Skip to content

    Under active development. New calculators and features ship every week. Everything currently runs free while the site matures — pricing pages are a preview, checkout is not open yet.

    Search ⌘K
    Calculates locally in your browser · growth-optimal sizing
    Quantitative finance / capital allocation

    Kelly criterion

    Kelly gives the stake that maximises long-run compound growth for a known edge. It is also the most over-applied formula in finance, because it assumes the edge is known — so this page reports the growth curve and the over-betting cliff alongside the number.

    Positive edgeYesOdds b1.20Expected value$459.38
    Full Kelly fraction
    17.50%
    Growth-optimal, maximum variance
    Applied stake
    $2,188
    8.75% of $25,000
    Expected log growth
    0.0137
    Per bet, at the applied fraction
    Edge per unit
    0.2100
    p·b − q, before sizing
    Stake sizing by Kelly multiple
    Growth rate and drawdown risk at each fraction of the full Kelly stake.
    8.8% of bankroll
    Stake sizing by Kelly multiple
    MultipleFractionStakeLog growthVerdict
    0.25×4.38%$1,0940.0080Under-betting
    0.50×8.75%$2,1880.0137Under-betting
    0.75×13.13%$3,2810.0171Under-betting
    1.00×17.50%$4,3750.0183Growth-optimal
    1.50×26.25%$6,5630.0136Over-betting

    Growth is flat near the optimum, so half Kelly gives up very little growth for roughly half the variance — which is why practitioners rarely bet full Kelly.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Growth rate vs stake fraction
    Concave near the optimum, then sharply negative past twice Kelly.
    Growth rate vs stake fraction0.0000.0040.0080.0120.0160.0200.00.01.81.83.53.55.35.37.07.08.88.810.510.514.014.017.517.521.021.0Log growth
    Stake at each Kelly multiple
    Dollars committed at quarter, half and full Kelly.
    Stake at each Kelly multiple$0$1.0k$2.1k$3.1k$4.1k$5.2k$1.1k0.25×0.25×$2.2k0.5×0.5×$3.3k0.75×0.75×$4.4k1×1×Stake
    Canonical equations
    The exact relationships implemented by this engine.

    f* = (p(b+1) − 1)/b · equivalently p − q/b

    g = p ln(1+fb) + q ln(1−f) · expected log wealth per bet

    f = k·f* · k < 1 trades growth for stability

    f > 2f* ⇒ g < 0 · past twice Kelly, growth turns negative

    finance methodology

    A clear path from inputs to results.

    Kelly maximises long-run compound growth given a known edge. It is also the most over-applied formula in finance, because the edge is never known — so this page reports the growth curve and the over-betting cliff, not just the number.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Enter the probability of winning.
    2. Enter the net odds: how much you win per unit staked.
    3. Set the bankroll.
    4. Choose the Kelly multiple — 0.5 is the usual practical choice.
    5. Check that the edge is positive before sizing anything.

    The formula

    f∗=p(b+1)−1b,g(f)=pln⁡(1+fb)+qln⁡(1−f)f^{*} = \frac{p(b+1)-1}{b}, \qquad g(f) = p\ln(1+fb) + q\ln(1-f)

    Reference: Kelly (1956); Thorp (2006) on the Kelly criterion in markets.

    Worked example

    55% win rate at 1.2:1 odds, $25,000 bankroll

    Inputs
    • Win probability 55%.
    • Net odds 1.2:1.
    • Bankroll $25,000.
    • Half Kelly.
    Calculation path
    1. 1

      Edge: 0.55 × 1.2 − 0.45 = 0.21 per unit staked.

    2. 2

      Full Kelly: (0.55 × 2.2 − 1) / 1.2 = 17.5% of bankroll.

    3. 3

      Half Kelly: 8.75%, or $2,187.50.

    4. 4

      Log growth at half Kelly is slightly below the full-Kelly maximum.

    5. 5

      Betting above 35% (twice Kelly) turns growth negative despite the edge.

    Result

    Full Kelly 17.5% · half Kelly 8.75% ($2,188) · over-betting threshold 35%.

    Interpreting the result

    Growth is flat near the optimum, which is the practical case for fractional Kelly: half Kelly gives up very little growth for roughly half the volatility and a much smaller chance of a catastrophic drawdown. The asymmetry at the top end is brutal — twice Kelly is zero growth, and beyond that the long-run outcome is ruin.

    Assumptions & limitations

    • The win probability and payoff are known and constant across bets.
    • Bets are independent and can be repeated indefinitely.
    • No transaction costs, taxes or position limits.
    • The bankroll is fully available and cannot be replenished.

    Frequently asked questions

    What if Kelly says the fraction is negative?

    There is no edge, and the correct stake is zero. A negative Kelly is the formula telling you to take the other side or walk away, not to bet a small amount.

    Does Kelly apply to stock portfolios?

    Yes, in continuous-time form it becomes the growth-optimal portfolio, which for a single risky asset is the excess return divided by variance. With multiple assets it is the mean-variance tangency portfolio.

    Why do practitioners use half Kelly?

    Because the inputs are estimates. If you overestimate your edge, full Kelly is already over-betting; half Kelly gives a margin of safety and halves the variance for a small growth cost.

    Why is over-betting so dangerous?

    Because log growth is concave and turns sharply negative past the optimum. At twice Kelly the expected growth rate is exactly zero, and any further increase guarantees a declining bankroll over time.

    Methodology last reviewed 2026-10-18. Results are informational and not financial or engineering advice — see our disclaimer.