f. The model chooses f to maximize p × log(1 + f(d − 1)) + (1 − p) × log(1 − f).
For the simple no-push, no-fee case, the interior solution is f = (p × d − 1) ÷ (d − 1). CashCord’s calculator clips a negative result to zero and allows a chosen fraction of the full result.
Work through a hypothetical input
At decimal 2.00 and an assumed 55% win chance, full Kelly is (0.55 × 2 − 1) ÷ 1 = 10%. On a hypothetical $500 amount, that is $50. Half Kelly scales the result to 5%, or $25. Neither amount is being recommended.
| Assumed chance | Full Kelly | Half Kelly on $500 |
|---|---|---|
| 50% | 0% | $0 |
| 52% | 4% | $10 |
| 55% | 10% | $25 |
The probability estimate carries the risk
If the true win chance were 50%, the 55% input would invent an advantage. Using a smaller fraction reduces exposure but does not turn an incorrect estimate into a real edge. A model output with many decimal places can still be poorly calibrated.
At actual 50/50 odds of 2.00, risking 10% on each of one win and one loss changes bankroll by 1.10 × 0.90 = 0.99: a 1% loss. This illustrates why expected logarithmic growth and a simple count of wins are different concepts.
Do not add individual recommendations into a portfolio
Several simultaneous bets share the same available money and may be correlated. Computing full Kelly separately for each and adding the allocations does not solve the combined portfolio problem. The single-bet formula also omits pushes, uncertain probabilities, fees, changing limits and personal affordability.
The calculator does not enforce an affordable budget or forecast drawdown. Its zero output means no positive allocation under the entered binary model, not that an alternative bet should be found or that a different input should be forced.
Try the numbers yourself.
Open the calculator with this guide’s example inputs. Change one assumption at a time and see what changes.
Open kelly criterion calculatorCommon questions
Does half Kelly make a bad estimate safe?
No. It scales exposure but preserves the same probability assumption. An error can still produce losses.
Is Kelly a guarantee of maximum cash profit?
No. Its objective is expected log growth within a model, with specific assumptions. It does not guarantee a realized outcome or satisfy every person’s risk constraints.
Sources and example scope
The source describes expected-logarithmic-growth optimization. The derivation, sizing examples and sensitivity cases here are original illustrations of the implemented binary formula.