Kelly Criterion: formula
J. L. Kelly Jr. (1956): A New Interpretation of Information Rate
Kelly Criterion: worked example
| Chance of winning | 55.0% |
|---|---|
| Net profit per $1 risked (b) | 1 |
| Bankroll | $10,000 |
| Kelly fraction to risk | 10.0% |
| Amount to risk at full Kelly | $1,000 |
| Amount to risk at half Kelly | $500 |
The result depends on the entered win probability and net payoff. Parameter error, correlated outcomes, changing payouts, taxes and transaction costs are not modeled. The fraction does not set a maximum drawdown or protect a stake from a losing sequence.
Kelly Criterion: changing Chance of winning
| Chance of winning | Kelly fraction to risk | Amount to risk at full Kelly | Amount to risk at half Kelly |
|---|---|---|---|
| 51.0% | 2.0% | $200 | $100 |
| 53.0% | 6.0% | $600 | $300 |
| 55.0% (example) | 10.0% | $1,000 | $500 |
| 57.0% | 14.0% | $1,400 | $700 |
| 59.0% | 18.0% | $1,800 | $900 |
Kelly Criterion: compare input ranges
| Input | Tested from | To | Kelly fraction to risk: low input | Kelly fraction to risk: high input | Swing |
|---|---|---|---|---|---|
| Net profit per $1 risked (b) | 0.9 | 1.1 | 5.0% | 14.1% | 9.1% (91%) |
| Chance of winning | 53.0% | 57.0% | 6.0% | 14.0% | 8.0% (80%) |
| Bankroll | $9,000 | $11,000 | 10.0% | 10.0% | 0.000% |
Kelly Criterion: individual input comparisons
| Input | Value | Kelly fraction to risk | Amount to risk at full Kelly | Amount to risk at half Kelly |
|---|---|---|---|---|
| Chance of winning | 51.0% | 2.0% | $200 | $100 |
| 53.0% | 6.0% | $600 | $300 | |
| 55.0% | 10.0% | $1,000 | $500 | |
| 57.0% | 14.0% | $1,400 | $700 | |
| 59.0% | 18.0% | $1,800 | $900 | |
| Net profit per $1 risked (b) | 0.8 | -1.2% | $0.00 | $0.00 |
| 0.9 | 5.0% | $500 | $250 | |
| 1 | 10.0% | $1,000 | $500 | |
| 1.1 | 14.1% | $1,409 | $705 | |
| 1.2 | 17.5% | $1,750 | $875 | |
| Bankroll | $8,000 | 10.0% | $800 | $400 |
| $9,000 | 10.0% | $900 | $450 | |
| $10,000 | 10.0% | $1,000 | $500 | |
| $11,000 | 10.0% | $1,100 | $550 | |
| $12,000 | 10.0% | $1,200 | $600 |
Kelly Criterion: combined input changes
| Chance of winning / Net profit per $1 risked (b) | 0.8 | 1 | 1.2 |
|---|---|---|---|
| 51.0% | -10.2% | 2.0% | 10.2% |
| 53.0% | -5.7% | 6.0% | 13.8% |
| 55.0% | -1.2% | 10.0% | 17.5% |
| 57.0% | 3.2% | 14.0% | 21.2% |
| 59.0% | 7.8% | 18.0% | 24.8% |
Kelly Criterion: which assumptions move the result?
Kelly Criterion: values and definitions
| Quantity and definition | Value used |
|---|---|
| Chance of winning | 55.0% |
| Net profit per $1 risked (b) — 1 means you win $1 for each $1 you risk. | 1 |
| Bankroll | $10,000 |
| Kelly fraction to risk | 10.0% |
| Amount to risk at full Kelly | $1,000 |
| Amount to risk at half Kelly | $500 |
Winning probability is the estimated chance of winning. Payoff odds give net profit per dollar risked. Winning $1 for each $1 staked means b = 1.
The estimate of probability is the hard part, since any error is magnified in the result. Bankroll is the capital the fraction applies to. Use conservative probabilities backed by a track record, not hopes.
Common error: Treating the fraction as a guarantee. Kelly maximizes growth on average and still allows deep losses along the way.
Investing & Returns calculators: Net Present Value (NPV) Calculator, Continuous Compounding Calculator, Compound Interest & Future Value Calculator, Investment Return Calculator, CAGR and Required Annual Return Calculator, Present Value Calculator.
A 10% stake at even-money odds multiplies wealth by 1.1 after a win and 0.9 after a loss. Under the assumed 55% win probability, expected log growth is 0.55 ln(1.1) + 0.45 ln(0.9), about 0.00501 per bet. At a true 50% win probability, the same stake instead gives about −0.00503.
An overstated probability can therefore turn the modeled growth advantage into a loss. Half Kelly reduces the stake, but does not supply evidence for the probability estimate.
Estimate the win probability. It is an estimate, and the result is only as good as that estimate.
Use the position size and risk-reward calculators to put the fraction into a trade, and the maximum drawdown page to see what a losing run does to the account. The option profit calculator covers payoffs for options trades. Treat Kelly as an upper limit, not a target.
Why use half Kelly?
Fractional Kelly reduces the modeled stake. Its exact expected log-growth and loss distribution depend on the payoff probabilities and outcomes; it does not guarantee protection when the estimated edge is wrong.
What does a negative Kelly fraction mean?
Your bet has negative expected value, so the formula says not to bet at all. It signals that the odds are against you.
Can I use Kelly for stocks?
Only loosely, since stock returns are not simple win-lose bets and probabilities are unknown. It is better as a way to think about sizing than a precise rule.