The 1956 paper
John L. Kelly Jr. published "A New Interpretation of Information Rate" in the Bell System Technical Journal in March 1956. The paper was nominally about Shannon's information theory applied to noisy telephone channels. It happened to derive — almost as a footnote — the optimal bet-sizing formula for a gambler with a known edge.
Kelly never gambled himself. The formula sat unused in the academic literature until 1962, when Edward Thorp introduced it to card counters in Beat the Dealer. From blackjack, Kelly spread to horse racing, sports betting, hedge funds, market-making, and quantitative finance generally. Today every algorithmic trading desk, every prop sportsbook, and every advantage bettor uses some variant of Kelly sizing.
The formula
# Kelly criterion for a binary bet f* = (bp − q) / b where f* = optimal fraction of bankroll to bet b = decimal odds − 1 (i.e., net payout per unit staked on a win) p = win probability (your estimate) q = 1 − p (loss probability)
The formula's elegance is its derivation: it maximizes the expected logarithm of wealth, which is equivalent to maximizing the geometric growth rate of bankroll. Logarithmic utility means the bettor weighs equal percentage gains and losses equally — a 50% drawdown hurts as much as a 50% gain helps. That property is mathematically what produces the optimal compounding rate over long horizons.
Worked examples
Example 1 — NFL spread bet
# Bet at -110 (decimal 1.909, b = 0.909) # Your model: 55% win probability b = 0.909; p = 0.55; q = 0.45 f* = (0.909 × 0.55 − 0.45) / 0.909 = (0.500 − 0.450) / 0.909 = 0.050 / 0.909 = 5.5% of bankroll # full Kelly quarter_Kelly = 1.4% # typical pro stake
Example 2 — Long underdog moneyline
# Lakers +250 (decimal 3.50, b = 2.50) # Your model: 35% win probability b = 2.50; p = 0.35; q = 0.65 f* = (2.50 × 0.35 − 0.65) / 2.50 = (0.875 − 0.650) / 2.50 = 9.0% of bankroll # full Kelly — much larger because of longshot odds quarter_Kelly = 2.25%
Example 3 — Marginal edge
# Player prop -110 (b = 0.909), model says 53% b = 0.909; p = 0.53 f* = (0.909 × 0.53 − 0.47) / 0.909 = 0.01177 / 0.909 = 1.3% of bankroll # full Kelly quarter_Kelly = 0.32% # tiny stake — barely worth executing
The variance problem — why full Kelly hurts
Full Kelly is the mathematically optimal long-run growth rate. The catch: the path is hideous. A bettor with documented +5% edge running full Kelly faces approximately:
- 50% probability of a 50% drawdown at some point.
- 33% probability of a 75% drawdown.
- 10% probability of an 89% drawdown.
- Expected log-growth: ~0.125% per bet (under perfect edge estimation).
For most bettors, a 75% drawdown is psychologically untenable. They quit before the long-run compounding has time to manifest. Fractional Kelly solves this:
| Fraction of Kelly | Long-run growth | Drawdown risk | Used by |
|---|---|---|---|
| 1.00 (full) | 100% of optimal | 50% chance of -50% DD | Theoretical optimum, rare in practice |
| 0.50 (half) | 75% of optimal | ~25% chance of -50% DD | Some hedge funds, aggressive sharps |
| 0.25 (quarter) | 43.75% of optimal | ~5% chance of -50% DD | Industry standard for sports betting pros |
| 0.10 (one-tenth) | 19% of optimal | ~1% chance of -50% DD | Conservative bankrollers, beginners |
Quarter Kelly delivers nearly half of full Kelly's growth at a tenth of the drawdown risk. The math: variance scales with the square of the fraction, but growth scales linearly. Cutting fraction by half cuts growth by half but cuts variance by 75%. This is why quarter Kelly is the practical sweet spot.
Edge uncertainty — Kelly's hidden assumption
The formula assumes you know your edge. In real sports betting you don't. Your model says 55% win probability but the true rate could be 51% or 59% (sample uncertainty, model misspecification, regime change). The robustness literature is unambiguous: overstated edge + full Kelly = ruin.
A bettor who consistently overestimates edge by 1 percentage point — believing he has 4% edge when he actually has 3% — running full Kelly will see growth roughly equal to running 0.75-Kelly at the true edge. Once overestimation reaches 3-4 percentage points, full Kelly produces negative log-growth despite a real positive edge. The bettor goes bankrupt while being mathematically right about the direction.
Fractional Kelly is a robustness adjustment. If you use quarter Kelly, your effective Kelly fraction is 0.25 × your edge estimate. Even if your edge estimate is overstated by 2x, you're still effectively running at half Kelly of the true edge — survivable and profitable.
Kelly with simultaneous bets
The basic formula assumes one bet at a time. In sports betting, pros often have 5-15 bets running simultaneously across Sunday's NFL slate. The independent-bets approximation: bet Kelly fractions on each, capped at total Kelly exposure ≤ 25-40% of bankroll.
For genuinely correlated bets (multiple legs of the same NFL game, multiple totals on the same weekend's weather-related games), pros use simultaneous Kelly — solve the joint optimization across all bets accounting for correlation. The math is messier (numerical optimization, not closed-form), but the principle is the same: maximize expected log-bankroll across the joint outcome distribution.
The compounding miracle
# Bettor with +3% average edge, quarter Kelly, 200 bets/year starting_bankroll = $10,000 # Conservative estimate of growth rate quarter_Kelly_growth ≈ 0.4% per bet annual_compounding ≈ (1.004)^200 = 2.22x year 1: $22,200 year 2: $49,400 year 3: $109,800 year 5: $543,000 year 10: $29.5 million # in theory
The catch is that books limit. A bettor compounding from $10K to $50K rarely makes it to $500K without account closures. Real-world pro careers involve constant book rotation, family/spouse accounts, and offshore relationships. The Kelly formula is mathematically beautiful and operationally messy.
When NOT to use Kelly
- Negative edge bets — Kelly returns negative, meaning "don't bet." Many bettors ignore this and bet for entertainment. Fine, but accept it's not a +EV strategy.
- Very thin edges (< 1%) — execution friction (line movement, slippage) eats edges this small. Kelly says bet ~0.3% of bankroll; in practice the bet's not worth placing.
- Highly uncertain edge estimates — if your model has wide confidence intervals, fractional Kelly should be reduced further or skipped entirely.
- Single-shot scenarios — Kelly is a long-run formula. For a one-time bet (you'll never bet again), expected value, not Kelly growth, is the right framework.
Sources & further reading
- Kelly, John L. Jr. "A New Interpretation of Information Rate." Bell System Technical Journal, vol. 35, 1956.
- Thorp, Edward O. "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." Handbook of Asset and Liability Management, 2006.
- Poundstone, William. Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. Hill and Wang, 2005.
- MacLean, Leonard C., Thorp, Edward O., Ziemba, William T. (eds.). The Kelly Capital Growth Investment Criterion: Theory and Practice. World Scientific, 2011.
- Buchdahl, Joseph. "Kelly Criterion: a benefits and dangers analysis." Football Data Blog, 2018.