Kelly Criterion Calculator

Defaults are Win Probability 55 percent, Decimal Odds 1.91, Bankroll 1000. Submit. Kelly % is 5.55% and Optimal Stake is 55.49. Here b = 1.91 − 1 = 0.91, p = 0.55, q = 0.45, f* = (b p − q)/b = (0.91×0.55 − 0.45)/0.91 = (0.5005 − 0.45)/0.91 = 0.0505/0.91 ≈ 0.0554945. Times 1000 is 55.49. That is full Kelly, the fraction that maximises the expected logarithm of wealth if p and b are exact and bets are i.i.d.

Half Kelly would be about 2.77% or 27.75 on this bankroll; the engine does not print fractions, so split the stake yourself if you want the conventional dampener. Drop p to 52.36% (the break-even at 1.91): f* goes to 0, stake 0. Below that, the formula would go negative and the engine clamps to 0. Raise p to 60% at 1.91: b p − q = 0.91×0.60 − 0.40 = 0.146, /0.91 ≈ 0.1604, stake 160.44 on 1000—aggressive, and sensitive to a 60% that is probably overconfident.

Bankroll is the dedicated roll, not monthly income. Changing 1000 to 4000 quadruples the stake to 221.98 and leaves Kelly % unchanged, because f* does not depend on wealth in the log-utility derivation. Odds 2.50 at 55%: b = 1.50, f* = (1.5×0.55 − 0.45)/1.5 = 0.375/1.5 = 0.25, stake 250 on 1000—full Kelly on a longshot with a claimed 55% is huge because the claimed edge is huge; distrust the 55% before you distrust the formula.

Reload 55, 1.91, 1000 to recover 5.55% and 55.49. Compare with the EV calculator’s 5.05 dollars on a 100 stake: Kelly is not “bet 100 because EV is plus.” It is a growth-optimal fraction given p and b. If your p is a guess, shrink f*. A worked half-Kelly: 0.5 × 55.49 = 27.75. Quarter-Kelly is 13.87. Those are not printed rows; they are the usual practical answers when the 55% came from a noisy model rather than a physical coin.

The Kelly criterion is a stake fraction f* that maximises E[log wealth] for a repeated bet with known odds and known win probability. John L. Kelly Jr. published it in a 1956 Bell System Technical Journal paper on information rate. Edward Thorp carried it into blackjack, sports, and then securities. Full Kelly is aggressive: it maximises median growth in the model and produces stomach-turning drawdowns in practice. Many practitioners use half or quarter Kelly.

Purpose of this calculator: turn p, decimal odds, and bankroll into f* and a cash stake for a binary bet. It is the sizing companion to the expected-value tool, which only reports the mean. Positive EV does not say how much to bet; Kelly does, under log utility and known p.

Limits: p is not known. Overestimating p is the main way Kelly blows up; the fraction is linear in the edge bp − q, so a 2-point fantasy in p is a large relative error when the true edge is 2 points. Simultaneous correlated bets need a multivariate Kelly that this one-shot formula is not. Minimum-odds clamps, table maxima, and book limits are ignored. Bankroll must be the true risk capital; including next month’s rent inflates f* into ruin.

History: Kelly’s paper is about horse-race-style side information and channel capacity; the gambling form f* = (bp − q)/b is the even-money generalisation with net decimal b = o − 1. Thorp’s Fortune’s Formula popularisation (via Poundstone’s later book of that name) and his market-making work made “Kelly” a slogan. Casinos do not care. Sportsbooks limit winners regardless of whether they used Kelly or units.

Not advice to bet 5.55% of wealth on a 1.91 ticket because a form defaulted to 55%. If p is 55% in truth and bets repeat, full Kelly is the growth-optimal fraction in the i.i.d. model. If p is 55% in hope, the optimal practical fraction is smaller, often much smaller. The clamp at zero when the edge is negative is the only “do not bet” the engine will ever utter.

The Kelly criterion is a stake fraction f-star that maximises the expected logarithm of wealth for a repeated bet with known odds and known win probability. John L. Kelly Jr. published it in a 1956 Bell System Technical Journal paper on information rate. Edward Thorp carried it into blackjack, sports, and then securities. Full Kelly is aggressive: it maximises median growth in the model and produces stomach-turning drawdowns in practice. Many practitioners use half or quarter Kelly. Positive expected value does not say how much to bet; this fraction does, under log utility and known p.

Derivation in one paragraph. Net odds b equal decimal odds minus one. Win probability p is the typed percent over 100, and q equals 1 − p. Wealth is multiplied by (1 + f b) with probability p and by (1 − f) with probability q. The growth rate G(f) equals p log(1 + f b) plus q log(1 − f). Setting the derivative to zero yields f-star = (b p − q) / b when b is positive, otherwise zero. If b p − q is not positive, the maximum on the unit interval is at zero and the worksheet holds the fraction at zero. The cash stake is bankroll times f-star. The numerator b p − q is the edge per unit staked, the same quantity that makes expected value equal to stake times (p o − 1). Kelly divides that edge by b.

The published defaults are 55 percent, decimal odds 1.91, and a bankroll of 1,000. Then b = 0.91, p = 0.55, q = 0.45. Numerator equals 0.91 × 0.55 − 0.45 = 0.5005 − 0.45 = 0.0505. Fraction equals 0.0505 / 0.91 ≈ 0.0554945, reported as 5.55 percent. Optimal stake equals 55.49 on a 1,000 roll. That 5.55 percent is not the folklore 55 percent at 2.10. At 2.10 one would have b = 1.10 and f-star = (1.10 × 0.55 − 0.45) / 1.10 = 0.155 / 1.10 ≈ 14.09 percent, a much fatter bet. The 5.55 percent figure is the 1.91 price, whose break-even is already 52.36 percent, so a 55 percent opinion is only a 2.64-point edge sitting on a short favourite.

Half Kelly would be about 2.77 percent or 27.75 on this bankroll; split the stake yourself if you want the conventional dampener. Drop p to 52.36 percent, the break-even at 1.91, and f-star goes to 0. Below that the formula would go negative and is clamped to 0. Raise p to 60 percent at 1.91: numerator 0.91 × 0.60 − 0.40 = 0.146, divided by 0.91 ≈ 0.1604, stake 160.44 on 1,000, aggressive and sensitive to a 60 percent that is probably overconfident. Bankroll is the dedicated roll, not monthly income. Changing 1,000 to 4,000 quadruples the stake to 221.98 without changing the percent.

Assumptions: known constant p, known b, logarithmic utility, infinitely divisible stakes, independent identically distributed repeats, no simultaneous correlated bets. Overestimating p is the main way Kelly blows up; the fraction is linear in the edge, so a two-point fantasy in p is a large relative error when the true edge is two points. The worksheet reports Kelly percent and optimal stake for a binary bet. It is full Kelly. It is not a promise that 55.49 is safe.