Kelly Criterion Calculator
Calculate the optimal bet size for maximum long-term growth using the Kelly Criterion.
Last reviewed: May 2026This Kelly Criterion calculator computes the bet fraction that maximizes the long-run geometric growth rate of your bankroll given a known win probability and payout. It returns full Kelly, half-Kelly, and quarter-Kelly amounts, the expected log growth rate per bet, the expected value per dollar wagered, and an explicit warning when the formula returns a negative number — the case where the correct action is not to bet at all. The sections below walk through the formula, a fully worked example with a growth-rate curve, the math behind why Kelly is the unique log-optimal rule, the practical reasons most professionals use fractional Kelly, and the failure modes that produce ruin even with a positive edge.
What This Calculator Does
The Kelly Criterion answers a single question: given a wager where you know the probability of winning and the payout on a win, what fraction of your current bankroll should you bet to maximize the long-run compound growth rate? The formula, derived by John L. Kelly Jr. at Bell Labs in 1956 in the paper "A New Interpretation of Information Rate," is deceptively compact:
f* = (b·p − q) / bHere f* is the optimal fraction of your bankroll to wager, b is the net payout per unit wagered (decimal odds minus one — a coin flip paying even money has b = 1, a 3-to-1 underdog has b = 3), p is the probability of winning, and q = 1 − p is the probability of losing. The numerator bp − q is your edge expressed in payout units; dividing by b normalizes it to a fraction of bankroll.
This calculator returns f* alongside the dollar amount of a full Kelly bet (f* × bankroll), the half-Kelly amount (a more practical default for noisy real-world probabilities), the quarter-Kelly amount (used by many professional sports bettors), the expected geometric growth rate per bet g = p·ln(1 + fb) + q·ln(1 − f), and the expected value per dollar wagered. When bp < q the formula produces a negative number; the calculator shows an explicit "do not bet" warning rather than displaying a misleading negative percentage. Inputs are validated and clamped: probabilities outside [0, 1] are pinned to the valid range, and odds must be strictly positive. The calculator runs entirely in your browser — no data leaves your device, no logins, no rate limits.
How to Use It
Enter your win probability as a decimal between 0 and 1 (for example, 0.6 for a 60% chance of winning) and the payout odds b as the net profit per $1 wagered. A coin flip paying even money has b = 1; a sports bet at -110 American odds (the standard line on most spreads) has b ≈ 0.909; a 3-to-1 long shot has b = 3. If you only know decimal odds in the European format (where 2.00 means double-or-nothing), subtract 1 to get b — decimal 2.00 corresponds to b = 1.
Optionally enter your current bankroll. The calculator immediately shows the optimal Kelly fraction as a percentage, the dollar amounts for full Kelly, half-Kelly, and quarter-Kelly sizing, the expected log growth rate per bet (positive means your bankroll grows on average in geometric mean; near zero or negative means do not bet), and your raw expected value per dollar. Adjust any input to see results update live; the share URL at the bottom encodes your inputs so you can bookmark or send a specific scenario. The reset button restores the default scenario (60% win probability, even-money payout, $10,000 bankroll), and the print button generates a clean printable summary.
If the probability or odds you enter imply a negative edge — meaning the house, sportsbook, or counterparty has the advantage — a red warning appears and the recommended bet drops to zero. This is the formula working correctly: Kelly's correct prescription for a losing bet is to not place it. The expected value figure shown in that case will be negative, confirming the math.
Worked Example: 60% Win Probability at Even-Money Odds
- Win probability (p)
- 0.60
- Loss probability (q)
- 0.40 (= 1 − p)
- Payout per $1 wagered (b)
- 1.0 (even money)
- Bankroll
- $10,000
Step 1 — Plug into the Kelly Formula
f* = (b·p − q) / b = (1 × 0.60 − 0.40) / 1 = 0.20. The optimal bet is 20% of your bankroll, or $2,000 on a $10,000 stake. Half-Kelly is $1,000 (10% of bankroll) and quarter-Kelly is $500 (5%).
Step 2 — Compute the Expected Growth Rate
g(f) = p·ln(1 + fb) + q·ln(1 − f). At f = 0.20: g = 0.60 × ln(1.20) + 0.40 × ln(0.80) = 0.60 × 0.1823 + 0.40 × (−0.2231) = 0.1094 − 0.0893 = +0.0201 = +2.01% per bet. Over 100 such bets your bankroll grows by approximately e^(100 × 0.0201) − 1 ≈ +644% in geometric mean — but with massive variance and frequent 30–50% drawdowns along the way.
Step 3 — Compare Half-Kelly and Quarter-Kelly
At half-Kelly (f = 0.10): g = 0.60 × ln(1.10) + 0.40 × ln(0.90) = 0.0572 − 0.0421 = +0.0151 = +1.51% per bet. Half-Kelly captures about 75% of full Kelly's growth rate while the standard deviation of returns drops by roughly 50%. At quarter-Kelly (f = 0.05): g ≈ +0.86% per bet (43% of full Kelly), with standard deviation cut by 75%.
Step 4 — Visualize the Growth-Rate Curve
The chart below plots the expected log growth rate g(f) against bet fraction f for this exact scenario (p = 0.60, b = 1). Notice three landmarks: the curve peaks at the Kelly fraction f* = 0.20, returns to zero at f = 0.40 (double-Kelly), and goes sharply negative for any f > 0.40. Betting more than double-Kelly produces negative geometric growth — your bankroll trends to zero with probability one.
Common Use Cases
Sports Betting Bankroll Management
Most professional sports bettors use Kelly sizing — almost always fractional Kelly — to manage their bankroll across a season of bets. The challenge in sports betting is that your edge estimate is rarely better than ±2–4 percentage points; calculating Kelly at a 56% estimated win rate on a -110 line and then sizing at full Kelly is dangerous because the true rate could easily be 53% or 59%. Quarter-Kelly with a 5% maximum on any single bet is a common professional convention.
Poker Bet Sizing and Game Selection
Kelly applies cleanly to cash-game poker buy-in selection and tournament entry decisions. If you have a measurable edge in a $5/$10 NL game (based on tracked win rate over thousands of hands) and a quantified bankroll, Kelly tells you how aggressively to play. Most poker bankroll guidance (the "20 buy-ins for cash, 100 buy-ins for tournaments" heuristic) is implicitly a fractional Kelly rule of thumb.
Position Sizing in Trading
Quantitative traders and discretionary investors use Kelly to size positions on edges identified by a model or thesis. Edward Thorp's Princeton-Newport Partners and later hedge funds applied fractional Kelly throughout their statistical-arbitrage and convertible-bond strategies. The key adaptation in trading is correlation: when multiple positions share factor exposure (sector, country, style), naive single-asset Kelly produces dangerous over-concentration, and correlations must be modeled jointly.
Multi-Position Allocation (Multi-Kelly)
When you face several independent positive-EV opportunities at once — say, three sports bets on uncorrelated games or three positions in unrelated sectors — single-bet Kelly is replaced by simultaneous Kelly: maximize E[log(1 + Σ f_i·X_i)] subject to Σ f_i ≤ 1. The resulting f_i values are typically smaller than the independent Kelly for each bet because diversification pulls everything toward proportional sizing. Correlated bets require explicit covariance modeling, which is why institutional Kelly implementations often start with risk-parity allocation as an approximation before refining.
Critical Warnings: Where Kelly Breaks
Kelly maximizes expected log growth — the geometric, not the arithmetic, mean. The arithmetic EV of a bet (p·b − q) can be positive while the log-growth optimum prescribes a much smaller fraction than naive intuition suggests, and the difference is exactly what protects you from ruin. Several specific failure modes deserve explicit attention.
Drawdowns Are Brutal at Full Kelly
Full Kelly produces a 50% drawdown roughly once per "Kelly multiple" of bets — meaning over hundreds of bets at full Kelly, you should expect to lose half your bankroll at least once. Most human investors cannot tolerate that psychologically, even when the math says recovery is statistically guaranteed. This is the single biggest reason fractional Kelly dominates real-world practice.
You Must Know p and b Precisely
Real-world probability estimates are noisy. A calculated edge of 4% can easily be a true edge of 1% (or zero, or negative). Because g(f) is steeply asymmetric — over-betting punishes you more than under-betting — noisy inputs systematically push you toward over-betting at full Kelly, which then causes ruin. The asymmetry is well-known among quantitative traders as the "Kelly fragility" problem.
Independence and Correlation Assumptions
Single-bet Kelly assumes independent wagers. Correlated bets — multiple positions in the same sector, multiple sports bets on outcomes that depend on the same weather or injury — break the formula. When correlations are positive, simultaneous Kelly fractions need to be lower than the single-bet formula suggests, sometimes dramatically so during market stress when correlations rise toward one.
You Must Be Able to Lose the Entire Bet
Kelly's derivation assumes that on a loss you forfeit f × bankroll completely. Markets with leverage, margin calls, or path-dependent payoffs (e.g., barrier options, certain prediction-market contracts) violate this assumption and require modified sizing rules. Path-dependent losses can exceed the nominal Kelly stake.
Infinite Divisibility Is Assumed
Kelly assumes you can wager any fraction of your bankroll. In practice, bet sizes are quantized — sports books have minimum bet sizes, stocks trade in shares, options have lot sizes — and very small bankrolls combined with high Kelly fractions can produce situations where the nearest tradable size is materially different from the theoretical optimum.
Why Half-Kelly or Quarter-Kelly?
When probability estimates are uncertain — which is essentially every real-world application outside idealized coin flips — fractional Kelly is the right framework. The argument from Edward Thorp's 2006 paper "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market" makes the trade-off precise: half-Kelly captures about 75% of full Kelly's expected log growth rate while cutting the standard deviation of log returns roughly in half. That is, you give up 25% of growth in exchange for roughly 50% less variance — a vastly improved Sharpe-like ratio.
Thorp himself recommends half-Kelly for trading applications, and his Princeton-Newport Partners (one of the most successful quantitative funds of the 1970s–80s) reportedly used half-Kelly or smaller throughout its convertible-bond arbitrage book. Most professional sports bettors use somewhere between 1/4-Kelly and 1/2-Kelly; the lower end is favored by bettors with less mature edge-estimation models, where the risk of over-estimating p is highest.
The math behind why fractional Kelly works as risk control is straightforward. The growth-rate curve g(f) is approximately parabolic near the peak: g(f) ≈ g(f*) × (1 − ((f − f*)/f*)²). At f = f*/2, the quadratic term is 1/4, so growth falls to roughly 75% of peak — but standard deviation of log returns scales linearly with f, so variance falls by 75% of its peak value as well. The trade is asymmetric in your favor whenever you have any doubt about the true probabilities.
An additional reason favoring fractional Kelly: real human utility functions are typically more risk-averse than pure log utility. The classic Cumulative Prospect Theory work of Kahneman and Tversky documents that humans weight losses roughly 2x as heavily as equivalent gains — an asymmetry log utility does not capture. Half-Kelly produces a sizing rule closer to most actual utility functions than full Kelly does, independent of any uncertainty about p.
The Math: Derivation from Log Utility
Why Maximize Expected Log Growth?
Bankroll evolution under repeated betting is multiplicative: B_n = B_0 × Π(1 + f·X_i), where X_i is the per-bet return (b on a win, −1 on a loss). The long-run growth rate is therefore (1/n) × log(B_n / B_0) = (1/n) × Σ log(1 + f·X_i), which by the strong law of large numbers converges to E[log(1 + f·X)] almost surely. Maximizing this expectation maximizes the geometric mean return, which is the only return measure that compounds correctly through time.
Daniel Bernoulli's Original Argument (1738)
The case for log utility predates Kelly by over 200 years. In his 1738 paper resolving the St. Petersburg paradox, Daniel Bernoulli argued that utility should be proportional to the logarithm of wealth rather than to wealth itself — implying diminishing marginal utility and, crucially, time-invariance under multiplicative dynamics. Bernoulli's log utility is the unique utility function for which the optimal bet fraction is invariant to bankroll level (you bet the same fraction whether you have $1,000 or $1 million), making it the natural choice for repeated-game decision-making.
Deriving the Kelly Formula
To find f*, differentiate g(f) = p·ln(1 + fb) + q·ln(1 − f) with respect to f and set equal to zero: g'(f) = p·b/(1 + fb) − q/(1 − f) = 0. Cross-multiplying: p·b·(1 − f) = q·(1 + fb). Expanding: pb − pbf = q + qbf, so pb − q = pbf + qbf = f·b·(p + q) = f·b (since p + q = 1). Therefore f* = (pb − q) / b, the canonical Kelly formula. The second derivative is uniformly negative on (0, 1), confirming this critical point is a maximum.
Shannon-McMillan-Breiman and Information Theory
Kelly's original 1956 paper was titled "A New Interpretation of Information Rate" because he derived the formula from Claude Shannon's information theory rather than from utility theory. The Shannon-McMillan-Breiman theorem connects the rate of information transmission through a noisy channel to the rate at which a gambler's bankroll can be made to grow by betting in proportion to subjective probabilities matching the true distribution. The Kelly fraction is the per-bet log-growth rate translated into a sizing rule — meaning Kelly is not just a betting heuristic but the unique rate-optimal coding/betting strategy.
Historical Notes and Practitioner Adoption
Kelly's formula gained immediate practical adoption when Edward Thorp applied it to blackjack card counting in the 1962 book Beat the Dealer, then to roulette wheel-bias exploitation, and finally to convertible bond arbitrage at Princeton-Newport Partners from the 1970s onward. Thorp's adaptation — always fractional, typically half-Kelly or smaller — became the template for systematic quantitative trading. William Poundstone's Fortune's Formula (2005) documents the surprising resistance Kelly faced from economists schooled in mean-variance optimization (Markowitz, Samuelson) before its eventual acceptance as the rate-optimal sizing rule for multiplicative-growth contexts.
Among practitioners today, Kelly underlies sizing in horse-racing syndicates (Bill Benter's syndicate famously used Kelly variants on Hong Kong races), professional sports betting bankrolls, several quantitative hedge funds' position-sizing layers, and high-frequency market-making capital allocation. The persistent gap between academic finance (mean-variance, CAPM) and practitioner finance (Kelly-style log-growth maximization) reflects the difference between explaining cross-sectional returns and actually compounding capital over time.
YMYL Disclaimer and Risk Acknowledgment
This calculator is a mathematical tool, not financial advice. The Kelly Criterion is a sizing framework based on assumptions that frequently fail in real-world settings: probabilities are precisely known, bets are independent, outcomes are binary, the entire stake can be lost on a loss, and the bankroll can be wagered in any fraction. Violating any of these assumptions invalidates the formula's optimality guarantee.
Gambling carries inherent and substantial risk of loss. Even with a true positive edge, full Kelly produces frequent 30–50% drawdowns; over-betting Kelly — which happens whenever your edge estimate is noisier than you realize — leads to bankroll ruin with probability one. Past performance of any betting or investing strategy does not predict future results. This calculator does not account for tax consequences, transaction costs, slippage, bookmaker juice, market-maker spreads, or the operational risks of placing real wagers.
If you are considering applying Kelly-based sizing to investment decisions, consult a qualified, licensed financial advisor who can model your full financial picture, tax situation, and risk tolerance. If you or someone you know is struggling with problem gambling, free confidential help is available 24/7 from the National Council on Problem Gambling at 1-800-522-4700 (U.S.) or your local equivalent. Nothing in this calculator or accompanying content should be construed as a recommendation to place any specific wager.
Frequently Asked Questions
What is the Kelly Criterion?
The Kelly Criterion is a mathematical formula derived by John L. Kelly Jr. at Bell Labs in 1956 that computes the bet fraction maximizing the expected logarithm of bankroll over many repeated wagers. For a bet that pays b-to-1 with win probability p and loss probability q = 1 − p, the optimal fraction is f* = (bp − q) / b. Kelly is the unique bet-sizing rule that maximizes the long-run geometric growth rate; betting more than Kelly causes growth to fall and eventually go negative, while betting less than Kelly produces lower growth with substantially lower variance. Kelly applies only when you have a positive edge (bp > q); when the formula returns a negative number, the correct action is not to bet at all.
Why does Kelly maximize geometric growth?
Bankroll evolution is multiplicative — each bet multiplies your stake by some factor — so the long-run outcome is determined by the product of returns, which by the law of large numbers converges to the expected value of the logarithm of those returns. Kelly maximizes E[log(1 + f·X)] where X is the per-unit return. Because log is concave, this objective penalizes large losses far more than it rewards equally-large gains, so the Kelly fraction is always strictly lower than the arithmetic expected-value-maximizing bet. Maximizing log growth is equivalent to maximizing the median terminal wealth and, by the Shannon-McMillan-Breiman theorem, also maximizes the information-theoretic rate at which your bankroll compounds.
What happens if I over-bet Kelly?
Over-betting Kelly is mathematically catastrophic. The growth-rate curve g(f) = p·ln(1 + fb) + q·ln(1 − f) is shaped like an inverted parabola peaking at f*; double-Kelly (2f*) drives the growth rate back to exactly zero, and anything above 2f* produces negative expected log growth — meaning your bankroll trends to zero with probability one over enough bets. At 1.5×Kelly your growth rate falls by roughly 75% of its peak. Worse, real-world probability estimates are never exact, so a calculated Kelly of 20% based on noisy data can easily be a true Kelly of 10%, meaning your "full Kelly" is already double-Kelly without you knowing it. This is the single most common reason professional bettors and traders use fractional Kelly.
Should I use half-Kelly or full Kelly?
Half-Kelly is the practical default for almost all real-world applications. Edward Thorp, who first applied Kelly to blackjack and later to the Princeton-Newport Partners hedge fund, recommends half-Kelly for trading because probability estimates in financial markets are noisy. Half-Kelly delivers about 75% of full Kelly's expected growth rate while cutting standard deviation of returns roughly in half — a dramatically better risk-adjusted outcome. Full Kelly is only justified when probabilities are known with mathematical certainty (idealized coin flips, perfectly modeled casino games with known house edges) and your psychology can tolerate 50% drawdowns. For sports betting, poker, and stock-position sizing — domains where your edge is estimated, not measured — half-Kelly or quarter-Kelly is the standard professional practice.
Can Kelly go negative?
Yes, and when it does the answer is unambiguous: do not place the bet. A negative Kelly value means bp − q is negative, which means your expected value per dollar wagered is negative — the house has the edge. The formula has no "short the bet" interpretation in casino games (you cannot become the casino), but in financial markets it can be read as a signal to take the opposite side of the trade if doing so is operationally feasible. In sports betting, a negative Kelly on a line you originally thought was favorable usually means your edge estimate was wrong or the line has moved against you; recheck your inputs before assuming the formula is malfunctioning. Most well-built Kelly calculators (including this one) display a "do not bet" warning rather than a negative number to prevent confusion.
How do I apply Kelly to multiple bets?
Single-bet Kelly assumes you have one wager at a time. With multiple simultaneous positive-EV bets, the problem becomes a constrained optimization: maximize E[log(1 + Σ f_i·X_i)] subject to Σ f_i ≤ 1 (you cannot bet more than 100% of your bankroll). When bets are independent, the simultaneous Kelly fractions are typically lower than the sum of individual Kelly fractions because of diversification interactions. When bets are correlated — for example, multiple positions in the same sector of the stock market — correlations must be modeled explicitly, and the simultaneous Kelly fractions can be much lower than independent Kelly would suggest. In practice, professional sports bettors and quantitative traders use fractional Kelly on each position (often 1/4 to 1/2) and cap total exposure as a safety margin against estimation error.
Does Kelly assume rational markets?
No — Kelly is agnostic to whether markets are rational. It only requires that you have an accurate estimate of the probability and payoff of a specific wager. In efficient markets, prices fully reflect available information and the Kelly fraction for any given bet is essentially zero, which is why passive index investing is rational for most participants. In inefficient corners — micro-cap stocks, illiquid sports betting lines, niche prediction markets — exploitable edges do exist, and Kelly tells you how to size positions to exploit them. Kelly cannot generate edge where none exists; it is purely a sizing rule once an edge is identified. Bad probability estimates produce bad Kelly fractions regardless of whether the broader market is rational.
Why do most investors not use full Kelly?
Three reasons dominate. First, drawdowns at full Kelly are severe: in a long-run simulation you can expect a 50% drawdown roughly once per Kelly multiple (about every 5–10 years for typical portfolios), which is psychologically intolerable for almost all human investors. Second, probability estimates in finance are noisy enough that calculated full Kelly is often actual 1.5×–2×Kelly, putting you near the zero-growth cliff. Third, real-world utility functions for most investors are more risk-averse than pure log utility — losing 50% of net worth at age 55 has consequences (delayed retirement, lifestyle compression) that log utility doesn't capture. The consensus among practitioners — including Warren Buffett's commentary in Berkshire Hathaway letters and Edward Thorp's writings — is that fractional Kelly (typically 1/4 to 1/2) is the appropriate sizing framework for investors with finite horizons and finite tolerance for drawdown.
Quick reference
| Aspect | Detail |
|---|---|
| Purpose | Kelly Criterion Calculator is a free browser calculator for fast, repeatable estimates. |
| Runs in | Any modern browser — no install, no signup. |
| Privacy | Inputs are computed locally; nothing is uploaded. |
| Best for | Quick estimates, daily reference, study, planning. |