Kelly Criterion: The Complete Guide to Optimal Bet Sizing
The Kelly Criterion is the gold standard for bet sizing in sports betting and investing. It answers the question every bettor eventually faces: given a bankroll of size B and a bet with known odds and estimated probability, what fraction of B should I risk to maximize my long-term growth rate? This guide covers everything from the original 1956 formula and the logic behind it, through fractional Kelly and its variance tradeoffs, drawdown expectations, multi-bet portfolios, the mistakes that blow up bankrolls, and the situations where Kelly is the wrong tool entirely.
The Original Kelly Formula
John L. Kelly Jr., a Bell Labs researcher, published his criterion in 1956 as a solution to a problem in information theory: given a noisy communication channel with known signal and noise characteristics, what bet size maximizes the long-term growth rate of capital? The same mathematics apply directly to sports betting, where the "signal" is the bettor's edge and the "noise" is the variance of individual outcomes. Edward Thorp later carried the idea from blackjack tables to Wall Street, and it has been the backbone of quantitative stake sizing ever since.
The full Kelly formula is f* = (bp − q) / b, where f* is the fraction of bankroll to bet, b is the net decimal odds minus one (the profit per dollar wagered on a win), p is the estimated true probability of winning, and q is the probability of losing (1 − p). When b = 1 (even money, decimal odds of 2.00), this simplifies to f* = 2p − 1, or the edge itself.
For a bet at +150 American (decimal 2.50) that the bettor estimates has a 50% chance of winning: b = 1.50, p = 0.50, q = 0.50. The Kelly stake is (1.50 × 0.50 − 0.50) / 1.50 = 0.25 / 1.50 = 0.1667, or 16.67% of bankroll. A $10,000 bankroll would place a $1,667 bet. This is the mathematically optimal bet size for maximizing the long-term growth rate of the bankroll, assuming the probability estimate is exactly correct. You can run any odds-and-probability combination through our Kelly Criterion Calculator to get full, half, and quarter Kelly stakes instantly.
One property deserves emphasis before anything else: if bp − q ≤ 0, the formula outputs zero or a negative number, and the correct stake is nothing. Kelly is not just a sizing tool — it is a filter. It refuses to stake money on bets without a positive edge, which is why it pairs naturally with an expected value calculation: EV tells you whether a bet is worth making, Kelly tells you how much.
The Kelly Formula
f* = (bp − q) / b f* = fraction of bankroll to stake b = net decimal odds (decimal odds − 1) p = your estimated win probability q = 1 − p (probability of losing) Even-money shortcut (b = 1): f* = 2p − 1 (your raw edge) Fractional Kelly (fraction c of full): stake = c × f* (c = 0.5 half, 0.25 quarter) No bet condition: if bp − q ≤ 0 → stake = 0
Where the Formula Comes From: Maximizing Log Wealth
The derivation is worth understanding because it explains both why Kelly works and why over-betting is catastrophic. Betting is a multiplicative process: each result scales your bankroll by a factor, and those factors compound. After a win at net odds b with fraction f staked, your bankroll is multiplied by (1 + bf); after a loss, by (1 − f). Long-run wealth is the product of all those factors — which means what matters is not the arithmetic average of outcomes but the geometric average.
Here is the trap the arithmetic average sets. Staking your entire bankroll on every positive-EV bet maximizes expected wealth after any fixed number of bets — on paper. But the strategy loses everything the first time a single bet fails, and with repeated betting that first failure arrives with probability approaching one. Expected wealth is dragged upward by vanishingly unlikely jackpot paths while the typical path goes to zero. To describe what happens to the typical bettor, you must average the logarithm of the growth factors instead.
So Kelly maximizes the expected log growth per bet: g(f) = p·ln(1 + bf) + q·ln(1 − f). Take the derivative with respect to f, set it to zero, and solve: pb/(1 + bf) = q/(1 − f), which rearranges to f* = (bp − q)/b — the formula above. Because g(f) is a concave curve with a single peak at f*, two facts follow immediately. First, betting slightly less than f* costs very little growth (the curve is flat near the peak). Second, betting more than f* is punished increasingly hard: in the continuous approximation, staking exactly double Kelly drives long-run growth to zero, and anything beyond double Kelly turns a genuinely winning edge into a strategy that grinds toward ruin. Over-betting is not a bolder version of Kelly — it is a mathematically different, losing regime.
This asymmetry — under-betting is cheap, over-betting is fatal — is the single most important practical lesson of the entire framework, and it motivates everything in the next section.
Full, Half, and Quarter Kelly: The Variance Tradeoff
The full Kelly formula produces stakes that most recreational and even professional bettors find uncomfortably large. A 16.67% stake means losing five such bets in a row cuts a $10,000 bankroll to about $4,000 — a 60% drawdown. While full Kelly grows wealth faster than any other fixed-fraction strategy in the long run, the short-term variance is extreme enough that most bettors cannot withstand the psychological pressure, and bettors who abandon a staking plan mid-drawdown usually do so at the worst possible moment.
More importantly, the formula assumes your probability estimates are perfectly accurate. In practice every estimate contains error, and the error compounds dangerously: a bettor who estimates p = 0.60 when the true probability is 0.55 will stake roughly double the correct Kelly fraction, because the calculation treats the estimate as ground truth. Recall from the derivation that double Kelly earns approximately zero growth. Overconfidence plus full Kelly is the fastest path to bankruptcy in sports betting — not because the math is wrong, but because the inputs are.
This is why professionals almost universally bet fractional Kelly — a fixed fraction c of the full recommendation, typically half (c = 0.5) or quarter (c = 0.25). The tradeoff is remarkably favorable. In the standard continuous approximation, betting fraction c of Kelly delivers a share 2c − c² of the maximum growth rate while scaling bankroll volatility down by the factor c. Half Kelly keeps 75% of the growth at half the volatility; quarter Kelly keeps about 44% of the growth at a quarter of the volatility. And fractional Kelly doubles as insurance against estimation error: if your true edge is only half what you believe, half Kelly on your estimate is exactly full Kelly on reality.
| Strategy | Stake (× full Kelly) | Share of max growth rate | Bankroll volatility | P(ever hitting a 50% drawdown) |
|---|---|---|---|---|
| Full Kelly | 1.00× | 100% | 100% | ≈ 50% |
| Half Kelly | 0.50× | 75% | 50% | ≈ 12.5% |
| Quarter Kelly | 0.25× | ≈ 44% | 25% | ≈ 0.8% |
Growth and drawdown figures use the continuous-time approximation popularized by Edward Thorp; discrete real-world betting deviates somewhat, and all figures assume your probability estimates are accurate.
A quarter Kelly bet on the +150 example above would be $417 instead of $1,667 — small enough to survive any realistic losing streak, while still scaling stakes proportionally to edge, which is the property that makes Kelly better than flat staking in the first place. For the full mathematics of choosing your fraction, see our guide to half and quarter Kelly sizing, and use the Bet Sizing Calculator to combine a Kelly fraction with a hard percentage cap per bet.
How Deep Will the Drawdowns Be?
Kelly betting is often sold on its growth rate, but the honest sales pitch must include the drawdown profile. A clean published result makes it concrete. In the continuous approximation, a bettor staking fraction c of full Kelly will, at some point in an indefinitely long betting career, see the bankroll fall to a fraction x of its starting value with probability approximately x^(2/c − 1). For full Kelly (c = 1) the exponent is 1, giving the famous rule of thumb: the probability of ever losing half your bankroll under full Kelly is about one half, the probability of ever losing 80% of it is about one in five, and so on — the probability of dipping to any level equals the level itself.
| Drawdown from starting bankroll | Full Kelly | Half Kelly | Quarter Kelly |
|---|---|---|---|
| Lose 20% (fall to 0.80×) | ≈ 80% | ≈ 51% | ≈ 21% |
| Lose 30% (fall to 0.70×) | ≈ 70% | ≈ 34% | ≈ 8% |
| Lose 50% (fall to 0.50×) | ≈ 50% | ≈ 12.5% | ≈ 0.8% |
| Lose 70% (fall to 0.30×) | ≈ 30% | ≈ 2.7% | ≈ 0.02% |
Probability of ever reaching each drawdown level over an indefinitely long betting horizon, per the continuous-approximation formula x^(2/c − 1). Treat these as orders of magnitude, not precise forecasts: discrete bets, changing edges, and estimation error all shift the real numbers.
Read that table slowly, because it reframes the fractional Kelly decision. Moving from full to half Kelly does not merely halve your drawdown risk — it cubes the decay: a 50% drawdown goes from a coin flip to a one-in-eight event. Moving to quarter Kelly makes it a once-in-a-career rarity. The growth you give up is real, but the survivability you gain is disproportionate. Before committing to any fraction, stress-test it: the Staking Plan Simulator runs flat, percentage, and Kelly staking over hundreds of simulated bets so you can see the drawdown paths for yourself, and the Risk of Ruin Calculator puts an exact probability on busting under your specific edge, odds, and stake size.
Worked Examples
Odds +150 (decimal 2.50), your estimate p = 0.50. b = 1.50, q = 0.50. f* = (1.50 × 0.50 − 0.50) / 1.50 = 16.7% full Kelly. On $10,000: full $1,667, half $833, quarter $417. Most professionals would bet the quarter-Kelly figure.
Odds 1.70 (−143), your estimate p = 0.62. b = 0.70, q = 0.38. f* = (0.70 × 0.62 − 0.38) / 0.70 = 0.054 / 0.70 = 7.7% full Kelly. On $10,000: full $771, quarter $193. Note the EV is +5.4% per unit — a good bet, but Kelly sizes it far below the underdog example because the payout is smaller.
Odds 1.83 (−120), your estimate p = 0.55. b = 0.83, q = 0.45. f* = (0.83 × 0.55 − 0.45) / 0.83 ≈ 0.8% full Kelly — just $78 on $10,000, and about $20 at quarter Kelly. Kelly automatically shrinks stakes on marginal edges; if the calculation feels disappointingly small, the edge was small.
Odds 1.91 (−110), your estimate p = 0.52. b = 0.91, q = 0.48. bp − q = 0.4732 − 0.48 = −0.0068. Negative numerator → f* < 0 → stake nothing. At standard −110 juice you need roughly p > 52.4% just to break even.
Multiple Simultaneous Bets: The Kelly Portfolio Problem
Real bettors rarely place one bet at a time. On a Sunday with a full NFL slate, a serious bettor might have 8 to 12 active wagers simultaneously. When bets overlap in time, the naive approach of applying Kelly independently to each bet over-stakes the bankroll, because it ignores the fact that multiple bets can lose at once. Ten independent bets of 6% each put 60% of the bankroll in play — an exposure level full Kelly would never sanction for a single position.
The formally correct approach is the Kelly portfolio problem: choose the vector of stakes that maximizes expected log growth across the whole set of simultaneous bets. For a handful of independent bets, the solution slightly shrinks each individual stake relative to its standalone Kelly figure. For correlated bets the mathematics require covariance estimates and numerical optimization, which is why professional syndicates run custom software rather than a formula.
For everyone else, two heuristics capture most of the benefit. First, treat correlated bets as one bet: two legs on the same game, a moneyline and a player prop on the same team, or the same side taken at two books are effectively a single position — compute one Kelly stake for the combined exposure and divide it among the tickets, rather than sizing each independently. Second, cap total open exposure: if five bets individually call for 6% each, scale them down so the sum of all live stakes stays within roughly 20–25% of bankroll. This costs little growth and removes the scenario where one bad slate produces a full-Kelly-sized loss across the whole portfolio. Logging every open position in a bankroll tracker makes it obvious when your aggregate exposure has crept past the cap.
Common Kelly Mistakes
Most Kelly failures in practice are not failures of the formula — they are failures of its inputs or of discipline. The recurring ones:
- Betting full Kelly on estimated probabilities. The formula is calibrated for true probabilities. Estimates carry error, error inflates the computed stake, and over-betting destroys growth faster than under-betting sacrifices it. If you have not verified your edge over hundreds of bets against closing lines, full Kelly is overconfidence in mathematical clothing.
- Using edge computed from vigged odds. Your probability must be compared against the price you are actually paid. Run the numbers through an EV calculator first; if the expected value at the offered odds is not positive, the Kelly fraction is not positive either, no matter how good the team looks.
- An undefined bankroll. Kelly percentages are meaningless if "bankroll" is sometimes your betting account and sometimes your net worth. Define a dedicated betting bankroll you could lose entirely without hardship, and compute every stake against that figure.
- Sizing correlated bets independently. Same-game legs, same-team props, and duplicated positions across books share fate. Independent Kelly sizing on each silently multiplies your true exposure to a single outcome.
- Abandoning the plan mid-drawdown. The drawdown table above shows that painful losing runs are an expected feature, not evidence the method failed. Bettors who quit or, worse, double their stakes to "recover" during a drawdown convert a survivable event into ruin. If a plan's normal drawdowns exceed what you can emotionally tolerate, the fix is a smaller Kelly fraction chosen in advance — not improvisation under stress.
- Rounding stakes up "for conviction." Conviction is already in the formula — it is p. Adding an extra bump on top double-counts your confidence and pushes you toward the over-betting regime where growth turns negative.
When NOT to Use Kelly
Kelly is the right tool for a specific job: repeated bets, a quantifiable edge, and a long horizon. Outside those conditions, it is the wrong tool.
- You cannot estimate probabilities. Kelly's inputs are odds and your win probability. If your probability is a gut feeling rather than the output of a model or a verified record, Kelly will faithfully optimize garbage. Flat staking of 1–2% per bet is more robust for bettors still proving whether they have an edge at all — compare the approaches in the staking simulator before choosing.
- One-off or rare bets. Kelly maximizes the growth rate over a long sequence of compounding bets. For a single bet that will not repeat, the asymptotic argument does not apply, and the right stake is a judgment about how much loss you can absorb, not a growth-rate optimization.
- No positive edge. This is worth restating as its own rule: Kelly with p at or below the break-even probability outputs zero. No staking plan manufactures profit from negative-EV bets — it can only change the shape of the losing.
- The bankroll is not really risk capital. If a 50% drawdown would force you to withdraw funds for rent, the mathematical assumptions (indefinite horizon, fully re-investable bankroll) are already violated. Size down or step away.
- Bet limits bind anyway. Sharp bettors with large bankrolls often find the book's maximum stake is below their quarter-Kelly figure. At that point the sizing question is moot — bet the limit and let the constraint do the risk management.
Even when Kelly does apply, remember its place in the workflow: find the edge first, verify it, then size it. Kelly is the last step of a disciplined process, not a substitute for one.
Frequently Asked Questions
What is the Kelly Criterion formula?
The Kelly formula is f* = (bp − q) / b, where f* is the fraction of your bankroll to stake, b is the net decimal odds (decimal odds minus 1), p is your estimated probability of winning, and q = 1 − p. At even money (decimal 2.00), it simplifies to f* = 2p − 1, which is simply your edge.
What is half Kelly and why do professionals prefer it?
Half Kelly means staking 50% of the full Kelly recommendation. In the standard continuous approximation it preserves about 75% of the maximum long-run growth rate while halving the volatility of your bankroll, and it dramatically reduces deep drawdowns. It also protects against estimation error: if your true edge is only half of what you think, half Kelly on your estimate is full Kelly on reality, whereas full Kelly on your estimate would be double Kelly — which earns roughly zero long-run growth.
How big are drawdowns under full Kelly betting?
Large. In the continuous approximation analyzed by Edward Thorp, a full-Kelly bettor has roughly a 50% probability of at some point losing half the bankroll, and roughly an x probability of ever falling to a fraction x of it. Under half Kelly the probability of a 50% drawdown drops to about 12.5%, and under quarter Kelly to under 1%. Real-world discrete betting differs somewhat, but the ordering and rough magnitudes hold.
Can the Kelly Criterion tell me not to bet?
Yes — and this is one of its most valuable outputs. If bp − q is zero or negative, you have no edge at the offered odds and the optimal stake is zero. A negative Kelly fraction is not an instruction to bet the other side (the other side has its own vig); it is an instruction to pass.
How do I apply Kelly to several simultaneous bets?
Independent Kelly staking on each bet over-exposes the bankroll because several bets can lose at once. Practical rules: treat correlated bets (same game, same team across markets) as a single position and size it once; reduce each stake when many bets run at the same time; and cap total simultaneous exposure — many practitioners keep the sum of all open stakes below roughly 20–25% of bankroll unless they run a full portfolio optimizer.
Is Kelly betting guaranteed to be profitable?
No. Kelly maximizes long-run growth only if your probability estimates are accurate and your edge is real. With no edge, no staking plan — Kelly included — creates profit. Kelly is a stake-sizing layer on top of an edge, not a source of edge. Validate your edge first with expected value analysis and closing line value tracking.