[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"term-betting-kelly-criterion-en":3,"related-kelly-criterion-en":49,"mdc--61pz5x-key":65},{"id":4,"slug":5,"status":6,"section":7,"category":8,"difficulty":9,"aliases":10,"related_terms":16,"related_calculators":22,"term":25,"definition":26,"content":27,"example":28,"faq":29,"availableLocales":42},"10d4dace-0c02-4025-80ab-49d8488a472b","kelly-criterion","published","betting","strategies","advanced",[11,12,13,14,15],"Kelly formula","Kelly strategy","Kelly staking","Kelly bet","optimal f",[17,18,19,20,21],"bankroll","expected-value","edge","stake","bankroll-management",[23,24,24],"\u002Fbetting\u002Fkelly-calculator","\u002Fbetting\u002Fbankroll-growth-calculator","Kelly Criterion","The Kelly Criterion is a mathematical formula that calculates the optimal percentage of your bankroll to wager on a bet based on your edge and the odds offered. Developed by John Kelly at Bell Labs in 1956, it maximizes long-term bankroll growth while minimizing the risk of ruin. Professional bettors use fractional Kelly (25-50%) to reduce volatility.","# Kelly Criterion\n\nThe **Kelly Criterion** is the mathematically optimal formula for bet sizing, calculating exactly what percentage of your bankroll to wager based on your edge and the odds. Unlike flat staking or arbitrary unit systems, Kelly maximizes the geometric growth rate of your bankroll over time. It's the gold standard for professional bettors, investors, and anyone managing risk with positive expected value opportunities.\n\n## Table of Contents\n\n- [Understanding Kelly Criterion](#understanding)\n- [The Kelly Formula](#formula)\n- [Step-by-Step Calculation](#calculation)\n- [Fractional Kelly](#fractional)\n- [Kelly for Multiple Bets](#multiple-bets)\n- [Common Mistakes](#mistakes)\n\n## Understanding Kelly Criterion {#understanding}\n\nImagine you have a coin that lands heads 60% of the time, and someone offers you even money (2.0 odds) on heads. You have an edge—but how much should you bet?\n\n- Bet too little: You don't capitalize on your advantage\n- Bet too much: One bad run wipes out your bankroll\n- Bet optimally (Kelly): Maximum long-term growth\n\n**The Kelly Criterion answers:** Given your edge and the odds, what bet size maximizes wealth over time?\n\n**Key Insight:** Kelly doesn't maximize expected profit—it maximizes expected logarithmic utility, which translates to maximum geometric growth rate. This distinction is crucial for long-term wealth accumulation.\n\n### Why Kelly Works\n\n| Strategy | Short-term | Long-term |\n|----------|------------|-----------|\n| Bet everything | High variance | Bankrupt |\n| Flat 1% stakes | Low growth | Slow accumulation |\n| Kelly optimal | Balanced | Maximum growth |\n\nKelly finds the perfect balance between growth and survival. Understanding your [risk of ruin](\u002Fblog\u002Frisk-of-ruin-calculator) helps determine optimal Kelly fraction.\n\n## The Kelly Formula {#formula}\n\n### Basic Kelly Formula\n\n```math\nf^* = \\frac{bp - q}{b}\n```\n\nWhere:\n- f* = Fraction of bankroll to bet\n- b = Decimal odds - 1 (net odds)\n- p = Probability of winning\n- q = Probability of losing (1 - p)\n\n### Simplified Formula\n\nFor decimal odds:\n\n```math\n\\text{Kelly \\%} = \\frac{p \\times \\text{Odds} - 1}{\\text{Odds} - 1}\n```\n\nOr even simpler:\n\n```math\n\\text{Kelly \\%} = \\frac{\\text{Edge \\%}}{\\text{Odds} - 1}\n```\n\nWhere Edge % = (Probability × Odds) - 1\n\n### Kelly Formula Derivation\n\nThe formula maximizes expected log wealth:\n\n```math\nG = p \\times \\log(1 + f \\times b) + q \\times \\log(1 - f)\n```\n\nTaking the derivative and setting to zero yields the Kelly formula.\n\n## Step-by-Step Calculation {#calculation}\n\n### Example 1: Football Match\n\n**Scenario:** You estimate Liverpool has 55% chance to beat Chelsea. Bookmaker offers odds of 2.10.\n\n**Step 1:** Identify variables\n- Odds = 2.10\n- p (your probability) = 0.55\n- q = 1 - 0.55 = 0.45\n- b = 2.10 - 1 = 1.10\n\n**Step 2:** Calculate edge\n```math\n\\text{Edge} = (0.55 \\times 2.10) - 1 = 1.155 - 1 = 0.155 = 15.5\\%\n```\n\n**Step 3:** Apply Kelly formula\n```math\nf^* = \\frac{0.55 \\times 2.10 - 1}{2.10 - 1} = \\frac{0.155}{1.10} = 0.141 = 14.1\\%\n```\n\n**Result:** Bet 14.1% of your bankroll.\n\n### Example 2: Tennis Match (Underdog)\n\n**Scenario:** You estimate underdog has 35% chance. Odds are 3.50.\n\n- Edge = (0.35 × 3.50) - 1 = 0.225 = 22.5%\n- Kelly = 0.225 \u002F (3.50 - 1) = 0.225 \u002F 2.50 = **9%**\n\n### Example 3: No Edge (Negative Kelly)\n\n**Scenario:** True probability 45%, odds 2.00.\n\n- Edge = (0.45 × 2.00) - 1 = -0.10 = -10%\n- Kelly = -0.10 \u002F 1.00 = **-10%**\n\n**Negative Kelly means don't bet.** If you could bet against this outcome, you would.\n\n### Kelly Calculation Table\n\n| True Prob | Odds | Edge | Kelly % |\n|-----------|------|------|---------|\n| 50% | 2.00 | 0% | 0% |\n| 50% | 2.20 | 10% | 8.3% |\n| 55% | 2.00 | 10% | 10% |\n| 55% | 1.90 | 4.5% | 5% |\n| 60% | 1.80 | 8% | 10% |\n| 40% | 3.00 | 20% | 10% |\n| 30% | 4.00 | 20% | 6.7% |\n\n## Fractional Kelly {#fractional}\n\n### Why Use Fractional Kelly?\n\nFull Kelly assumes you know your exact edge. In reality:\n- Your probability estimates have errors\n- Sample sizes are limited\n- Edge can change over time\n\n**Fractional Kelly** (betting a fraction of full Kelly) addresses these issues.\n\n### Fractional Kelly Performance\n\n```math\n\\text{Fractional Kelly Growth} = f \\times G_{Kelly} \\times (2 - f)\n```\n\n| Fraction | Growth vs Full Kelly | Variance Reduction |\n|----------|---------------------|-------------------|\n| 100% (Full) | 100% | None |\n| 75% | 93.75% | Significant |\n| 50% (Half) | 75% | Major |\n| 25% (Quarter) | 43.75% | Very large |\n\n**Half Kelly achieves 75% of optimal growth with far less risk.**\n\n### Recommended Fractions by Confidence\n\n| Your Edge Confidence | Recommended Fraction |\n|---------------------|---------------------|\n| Very high (verified model, 1000+ bets) | 50-75% |\n| High (good model, 500+ bets) | 33-50% |\n| Medium (reasonable estimates) | 25-33% |\n| Low (uncertain) | 10-25% |\n\n### Fractional Kelly Example\n\nFull Kelly says bet 14.1%. Using half Kelly:\n\n```math\n\\text{Actual Bet} = 14.1\\% \\times 0.50 = 7.05\\%\n```\n\n## Kelly for Multiple Bets {#multiple-bets}\n\n### Simultaneous Independent Bets\n\nWhen placing multiple bets at once, reduce each bet's Kelly fraction:\n\n```math\n\\text{Adjusted Kelly}_i = \\frac{f_i^*}{\\sum f_j^*} \\times k\n```\n\nWhere k is your target total exposure (often capped at 30-50% of bankroll).\n\n### Practical Approach: Diversification\n\n| Number of Bets | Max Single Bet | Max Total Exposure |\n|----------------|----------------|-------------------|\n| 1 | Full Kelly | Full Kelly |\n| 2-3 | 2\u002F3 Kelly each | 50% total |\n| 4-5 | 1\u002F2 Kelly each | 50% total |\n| 6+ | 1\u002F3 Kelly each | 50% total |\n\n### Sequential vs Simultaneous\n\n**Sequential bets** (one after another resolves): Use full calculated Kelly each time—your bankroll updates.\n\n**Simultaneous bets** (all pending at once): Reduce allocation to prevent over-exposure.\n\n## Common Mistakes {#mistakes}\n\n### Mistake 1: Overestimating Your Edge\n\nThe most dangerous mistake. If you think your edge is 10% but it's actually 2%, full Kelly will devastate your bankroll.\n\n**Solution:**\n- Track your actual results over 500+ bets\n- Compare to closing line value (CLV)\n- Use fractional Kelly\n\n### Mistake 2: Ignoring Correlation\n\nKelly assumes independent bets. Betting on related outcomes (same game, same team) violates this assumption.\n\n**Example of correlated bets:**\n- Liverpool to win\n- Liverpool over 1.5 goals\n- Liverpool clean sheet\n\nThese aren't independent—treat as one large bet.\n\n### Mistake 3: Not Recalculating Bankroll\n\nKelly percentages should apply to your **current** bankroll, not starting bankroll.\n\n| Bankroll | 10% Kelly Bet |\n|----------|---------------|\n| $1,000 (start) | $100 |\n| $1,200 (after wins) | $120 |\n| $800 (after losses) | $80 |\n\nThis automatic adjustment is part of Kelly's power—you bet more when winning, less when losing.\n\n### Mistake 4: Applying to Negative EV\n\nKelly never recommends betting on negative expected value. If your calculation gives a negative number or zero, **don't bet**.\n\n### Mistake 5: Ignoring Practical Constraints\n\nKelly might suggest betting 25% of bankroll. Practical issues:\n- Bookmaker limits\n- Liquidity (can you actually get the bet down?)\n- Account preservation\n\n## Kelly Criterion in Practice {#practice}\n\n### The Professional Approach\n\n1. **Calculate theoretical Kelly**\n2. **Apply uncertainty discount** (typically 25-50% of Kelly)\n3. **Cap maximum bet** (usually 5-10% regardless of Kelly)\n4. **Consider correlation** with other bets\n5. **Track and adjust** based on results\n\n### Kelly with Bankroll Constraints\n\n| Constraint | Adjustment |\n|------------|------------|\n| Limited bankroll | Use absolute minimums ($10 regardless of %) |\n| Bookmaker limits | May need to distribute across books |\n| Account longevity | Sometimes underbetting preserves access |\n\n### Sample Kelly Betting Log\n\n| Date | Bet | Prob | Odds | Edge | Full Kelly | Actual Bet | Result |\n|------|-----|------|------|------|------------|------------|--------|\n| Jan 1 | Liverpool | 55% | 2.10 | 15.5% | 14.1% | 7% | Win |\n| Jan 2 | Man City | 70% | 1.50 | 5% | 10% | 5% | Win |\n| Jan 3 | Chelsea | 45% | 2.40 | 8% | 5.7% | 3% | Loss |\n\n## Kelly Criterion Visualized {#visualization}\n\n### Growth Rate by Bet Size\n\nFor a bet with 10% edge at 2.0 odds (Kelly = 10%):\n\n| Bet Size | Expected Growth Rate |\n|----------|---------------------|\n| 0% (no bet) | 0% |\n| 5% | ~0.45% per bet |\n| 10% (Kelly) | ~0.50% per bet |\n| 15% | ~0.45% per bet |\n| 20% (2x Kelly) | 0% per bet |\n| 25%+ | Negative growth |\n\n**At 2x Kelly, expected growth drops to zero.** Beyond that, you're mathematically expected to lose money despite having an edge.\n\n### Risk of Ruin by Strategy\n\n| Strategy | Risk of 50% Drawdown |\n|----------|---------------------|\n| Full Kelly | ~50% eventually |\n| Half Kelly | ~11% |\n| Quarter Kelly | ~1% |\n\n## Learn More\n\nFor a comprehensive guide including real-world examples, simulation results, and spreadsheet templates, read our [complete Kelly Criterion guide](\u002Fblog\u002Fkelly-criterion-explained).\n\n## Related Calculators\n\nApply Kelly Criterion with these tools:\n\n- [Kelly Criterion Calculator](\u002Fbetting\u002Fkelly-calculator) - Calculate optimal bet size\n- [Bankroll Management](\u002Fbetting\u002Fbankroll-growth-calculator) - Plan your bankroll strategy\n- [Bankroll Growth Calculator](\u002Fbetting\u002Fbankroll-growth-calculator) - Project growth over time",null,[30,33,36,39],{"answer":31,"question":32},"Betting more than Kelly ('overbetting') dramatically increases your risk of ruin without proportionally increasing returns. At 2x Kelly, your expected growth drops to zero. Above 2x Kelly, you're mathematically expected to go bankrupt regardless of your edge.","What happens if I bet more than Kelly suggests?",{"answer":34,"question":35},"Full Kelly assumes you know your exact edge, which you never do. Fractional Kelly (typically 25-50%) accounts for uncertainty in your edge estimates, reduces variance, and provides nearly the same long-term growth with much less risk. A 50% Kelly bettor achieves 75% of full Kelly growth with far fewer drawdowns.","Why do professional bettors use fractional Kelly?",{"answer":37,"question":38},"Technically yes, but since casino games have negative expected value (house edge), Kelly would recommend betting $0. Kelly only works when you have a positive edge. The only casino exception is card counting in blackjack, where skilled players can achieve positive EV.","Can Kelly Criterion be used for casino games?",{"answer":40,"question":41},"Kelly % = Edge \u002F Odds. More precisely: Kelly % = (Probability × Odds - 1) \u002F (Odds - 1). It tells you to bet more when your edge is larger and less when odds are longer, balancing growth against risk.","What is the Kelly Criterion formula in simple terms?",[43,44,45,46,47,48],"pt","en","ru","de","es","tr",[50,54,58,61],{"slug":17,"section":7,"category":8,"difficulty":51,"term":52,"definition":53},"beginner","Bankroll","A bankroll is the total amount of money a bettor has set aside exclusively for betting, completely separate from personal finances. Proper bankroll management determines stake sizes, protects against variance, and is considered the most important factor in long-term betting success. Without disciplined bankroll management, even skilled bettors with positive expected value will eventually go broke.",{"slug":21,"section":7,"category":8,"difficulty":55,"term":56,"definition":57},"intermediate","Bankroll Management","Bankroll management is the systematic approach to sizing bets and protecting betting funds to survive variance and maximize long-term growth. It determines how much to wager on each bet based on edge size, odds, and risk tolerance. Without proper bankroll management, even profitable bettors face ruin—a 10-bet losing streak at 10% stakes destroys 65% of funds. Proper staking ensures survival through inevitable downswings.",{"slug":18,"section":7,"category":8,"difficulty":55,"term":59,"definition":60},"Expected Value (EV)","The average profit or loss you can expect from a bet over the long run, calculated by multiplying each outcome's value by its probability — the single most important number that separates winning bettors from everyone else.",{"slug":20,"section":7,"category":62,"difficulty":51,"term":63,"definition":64},"fundamentals","Stake","A stake is the amount of money wagered on a single bet—your financial risk on that specific outcome. Stake sizing is the foundation of bankroll management, determining how much you risk per bet relative to your total betting fund. Professional bettors typically stake 1-5% of their bankroll per bet, with the exact percentage determined by their edge and confidence level.",{"data":66,"body":67},{},{"type":68,"children":69},"root",[70,77,90,96,156,162,167,185,195,205,212,298,311,317,323,629,634,657,663,668,923,928,1138,1143,1149,1154,1520,1525,1531,1537,1547,1557,1580,1590,1909,1919,2407,2417,2423,2432,2450,2456,2465,2483,2493,2499,2681,2687,2693,2698,2716,2725,2731,3047,3145,3153,3159,3233,3239,3244,3407,3413,3419,3424,3900,3905,3911,4006,4012,4022,4032,4038,4044,4049,4057,4075,4081,4086,4094,4112,4117,4123,4135,4195,4200,4206,4218,4224,4229,4247,4253,4259,4311,4317,4377,4383,4552,4558,4564,4569,4666,4676,4682,4741,4747,4759,4765,4770],{"type":71,"tag":72,"props":73,"children":74},"element","h2",{"id":5},[75],{"type":76,"value":25},"text",{"type":71,"tag":78,"props":79,"children":80},"p",{},[81,83,88],{"type":76,"value":82},"The ",{"type":71,"tag":84,"props":85,"children":86},"strong",{},[87],{"type":76,"value":25},{"type":76,"value":89}," is the mathematically optimal formula for bet sizing, calculating exactly what percentage of your bankroll to wager based on your edge and the odds. Unlike flat staking or arbitrary unit systems, Kelly maximizes the geometric growth rate of your bankroll over time. It's the gold standard for professional bettors, investors, and anyone managing risk with positive expected value opportunities.",{"type":71,"tag":72,"props":91,"children":93},{"id":92},"table-of-contents",[94],{"type":76,"value":95},"Table of Contents",{"type":71,"tag":97,"props":98,"children":99},"ul",{},[100,111,120,129,138,147],{"type":71,"tag":101,"props":102,"children":103},"li",{},[104],{"type":71,"tag":105,"props":106,"children":108},"a",{"href":107},"#understanding",[109],{"type":76,"value":110},"Understanding Kelly Criterion",{"type":71,"tag":101,"props":112,"children":113},{},[114],{"type":71,"tag":105,"props":115,"children":117},{"href":116},"#formula",[118],{"type":76,"value":119},"The Kelly Formula",{"type":71,"tag":101,"props":121,"children":122},{},[123],{"type":71,"tag":105,"props":124,"children":126},{"href":125},"#calculation",[127],{"type":76,"value":128},"Step-by-Step Calculation",{"type":71,"tag":101,"props":130,"children":131},{},[132],{"type":71,"tag":105,"props":133,"children":135},{"href":134},"#fractional",[136],{"type":76,"value":137},"Fractional Kelly",{"type":71,"tag":101,"props":139,"children":140},{},[141],{"type":71,"tag":105,"props":142,"children":144},{"href":143},"#multiple-bets",[145],{"type":76,"value":146},"Kelly for Multiple Bets",{"type":71,"tag":101,"props":148,"children":149},{},[150],{"type":71,"tag":105,"props":151,"children":153},{"href":152},"#mistakes",[154],{"type":76,"value":155},"Common Mistakes",{"type":71,"tag":72,"props":157,"children":159},{"id":158},"understanding-kelly-criterion-understanding",[160],{"type":76,"value":161},"Understanding Kelly Criterion {#understanding}",{"type":71,"tag":78,"props":163,"children":164},{},[165],{"type":76,"value":166},"Imagine you have a coin that lands heads 60% of the time, and someone offers you even money (2.0 odds) on heads. You have an edge—but how much should you bet?",{"type":71,"tag":97,"props":168,"children":169},{},[170,175,180],{"type":71,"tag":101,"props":171,"children":172},{},[173],{"type":76,"value":174},"Bet too little: You don't capitalize on your advantage",{"type":71,"tag":101,"props":176,"children":177},{},[178],{"type":76,"value":179},"Bet too much: One bad run wipes out your bankroll",{"type":71,"tag":101,"props":181,"children":182},{},[183],{"type":76,"value":184},"Bet optimally (Kelly): Maximum long-term growth",{"type":71,"tag":78,"props":186,"children":187},{},[188,193],{"type":71,"tag":84,"props":189,"children":190},{},[191],{"type":76,"value":192},"The Kelly Criterion answers:",{"type":76,"value":194}," Given your edge and the odds, what bet size maximizes wealth over time?",{"type":71,"tag":78,"props":196,"children":197},{},[198,203],{"type":71,"tag":84,"props":199,"children":200},{},[201],{"type":76,"value":202},"Key Insight:",{"type":76,"value":204}," Kelly doesn't maximize expected profit—it maximizes expected logarithmic utility, which translates to maximum geometric growth rate. This distinction is crucial for long-term wealth accumulation.",{"type":71,"tag":206,"props":207,"children":209},"h3",{"id":208},"why-kelly-works",[210],{"type":76,"value":211},"Why Kelly Works",{"type":71,"tag":213,"props":214,"children":215},"table",{},[216,239],{"type":71,"tag":217,"props":218,"children":219},"thead",{},[220],{"type":71,"tag":48,"props":221,"children":222},{},[223,229,234],{"type":71,"tag":224,"props":225,"children":226},"th",{},[227],{"type":76,"value":228},"Strategy",{"type":71,"tag":224,"props":230,"children":231},{},[232],{"type":76,"value":233},"Short-term",{"type":71,"tag":224,"props":235,"children":236},{},[237],{"type":76,"value":238},"Long-term",{"type":71,"tag":240,"props":241,"children":242},"tbody",{},[243,262,280],{"type":71,"tag":48,"props":244,"children":245},{},[246,252,257],{"type":71,"tag":247,"props":248,"children":249},"td",{},[250],{"type":76,"value":251},"Bet everything",{"type":71,"tag":247,"props":253,"children":254},{},[255],{"type":76,"value":256},"High variance",{"type":71,"tag":247,"props":258,"children":259},{},[260],{"type":76,"value":261},"Bankrupt",{"type":71,"tag":48,"props":263,"children":264},{},[265,270,275],{"type":71,"tag":247,"props":266,"children":267},{},[268],{"type":76,"value":269},"Flat 1% stakes",{"type":71,"tag":247,"props":271,"children":272},{},[273],{"type":76,"value":274},"Low growth",{"type":71,"tag":247,"props":276,"children":277},{},[278],{"type":76,"value":279},"Slow accumulation",{"type":71,"tag":48,"props":281,"children":282},{},[283,288,293],{"type":71,"tag":247,"props":284,"children":285},{},[286],{"type":76,"value":287},"Kelly optimal",{"type":71,"tag":247,"props":289,"children":290},{},[291],{"type":76,"value":292},"Balanced",{"type":71,"tag":247,"props":294,"children":295},{},[296],{"type":76,"value":297},"Maximum growth",{"type":71,"tag":78,"props":299,"children":300},{},[301,303,309],{"type":76,"value":302},"Kelly finds the perfect balance between growth and survival. 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