Enter Values
Kelly Criterion Formulas
f* = p - q/b
f* = Kelly fraction |
p = Win prob |
q = 1-p |
b = Win/loss ratio
Leverage Warning:
Kelly exceeds 100% - this implies using leverage.
Negative Edge:
Kelly is negative - the math says don't bet or invest in this position.
Kelly Output
Full Kelly Fraction
10.00%
Conservative
Half Kelly
5.00%
Quarter Kelly
2.50%
Stake Size
$1,000
Log-Growth Per Bet
0.50%
Position Size Comparison
Formula Breakdown
f* = p - q/b (where q = 1 - p)
Model Assumptions
- True probabilities/returns known with certainty
- Independent trials with constant parameters
- Fractional betting allowed, no transaction costs
- All winnings reinvested
- Binary outcomes (win b x stake or lose stake)
- No shorting - negative Kelly means don't bet
Educational only. In practice, parameter estimation error is substantial. Most practitioners use Half Kelly or less.
Understanding the Kelly Criterion
What is the Kelly Criterion?
The Kelly Criterion is a mathematical formula for optimal position sizing that maximizes the expected long-term growth rate of your capital. Developed by John Kelly at Bell Labs in 1956 for information theory, it was later adopted by gamblers and investors to determine how much to bet or invest given a known edge.
Kelly Criterion Formulas
Discrete (Gambling): f* = p - q/b where q = 1-p
Continuous (Investment): f* = (mu - r) / sigma2
f* = optimal fraction of bankroll/portfolio
Continuous (Investment): f* = (mu - r) / sigma2
f* = optimal fraction of bankroll/portfolio
Fractional Kelly
Half Kelly
50% of full Kelly
Retains ~75% of expected growth while cutting volatility roughly in half. The most popular choice among practitioners.
Quarter Kelly
25% of full Kelly
Retains ~44% of expected growth with ~75% less volatility. Very conservative, good for uncertain estimates.
Key Warnings
- Negative Kelly (f* < 0): The math says don't bet. Your expected value is negative.
- Leveraged Kelly (f* > 100%): Implies borrowing to bet more than your bankroll. Extremely risky.
- Estimation Error: Kelly assumes you know the true probabilities. In practice, you don't.
Important: The Kelly Criterion maximizes long-term growth, not short-term profits. It can result in large drawdowns (50%+ losses are common at full Kelly). Most practitioners use Half Kelly or less to account for estimation error and psychological tolerance.
Frequently Asked Questions
The Kelly Criterion is a mathematical formula for optimal position sizing that maximizes the expected long-term growth rate of your capital. Developed by John Kelly at Bell Labs in 1956, it tells you what fraction of your bankroll to bet (or portfolio to allocate) given your edge and the odds. Unlike fixed-fraction betting, Kelly dynamically adjusts position size based on how favorable the opportunity is - betting more when you have a bigger edge and less when the edge is slim.
The Discrete Kelly formula (f* = p - q/b) applies to binary-outcome bets where you either win a fixed amount or lose your entire stake - like sports betting, blackjack, or coin flips. The Continuous Kelly formula (f* = (mu-r)/sigma squared) applies to investments with log-normally distributed price paths (geometric Brownian motion), where you're deciding what fraction of your portfolio to allocate to a risky asset versus a risk-free asset. Both formulas optimize the same objective - maximizing expected log-wealth growth - but for different return distributions.
Full Kelly maximizes expected long-term growth but comes with extreme volatility - drawdowns of 50% or more are common. Most practitioners use Half Kelly (50% of full Kelly) or Quarter Kelly (25%) because: (1) it dramatically reduces volatility with only modest impact on growth - in the continuous GBM model, Half Kelly retains 75% of expected excess growth while cutting volatility roughly in half; (2) parameter estimates are never perfect, and betting less protects against estimation error; (3) human psychology cannot tolerate the large swings that full Kelly produces.
A negative Kelly result means you have a negative expected value - the odds are against you. In gambling terms, "the math says don't bet." In investment terms, your expected return is less than the risk-free rate, so you should hold cash or bonds instead. This calculator clamps dollar amounts and growth to $0 for negative Kelly since shorting is not modeled. In theory, the magnitude of a negative Kelly indicates how much you would "short" if shorting were possible and costless.
A Kelly fraction above 100% implies using leverage - borrowing money to invest more than your total capital. For example, Kelly = 250% means investing $2.50 for every $1.00 you have, which requires borrowing $1.50. This only occurs in Continuous (investment) mode. While mathematically optimal under ideal conditions, leveraged Kelly is extremely risky in practice because: (1) borrowing costs reduce returns; (2) margin calls can force liquidation at the worst times; (3) estimation errors are magnified. Most practitioners cap their Kelly at 100% or use fractional Kelly.
The Kelly Criterion assumes you know the true probabilities and expected returns - in reality, these must be estimated and are often wrong. Other limitations include: (1) it ignores transaction costs, taxes, and borrowing costs; (2) it assumes you can rebalance continuously and costlessly; (3) it maximizes long-term growth, not utility - it doesn't account for your personal risk tolerance; (4) it assumes infinite time horizon, but real investors have finite horizons; (5) extreme Kelly fractions (>50%) produce intolerable drawdowns for most people. For these reasons, Kelly is best used as an upper bound, with actual position sizes scaled down significantly.
Disclaimer
This calculator is for educational purposes only. The Kelly Criterion assumes you know the true probabilities with certainty, which is rarely the case in practice. Actual betting and investing involve substantial estimation error. Most practitioners use Half Kelly or less. This tool should not be used for actual betting or trading decisions without understanding its limitations.
