Kelly Criterion: Method

Edge, Expected Value and the Kelly Criterion

Payoff and implied probability

Every payoff format states the same thing: how much net profit b a unit stake returns on a favorable outcome. Moneyline +150 means b = 1.5, −110 means b = 100/110; a decimal payoff is 1 + b; a fractional payoff is b itself. A payoff of 1 + b is fair when the chance of the favorable outcome is 1/(1 + b), the implied probability. A market prices both sides so that the two implied probabilities add to more than one; the excess is the pricing margin, and dividing each implied probability by the sum gives the fair, margin-free probabilities.

Edge and expected value

Against your own estimate p of the true probability, the expected value of a unit stake is b p − q with q = 1 − p: positive only when p exceeds the implied probability. The edge in percentage points is p − 1/(1 + b). Everything that follows assumes p is right; an estimate a few points too optimistic turns a favorable opportunity into a losing one, which is the reason for the Kelly multiplier.

Kelly criterion

Committing a fixed fraction f of the capital, the capital grows by the factor (1 + b f) on a favorable outcome and (1 − f) on an unfavorable one, so its logarithm grows on average by g(f) = p ln(1 + b f) + q ln(1 − f) per trial. The fraction that maximizes g is Kelly's f* = (b p − q)/b. Staking more than f* lowers the growth rate, and beyond 2 f* the capital shrinks on average despite the positive expectation of each trial. Half Kelly gives three quarters of the growth with half the variance, and is the usual choice when p is an estimate. When the expected value is negative the Kelly fraction is negative and no stake grows the capital.

The simulation

Two thousand series of N trials are run with the stake taken as the fraction of the current capital, and the final capital and the paths are summarized by their median and 10th and 90th percentiles. The chance of a drawdown to a chosen fraction of the start is counted over the whole path, not just at the end. The median outcome follows the growth rate; the mean is larger because a few paths compound far above the rest.