EV & Kelly Criterion Calculator

How to use this calculator

Give the calculator a fair probability and the odds you are offered. Enter the probability directly, type the fair odds instead, or switch to from market odds and paste a sharp bookmaker's full market: the calculator removes the vig and uses the resulting probability of the outcome you pick. Add a bankroll to turn the Kelly fraction into an actual stake, and select an exchange if commission applies.

Expected value

Expected value is what a bet is worth per unit staked, averaged over every way it can end:

EV per unit = p × (odds − 1) − (1 − p), which simplifies to p × odds − 1

where p is the fair probability and odds are decimal. With a fair probability of 55% and a price of 2.00, the expected value is 0.55 × 2.00 − 1 = 0.10: ten cents of profit for every unit staked, in the long run. The fair price for a 55% chance is 1.818, or -122 in American odds, so 2.00 is a better price than fair. Open this example.

If the exchange charges 5% commission on winnings, the calculator prices the bet after commission: the same 2.00 is worth 1.95, the expected value drops to 0.0725 per unit and the Kelly fraction with it.

Where the fair probability comes from

The number that matters most is p, and the calculator cannot tell you whether it is right. Two sources are defensible:

• A sharper market, devigged. Take the full market at the sharpest bookmaker or exchange you can find, remove the margin with the devig calculator, and use the fair probability of your outcome. The market mode of this calculator does that in one step.
• Your own model, if it has a record of being calibrated: outcomes you rate at 30% happen about 30% of the time.

Devigging a recreational bookmaker's prices recovers that bookmaker's opinion with the margin removed, not the truth. If that is the only market available, treat the result as an upper bound on your edge.

Edge, EV and ROI

The three terms describe the same thing at different scales. EV per unit is the profit per 1.00 staked, 0.10 in the example above. Edge is the same number as a percentage, 10%. ROI is what you observe after the fact: profit divided by total stakes over a run of bets, which converges on the edge only after many bets. A 10% edge at even money still loses 45% of the time. Positive expected value is a statement about the average, not about the next bet.

The Kelly criterion

Kelly's formula gives the fraction of bankroll that maximises long-run growth:

f = (b × p − q) / b

where b is the net odds (decimal odds minus one), p the fair probability and q = 1 − p. At 2.00 with a 55% chance, b = 1, so f = (0.55 − 0.45) / 1 = 0.10: bet 10% of bankroll. With a 1,000 bankroll that is a full Kelly stake of 100, half Kelly of 50 and quarter Kelly of 25. A bet with negative expected value gives a negative fraction, which the calculator reports as zero: do not bet.

Why full Kelly is too aggressive in practice

The formula assumes p is exactly right. It never is, and Kelly punishes overestimates harder than underestimates: betting twice the true Kelly fraction has an expected growth rate of zero, and anything above that loses money over time even with a real edge. Full Kelly also produces swings most people cannot stomach. There is a one-in-three chance of halving the bankroll at some point before it doubles.

Fractional Kelly

Half Kelly keeps about three quarters of the growth rate with half the variance, and quarter Kelly is common among people who bet for a living. The calculator shows all three so the choice is explicit. A sensible rule is to size the fraction to how much you trust the probability: quarter Kelly for a model you are still testing, half Kelly for a devigged sharp line.

Several bets at once

The formula assumes one bet at a time. Several open bets compete for the same bankroll, and full Kelly on each overbets in total. A practical fix is to size each bet against the bankroll that would remain if every other open bet lost. Correlated bets (two legs on the same match) need more caution still, because they can lose together.

Frequently asked questions

What is a good Kelly fraction?

Half Kelly for probabilities from a devigged sharp market, quarter Kelly for your own model until it has a track record. Full Kelly is the theoretical maximum growth rate, not a recommendation.

Why is my Kelly stake zero or negative?

The fair probability times the offered odds is below one, so the bet has negative expected value. A 60% chance at 1.60 has an EV of 0.60 × 1.60 − 1 = −0.04, and Kelly says not to bet. Check the price and the probability; if both are right, the bet is simply not value.

Half Kelly or quarter Kelly?

Quarter Kelly if your probability estimates have not been tested against results, or if several bets are open at once. Half Kelly if the edge comes from a sharp closing line and you can tolerate the swings.

The expected value and Kelly criterion guide works through the growth-rate maths and the reasoning behind fractional Kelly. To check a price against every devig method at once, use the price-check panel of the devig calculator; to translate the odds first, use the odds converter.