Kelly criterion

A result from information theory, published by John Kelly at Bell Labs in the nineteen fifties and carried into finance afterwards. It answers a narrow question: for a bet that repeats, with a known probability of winning and a known payoff per unit staked, what fraction of the current bankroll makes the bankroll grow fastest over the long run. The answer is a single number between zero and one, and it rises with the probability of winning and with the size of the win relative to the loss. Where the edge is zero or negative the formula returns zero or less, which is its way of saying that no stake satisfies the objective it was built for.

The objective matters, because it is often skipped. The fraction Kelly returns maximises the expected logarithm of wealth, not the expected wealth itself, and those are different targets: the logarithmic objective penalises large losses far more heavily, because a loss is repaired by a proportionally larger gain. That is a statement about a mathematical model of repeated independent bets under stated assumptions, not a statement about any market. Applied to trading, the win probability and the payoff are replaced with estimates drawn from a backtest or a trading record, and the formula inherits every error in those estimates.

The disagreement among practitioners is not about the algebra, which is settled, but about whether the inputs a trading record supplies are stable enough to feed it. Two objections recur. The first is estimation error: an overstated win probability produces an overstated fraction, and the resulting path of a bankroll is far more volatile than the estimate suggested, which is why fractional Kelly, meaning a stated fraction of the number the formula returns, is the more common form in practice. The second is structural: the two outcome version assumes every win is the same size and every loss is the same size, and a discretionary record almost never looks like that, so a continuous form or a simulation is used instead by those who use it at all.

The usual confusion is treating the output as a risk limit. It is neither a stop distance nor a loss cap: it describes the proportion of a bankroll placed at stake, and it says nothing about where a position is closed, what the drawdown along the way would look like, or what happens when the estimates change. It also assumes losses are bounded by the amount staked, which is not true of a contract calculated on the full contract value, where losses are calculated on that value and are not limited to the amount deposited.