{
  "id": "kelly-criterion",
  "version": "b9eb6db7ff32",
  "status": "published",
  "name": "Kelly Criterion Calculator",
  "question": "What is the Kelly criterion bet size?",
  "summary": "Works out the Kelly criterion stake for a bet from your win probability and the odds, as a share of your bankroll and in money, with the edge and growth rate.",
  "category": "game",
  "subcategory": "betting",
  "url": "https://www.acalculator.org/game/kelly-criterion-calculator",
  "markdown": "https://www.acalculator.org/game/kelly-criterion-calculator.md",
  "kind": "function",
  "method": "b = decimal odds − 1 (or American / 100, or 100 / |American|); q = 1 − p; edge = b × p − q; f* = edge ÷ b when the edge is above 0, else 0; stake = bankroll × f* × multiplier.",
  "assumptions": [
    "The chance to win is your own estimate; the Kelly stake is only as good as it.",
    "One bet at a time, with the stake lost in full on a loss and no push.",
    "The growth rate is the expected change in the log of the bankroll per bet, at the share staked."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "p": {
        "title": "Chance to win",
        "description": "Your own estimate of the chance the bet wins.",
        "type": "number",
        "x-unit": "percent",
        "exclusiveMinimum": 0,
        "exclusiveMaximum": 100
      },
      "fmt": {
        "title": "Odds format",
        "description": "Whether the odds are decimal (2.10) or American (+110 or −150).",
        "type": "string",
        "enum": [
          "decimal",
          "american"
        ]
      },
      "odds": {
        "title": "Decimal odds",
        "description": "The decimal odds: the total paid back for each 1 staked, stake included.",
        "type": "number",
        "exclusiveMinimum": 1,
        "maximum": 1000000
      },
      "us": {
        "title": "American odds",
        "description": "The American odds: +150 wins 150 per 100 staked, −200 stakes 200 to win 100.",
        "type": "number",
        "minimum": -100000000,
        "maximum": 100000000
      },
      "bank": {
        "title": "Bankroll",
        "description": "The money you set aside for betting.",
        "type": "number",
        "x-unit": "USD",
        "exclusiveMinimum": 0,
        "maximum": 1000000000000
      },
      "mult": {
        "title": "Kelly multiplier",
        "description": "The share of the full Kelly stake to bet: 100% is full Kelly, 50% is half Kelly.",
        "type": "number",
        "x-unit": "percent",
        "exclusiveMinimum": 0,
        "maximum": 100
      }
    }
  },
  "outputs": {
    "stake": {
      "label": "Stake",
      "description": "Bankroll × Kelly fraction × multiplier; 0 when the bet has no edge.",
      "format": "money"
    },
    "kelly": {
      "label": "Full Kelly fraction",
      "description": "f* = p − q ÷ b: the share of the bankroll full Kelly stakes (0 when f* is not above 0).",
      "format": "percent"
    },
    "share": {
      "label": "Share of bankroll to stake",
      "description": "The full Kelly fraction × the multiplier.",
      "format": "percent"
    },
    "edge": {
      "label": "Edge",
      "description": "The expected profit per 1 staked: b × p − q.",
      "format": "percent"
    },
    "breakEven": {
      "label": "Break-even chance",
      "description": "The chance to win at which the bet has no edge: 1 ÷ (1 + b).",
      "format": "percent"
    },
    "growth": {
      "label": "Expected growth per bet",
      "description": "p × ln(1 + f × b) + q × ln(1 − f) for the share staked, as a percent of the bankroll.",
      "format": "percent"
    },
    "verdict": {
      "label": "Verdict",
      "description": "Whether the Kelly criterion says to bet.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "p": 55,
      "fmt": "decimal",
      "odds": 2.1,
      "us": 110,
      "bank": 1000,
      "mult": 100
    },
    "outputs": {
      "stake": 140.9090909090909,
      "kelly": 14.090909090909092,
      "share": 14.090909090909092,
      "edge": 15.5,
      "breakEven": 47.61904761904762,
      "growth": 1.0908950103842672,
      "verdict": "Bet"
    },
    "text": "At a 55% chance to win, the Kelly criterion stakes $140.91 (14.09% of the bankroll)."
  },
  "examples": [
    {
      "given": {
        "p": 60,
        "fmt": "decimal",
        "odds": 2,
        "bank": 1000,
        "mult": 100
      },
      "expect": {
        "kelly": 20,
        "stake": 200,
        "edge": 20,
        "breakEven": 50,
        "growth": 2.0135513550688864,
        "verdict": "Bet"
      },
      "source": "Wikipedia, Kelly criterion (f* = p − q ÷ b; 60% to win at even odds stakes 20%), https://en.wikipedia.org/wiki/Kelly_criterion (retrieved 2026-10-05); J. L. Kelly Jr., A New Interpretation of Information Rate, Bell System Technical Journal 35(4), 1956, https://doi.org/10.1002/j.1538-7305.1956.tb03809.x (retrieved 2026-10-05); hand calculation in content.mdx"
    },
    {
      "given": {
        "p": 55,
        "fmt": "decimal",
        "odds": 2.1,
        "bank": 1000,
        "mult": 50
      },
      "expect": {
        "kelly": 14.090909090909092,
        "share": 7.045454545454546,
        "stake": 70.45454545454545,
        "edge": 15.5
      },
      "source": "hand calculation in content.mdx: (1.1 × 0.55 − 0.45) ÷ 1.1; Wikipedia, Kelly criterion (f* = p − q ÷ b; 60% to win at even odds stakes 20%), https://en.wikipedia.org/wiki/Kelly_criterion (retrieved 2026-10-05)"
    },
    {
      "given": {
        "p": 40,
        "fmt": "american",
        "us": 150,
        "bank": 500,
        "mult": 100
      },
      "expect": {
        "kelly": 0,
        "stake": 0,
        "edge": 0,
        "breakEven": 40,
        "verdict": "No bet: the odds give no edge at this chance to win"
      },
      "source": "hand calculation in content.mdx: b = 1.5, 1.5 × 0.4 − 0.6 = 0; Wikipedia, Kelly criterion (f* = p − q ÷ b; 60% to win at even odds stakes 20%), https://en.wikipedia.org/wiki/Kelly_criterion (retrieved 2026-10-05)"
    },
    {
      "given": {
        "p": 75,
        "fmt": "american",
        "us": -200,
        "bank": 1000,
        "mult": 100
      },
      "expect": {
        "kelly": 25,
        "stake": 250,
        "edge": 12.5
      },
      "source": "hand calculation in content.mdx: b = 0.5, 0.75 − 0.25 ÷ 0.5 = 0.25; Wikipedia, Kelly criterion (f* = p − q ÷ b; 60% to win at even odds stakes 20%), https://en.wikipedia.org/wiki/Kelly_criterion (retrieved 2026-10-05)"
    }
  ],
  "sources": [
    "J. L. Kelly Jr., A New Interpretation of Information Rate, Bell System Technical Journal 35(4), 1956. https://doi.org/10.1002/j.1538-7305.1956.tb03809.x (retrieved 2026-10-05)",
    "Wikipedia, Kelly criterion: f* = p − q ÷ b and the 60% at even odds example. https://en.wikipedia.org/wiki/Kelly_criterion (retrieved 2026-10-05)"
  ],
  "related": [
    "odds",
    "implied-probability",
    "betting-odds"
  ],
  "changelog": []
}