Finance & Economics · Quantitative Trading & Crypto · Position Sizing
Kelly Criterion Calculator
Calculates the optimal fraction of capital to bet or invest using the Kelly Criterion formula, given win probability and win/loss payoff ratio.
Calculator
Formula
f* is the optimal fraction of capital to allocate. p is the probability of a win (between 0 and 1). q = 1 - p is the probability of a loss. b is the net odds received on the bet (i.e., the amount won per unit risked — a payoff ratio of 2 means you win $2 for every $1 risked). The formula tells you to bet a larger fraction when your edge is greater and the odds are more favorable.
Source: Kelly, J.L. (1956). A New Interpretation of Information Rate. Bell System Technical Journal, 35(4), 917–926.
How it works
The central insight of the Kelly Criterion is that both over-betting and under-betting are suboptimal in the long run. Bet too small and you leave compounding growth on the table. Bet too large — especially beyond the Kelly fraction — and the variance can destroy your capital even if you have a positive expected value on each individual bet. The Kelly formula strikes the mathematically optimal balance between growth and safety over an infinite sequence of identical bets.
The formula f* = (bp − q) / b requires three quantities: p, the probability of winning; q = 1 − p, the probability of losing; and b, the net payoff ratio (how much you gain per unit wagered on a win). For example, if a bookmaker offers 2-to-1 odds, then b = 2 — you win $2 for every $1 staked. If you assess your win probability as 55%, the Kelly fraction is ((2 × 0.55) − 0.45) / 2 = 0.325, meaning you should bet 32.5% of your bankroll. Note that a negative Kelly output means you have no edge and should not bet at all.
In practice, most professional gamblers and traders use a fractional Kelly strategy — typically half-Kelly (multiplying the full Kelly fraction by 0.5). This approach sacrifices some theoretical growth rate but dramatically reduces drawdowns and variance, making it far more psychologically manageable and robust to errors in probability estimation. The expected value per unit risked (bp − q) is also reported here: any positive figure confirms you have an edge, but only the Kelly fraction tells you how much to deploy.
Worked example
Suppose you are a sports bettor who has assessed that a particular team has a 60% probability of winning (p = 0.60). The bookmaker is offering decimal odds of 2.10, which translates to net payoff ratio b = 1.10 (you win $1.10 per $1.00 staked). Your bankroll is $5,000 and you decide to use a half-Kelly multiplier of 0.5.
Step 1 — Compute the full Kelly fraction:
f* = (bp − q) / b = (1.10 × 0.60 − 0.40) / 1.10 = (0.66 − 0.40) / 1.10 = 0.26 / 1.10 ≈ 0.2364 (23.64%)
Step 2 — Apply the half-Kelly multiplier:
Adjusted fraction = 0.2364 × 0.5 = 0.1182 (11.82%)
Step 3 — Compute the recommended bet size:
Recommended bet = $5,000 × 0.1182 = $591.00
Step 4 — Compute the expected value per unit:
EV = (1.10 × 0.60) − 0.40 = 0.66 − 0.40 = +0.26 per $1 risked.
This means you have a confirmed positive edge of 26 cents per dollar staked, and the half-Kelly strategy advises risking $591 of your $5,000 bankroll on this particular bet.
Limitations & notes
The Kelly Criterion assumes that your estimates of p and b are precise and stable — an assumption that is rarely perfectly met in real markets or sports betting. Even small overestimates of your win probability can lead to overbetting and severe drawdowns. This is the primary reason practitioners favour fractional Kelly strategies. The formula also assumes that each bet is independent and that you can size positions continuously, neither of which holds perfectly in practice — discrete bet sizes, correlated positions, and capital constraints all introduce real-world friction. Additionally, Kelly optimises for long-run median wealth, which means short-term variance can be very high; a sequence of losses at the full Kelly fraction can still temporarily cut your bankroll in half before recovery. Finally, the formula as presented here is designed for binary outcomes; for investments with continuous or multi-outcome return distributions, a generalised Kelly formula requiring the full probability distribution is required.
Frequently asked questions
What does a negative Kelly fraction mean?
A negative Kelly fraction means you have no mathematical edge on this bet or trade — the odds are not in your favour given your assessed win probability. The correct action according to Kelly is to bet nothing (f* = 0). You should never bet when the Kelly output is negative, as doing so has a negative expected logarithmic return and will erode your capital over time.
Why do professionals use half-Kelly instead of full Kelly?
Full Kelly maximises long-run geometric growth in theory, but it produces very high variance in practice. A sequence of losses can temporarily halve your bankroll even if you are betting correctly. Half-Kelly (multiplying the fraction by 0.5) reduces the growth rate by only about 25% while cutting variance roughly in half, making it far more psychologically tolerable and more robust to errors in probability estimation. Many professional gamblers and quantitative funds use between 25% and 50% Kelly.
What is the payoff ratio and how do I calculate it?
The payoff ratio b is the net amount you win per unit risked. For a bet at decimal odds of 2.50, you win $1.50 for every $1 staked, so b = 1.50. For a stock trade where you target a 3:1 reward-to-risk ratio, b = 3. In financial markets, b is often estimated from historical average win and loss sizes. The ratio b does not include the return of your original stake — it is purely the profit portion.
Can I use the Kelly Criterion for stock market investing?
Yes, but with important caveats. The standard binary Kelly formula applies most cleanly to bets with two outcomes. For stocks, practitioners typically estimate p as the historical win rate of a strategy and b as the average gain divided by the average loss. Some quant funds use the continuous Kelly formula derived from expected return and variance: f* = μ / σ², where μ is the expected excess return and σ² is the variance. Accurate estimation of these inputs is the central challenge.
Does Kelly Criterion guarantee no ruin?
In the theoretical model with continuous bet sizing and an infinite sequence of bets, Kelly betting on a positive-expectation game ensures that your bankroll grows to infinity and never reaches zero. However, in practice, with discrete bet sizes and finite sequences, ruin is still possible — especially if your probability estimates are wrong. The Kelly formula minimises the probability of ruin asymptotically, but it is not a guarantee in real-world settings.
Last updated: 2025-01-15 · Formula verified against primary sources.