Kelly Criterion: Optimal Bet Sizing
The mathematically optimal strategy for sizing your bets to maximize long-term bankroll growth.
You've found a +EV bet. Now what? Bet too little and you leave growth on the table. Bet too much and a bad streak wipes you out. The Kelly Criterion is the mathematically proven answer to this dilemma—a formula that maximizes the long-term growth rate of your bankroll.
The Problem Kelly Solves
Imagine you have a coin that lands heads 60% of the time, paying even money. You clearly have an edge. But how much of your bankroll should you wager on each flip?
- Bet 1% — Safe, but your bankroll barely grows
- Bet 50% — Aggressive, but a few losses in a row devastate you
- Bet 100% — One loss and you're done forever
Kelly tells you the exact fraction that maximizes the geometric growth rate of your wealth. Not the arithmetic average—the geometric rate, which accounts for compounding and the asymmetry between gains and losses.
The Derivation: Maximizing E[log(W)]
The key insight is that we want to maximize E[log(W)]—the expected value of the logarithm of wealth. Why logarithm? Because wealth compounds multiplicatively, not additively. Doubling and then halving leaves you at the same place, not up by the average gain.
For a bet with win probability p, loss probability q = 1 − p, and net odds b (you win b units for every 1 unit risked), the expected log-growth when betting fraction f of your bankroll is:
Taking the derivative, setting it to zero, and solving for f gives us the Kelly fraction.
Setting dG/df = 0 and solving:
Where p = win probability, q = 1 − p (loss probability), and b = net odds (decimal odds − 1). This is equivalent to f* = (edge) ÷ (odds).
Good to Know
Key Properties of Kelly
- Maximizes geometric growth — No other fixed-fraction strategy grows your bankroll faster in the long run
- Never bets the entire bankroll — Since f* < 1 always, you can never go to zero on a single bet
- Says "don't bet" on −EV wagers — When edge ≤ 0, Kelly returns f* ≤ 0
- Scales with edge — Bigger edge = bigger bet, proportionally
- Self-correcting — As your bankroll grows, bets grow; as it shrinks, bets shrink
Worked Example 1: Standard −110 Bet
You estimate a 55% win probability on a −110 line (decimal odds 1.909, so b = 0.909).
With a $1,000 bankroll, Kelly says wager $55. This is a relatively modest edge, so Kelly recommends a modest bet.
Worked Example 2: +200 Longshot
You believe a +200 underdog (decimal odds 3.0, b = 2.0) has a 40% true probability to win.
The edge is large (expected value = 0.40 × 3.0 − 1 = +20%), so Kelly recommends a substantial 10% of your bankroll. But full Kelly here is extremely aggressive in practice.
Strategy Insight
Why Full Kelly Is Too Aggressive
Full Kelly is mathematically optimal in theory, but in practice it produces stomach-churning drawdowns:
- The median drawdown from peak during full Kelly betting is roughly 50%
- You'll experience a 90% drawdown at some point with meaningful probability
- Kelly assumes you know the exact probabilities—any overestimation of your edge leads to over-betting, which is far more damaging than under-betting
Warning
Fractional Kelly: The Practical Solution
Nearly all professional gamblers and institutional investors use fractional Kelly—typically betting a fraction of what full Kelly recommends.
Fractional Kelly Comparison
| Strategy | Growth Rate | Max Drawdown | Best For |
|---|---|---|---|
| Full Kelly (100%) | 100% of max | ~50% median | Theoretical maximum only |
| Half Kelly (50%) | ~75% of max | ~25% median | Most professionals use this |
| Quarter Kelly (25%) | ~44% of max | ~12% median | Conservative / uncertain edge |
| Tenth Kelly (10%) | ~19% of max | ~5% median | Very high variance bets |
The key insight: half Kelly gives you 75% of the maximum growth rate while cutting drawdowns roughly in half. This is an extraordinary risk-reward tradeoff, which is why half Kelly is the gold standard among professionals.
Strategy Insight
Simultaneous Kelly for Multiple Bets
When you have multiple simultaneous +EV bets (e.g., several games on the same day), the independent Kelly fractions don't simply add up. The total amount wagered must be constrained to prevent over-exposure.
For independent bets, a practical approach:
- Calculate individual Kelly fractions for each bet
- If the sum exceeds your comfort level (e.g., 20–30% of bankroll), scale them all down proportionally
- For correlated bets, reduce further—correlated losses compound the damage
This proportional scaling preserves the relative sizing between bets (better edge = bigger bet) while capping total risk.
Common Kelly Mistakes
Overestimating Your Edge
If your true win rate is 53% but you think it's 58%, you're over-betting by roughly 2×. This is why fractional Kelly is essential—it provides a buffer against estimation error.
Ignoring Correlation
Betting on multiple outcomes in the same game? Those bets are correlated. Treating them as independent overestimates diversification and leads to over-betting.
Applying Kelly to −EV Bets
Kelly only works when you have a genuine edge. No amount of clever bet-sizing turns a −EV bet into a winning proposition. Step one is always finding +EV opportunities.
Kelly and Bankroll Management
Kelly naturally integrates with bankroll management:
- Your "bankroll" is the amount you can afford to lose entirely—never your rent money
- Recalculate after every bet (your bankroll changes, so bet size should too)
- In practice, update bet sizes daily or weekly rather than after every single wager
| Strategy | % of Roll | Bet Size | Growth |
|---|---|---|---|
| Full Kelly | 10.0% | $100 | 0.50% |
| Half Kelly ★ | 5.0% | $50 | 0.38% |
| Quarter Kelly | 2.5% | $25 | — |
Half Kelly (★) is recommended — 75% of the growth rate with roughly half the drawdown risk.
Good to Know
Use our Kelly Criterion Calculator for full Kelly, half Kelly, and quarter Kelly sizing with growth-rate projections. Pair it with the Risk of Ruin Calculator to understand your probability of going bust at different bet sizes.
Sources & References
- Kelly, J. L. (1956). "A New Interpretation of Information Rate." Bell System Technical Journal, 35(4), 917–926. The original paper deriving the Kelly criterion from information theory.
- Thorp, E. O. (2006). "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." Handbook of Asset and Liability Management, Volume 1. Comprehensive treatment of practical Kelly applications in gambling and finance.
- MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (2011). "The Kelly Capital Growth Investment Criterion." World Scientific. Definitive academic collection covering fractional Kelly, simultaneous bets, and estimation error.
- Growth-rate and drawdown comparisons for fractional Kelly strategies are standard results derivable from the Kelly log-wealth framework. Half Kelly achieving ~75% growth at ~50% drawdown reduction is independently verifiable.
Mathematical claims are independently verifiable. BonusBell platform analysis reflects data from 220+ tracked platforms as of March 2026.
Key Takeaways
- 1Kelly maximizes long-term bankroll growth by betting f* = (bp − q) ÷ b of your bankroll
- 2Full Kelly is too aggressive in practice—use half Kelly for 75% growth with ~50% less drawdown
- 3Over-betting is far more dangerous than under-betting—at 2× Kelly, growth drops to zero
- 4Kelly only works on +EV bets—no sizing strategy can overcome a negative edge
- 5For multiple simultaneous bets, scale individual Kelly fractions down to cap total exposure