The Kelly Criterion
A formula for the position size that maximises long-run growth. Mathematically clean, extremely sensitive to inputs nobody knows, and almost never used at full size.
MadStockAlerts Research · Updated August 28, 2026
What to take away
- It gives the fraction of capital that maximises the growth rate over many repetitions.
- The inputs are the probability of winning and the payoff ratio.
- Overestimating the edge produces catastrophic oversizing.
- Full Kelly produces drawdowns most people cannot hold.
- Fractional Kelly is what is used in practice, and it is an admission about the inputs.
MAD Academy Training Video · 0:45
The Optimal Bet Nobody Should Take
Kelly gives the mathematically growth-optimal position size, and almost every practitioner uses a fraction of it, for good reasons.
This lesson is part of a Stock Alerts + Tools plan.
The formula
f = p - (1 - p) / b
- p is the probability of a favourable outcome
- b is the ratio of the gain to the loss
- f is the fraction of capital to commit
At a 55 percent probability with a one-to-one payoff, the formula gives ten percent of capital. At 40 percent with a three-to-one payoff, it gives twenty percent. Both are far larger than any conventional risk rule.
The result maximises the expected logarithm of wealth, which over many repetitions maximises the growth rate. It does not maximise the expected value of any single outcome, and the distinction is the whole of what makes it interesting.
Why it is dangerous in practice
The formula assumes the inputs are known. In markets they are estimated from a limited record, and the sizing is extremely sensitive to the estimate of the edge.
Scroll the chart sideways to see all of it.
Because the curve is asymmetric, an edge that is overestimated by a factor of two produces zero growth and one overestimated further produces ruin. Half Kelly delivers three quarters of the growth with far less sensitivity, which is why fractional sizing is the practical form.
The drawdowns it accepts
Full Kelly sizing is designed to maximise growth and is indifferent to the path. The drawdowns it produces are severe: declines of half the capital are an ordinary feature rather than an exception.
That is the practical reason it is rarely used at full size. A mathematically optimal rule that will be abandoned during its ordinary drawdowns is not optimal for the person applying it, which is the same constraint the allocation article describes.
What it is useful for
- As an upper bound: sizing beyond the Kelly fraction reduces long-run growth, which is a definitive result.
- As a demonstration that position size has an optimum rather than being a preference.
- As a reminder that the sensitivity to the edge estimate is severe.
- As a framework for thinking about correlated positions, where the sizing must account for the whole book.
Extending it to a portfolio
The simple formula assumes one bet at a time. With several positions open the sizing has to account for how they relate, and the extension is where the practical difficulty lies.
- Positions that are correlated must be sized jointly, since they are closer to one larger bet.
- The joint calculation requires a covariance estimate, which carries the same error as any other.
- Total exposure across positions is capped by the same growth-maximising logic that caps a single one.
- In practice this reduces to the portfolio heat constraint, computed by exposure rather than by position.
The fourth item is the useful conclusion. The formal extension requires estimates nobody has; the practical version is a cap on total risk across correlated exposures, which is what portfolio heat measures.
That connection is worth making explicit, because it means the account-level rule most trading plans omit has a formal justification rather than being merely prudent.