Kelly Criterion Calculator

The Kelly Criterion: A Deep Dive for Serious Bettors

What Is the Kelly Criterion?

The Kelly Criterion is an optimal bet-sizing formula that maximizes the expected logarithm of wealth over a sequence of wagers. Developed by John L. Kelly Jr. at Bell Labs in 1956, originally for information theory, not gambling, it answers a precise question: given a known edge and known odds, what fraction of your bankroll should you stake on each bet to achieve the fastest possible long-term growth rate? Unlike "maximize expected value" strategies, which ignore the risk of ruin, Kelly optimizes for geometric growth, which naturally penalizes overbetting because a single catastrophic loss can undo dozens of winning bets.

The Intuition Behind Kelly

Imagine you have a coin that lands heads 60% of the time, paying even money. Betting your entire bankroll maximizes expected value per bet but guarantees eventual ruin. One tails wipes you out. Betting 1% of your bankroll is safe but painfully slow. Kelly finds the sweet spot by maximizing E[log(wealth)], the expected logarithm of your bankroll after each bet. Logarithmic utility captures the real-world fact that doubling your money is not twice as valuable as gaining 50%, and losing 50% is far more painful than gaining 50% is pleasurable. The Kelly fraction is the unique betting size that maximizes your compound growth rate.

The Formula

For a simple win/lose bet, the Kelly fraction is:

f* = (bp - q) / b

Where b is the net odds received (decimal odds minus 1, or the profit per dollar risked), p is the probability of winning, and q = 1 - p is the probability of losing. If f* is negative, you have no edge, do not bet. If f* exceeds your comfort zone (common for large edges on short odds), fractional Kelly is the answer.

Why Full Kelly Is Too Volatile

Maximizing expected log growth does not minimize drawdowns or guarantee a particular result over a fixed number of bets. A lower fraction reduces the stake exposed on each bet. Under a small-edge approximation, half Kelly retains about 75% of optimal log growth and quarter Kelly about 44%; these are approximations, not fixed performance promises. Exact growth is nonlinear in the stake fraction. The simulation illustrates possible paths under the entered assumptions, not a forecast or a bound on future losses.

When NOT to Use Kelly

The Kelly Criterion assumes you know your true edge precisely. In practice, edge estimation is the hardest part of sports betting, and even small overestimates of your win probability lead to severe overbetting. If you think you win 55% but actually win 52%, full Kelly will have you betting roughly double what you should. Other scenarios where Kelly breaks down: correlated bets (parlays, same-game props) violate the independence assumption; small bankrolls cannot be divided finely enough to follow Kelly sizing; simultaneous bets require a portfolio version of Kelly that accounts for overlap in exposure. In all these cases, conservative fractional Kelly with a healthy margin of safety is the professional approach.

Mathematical Derivation

After one bet, your wealth is either W(1+fb) with probability p, or W(1-f) with probability q = 1-p, where f is the fraction wagered and b is the net odds. After N independent bets with w wins and l = N-w losses, your wealth is:

W_N = W₀ × (1 + fb)^w × (1 - f)^l

Taking the logarithm and dividing by N gives the average growth rate per bet:

G(f) = (w/N) × ln(1 + fb) + (l/N) × ln(1 - f)

By the Law of Large Numbers, as N grows, w/N → p and l/N → q. So the expected growth rate is G(f) = p × ln(1 + fb) + q × ln(1 - f). To maximize, take the derivative and set it to zero:

dG/df = pb/(1 + fb) - q/(1 - f) = 0

Solving: pb(1-f) = q(1+fb), which gives pb - pbf = q + qfb, so f(pb + qb) = pb - q, and finally:

f* = (pb - q) / b = (p(b+1) - 1) / b

This is the Kelly Criterion. The second derivative is negative everywhere, confirming this is a global maximum. The growth rate at this fraction is strictly positive whenever you have a genuine edge (pb > q). Staking more than the optimum lowers expected log growth and can make it negative. The zero-growth threshold depends on the probability and odds; it is not universally twice Kelly.

Common Mistakes

1. Using full Kelly with uncertain edges. The formula assumes you know your true win probability exactly. In practice, you don't. If you overestimate your edge by even 3-5 percentage points and bet full Kelly, you are overbetting, which reduces growth more than underbetting by the same margin in many common settings. Fractional sizing reduces exposure to estimation errors but does not establish an edge.

2. Ignoring the unit of the formula. Kelly outputs a fraction of your current bankroll, not a fixed dollar amount. Your bet size must change as your bankroll changes. Betting a fixed $100 because Kelly once told you 10% of $1,000 defeats the entire purpose.

3. Applying single-bet Kelly independently to simultaneous bets. Their combined exposure and joint outcomes matter. Exact portfolio sizing requires a model of those joint outcomes, including correlations; a lower fraction alone does not model them.

4. Confusing edge with stake size. For a binary win/lose bet with valid decimal odds above 1, expected profit per unit is pb − q and uncapped Kelly is (pb − q) / b. Since b > 0, their signs agree. Positive EV therefore means positive uncapped Kelly under this model. Stake caps and rounding may still reduce the displayed amount.

5. Chasing losses with increased Kelly fractions. After a drawdown, Kelly automatically reduces your bet size (because your bankroll shrinks). Fighting this by keeping bet sizes constant or increasing them is the fastest path to ruin.

Worked Example

You believe a team has a 55% chance to win at -110 American odds. The decimal odds are 1.909, so b = 0.909 (net odds). Your edge: p = 0.55, q = 0.45.

f* = (0.909 × 0.55 - 0.45) / 0.909 = (0.500 - 0.45) / 0.909 = 0.055

Full Kelly says bet 5.5% of your bankroll. On a $10,000 bankroll, that is $550. At half Kelly, you bet $275. At quarter Kelly, $137.50. The expected profit per unit staked is 5%; that edge is distinct from the 5.5% full Kelly bankroll fraction. These values assume the estimated 55% win probability is accurate. They illustrate sizing, not evidence that this probability estimate is correct or that the bet will win.

Method and source

The single-bet calculation uses f = (p × b − (1 − p)) / b, where b is decimal odds minus one. For a 55% estimated win probability at decimal odds 2.00, full Kelly is 10% of bankroll: $100 from $1,000. Half Kelly is $50. This assumes the probability estimate is correct and excludes fees and correlated exposure. A positive fraction does not guarantee a winning bet or prevent drawdowns.

Primary source: J. L. Kelly Jr., A New Interpretation of Information Rate (1956). The calculator applies the binary-outcome formula; it does not estimate your true win probability.

Related Tools

Verify your edge before sizing with the EV Calculator, model long-term bankroll trajectories with the Bankroll Calculator, or explore alternative sizing systems with the Staking Plan Calculator.