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

Full Kelly is mathematically optimal in the long run, but "long run" means thousands of bets. In the short run, full Kelly produces stomach-churning drawdowns. The expected maximum drawdown of full Kelly betting is roughly 50% of your peak bankroll at some point in any extended sequence. That means even with a real, verified edge, you should expect your bankroll to halve before recovering. Most humans, and most bankrolls, cannot survive that. This is why fractional Kelly exists. Half Kelly (f*/2) achieves approximately 75% of the full Kelly growth rate while cutting maximum expected drawdown roughly in half. Quarter Kelly (f*/4) is even more conservative: about 56% of the growth rate but with dramatically smoother equity curves. The trade-off between growth rate and volatility is nearly linear in the fraction, making it easy to tune to your risk tolerance.

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), and overbetting beyond 2f* actually makes your bankroll shrink in the long run, a result that surprises many.

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. This asymmetry is why half Kelly is the professional default.

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 Kelly to correlated bets. If you bet Kelly on Game A and also Kelly on Game B, and both games are on the same day, your total exposure exceeds Kelly. Simultaneous Kelly requires a multivariate version (the Kelly portfolio) that accounts for correlations. Most bettors handle this by reducing to quarter or eighth Kelly when placing multiple bets.

4. Confusing positive EV with positive Kelly. A bet can have positive expected value but negative Kelly fraction if the odds are extremely short and your edge is tiny. Kelly protects you from bets where the edge-to-risk ratio is too poor to warrant capital allocation.

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. Most sharp bettors with moderate confidence in their model would choose half Kelly here: the 5.5% edge at -110 is a strong signal but not so large that full Kelly is warranted. If your edge estimate is uncertain (say, you think the true probability is somewhere between 53% and 57%), quarter Kelly provides insurance against the downside scenario.

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.