Kelly Criterion Position Sizing: What f* Is a Fraction Of

The Kelly formula returns a single number. Feed it a win rate and a payoff ratio and it answers with a fraction, and the arithmetic takes about ten seconds.

The hard part arrives afterwards. A fraction is meaningless until something is named as the thing it is a fraction of, and four widely read explanations of this formula each name a different one.

One stakes a share of a bankroll. One sets a weight inside an equity portfolio. One reserves a share of options buying power. One caps a percentage of account capital and then converts it into lots through margin. The same two inputs travel through all four and arrive at four different position sizes.

This page states what the fraction is a fraction of, why a leveraged currency position does not supply that denominator until a stop is placed, and what the formula assumes about an edge that a measured edge does not satisfy.

What the Kelly Fraction Computes

The formula takes two inputs. The first is the proportion of outcomes that are wins. The second is the ratio of the average win to the average loss. Written with W for the win rate and R for that ratio, the fraction is W minus the quantity one minus W divided by R.

Both inputs are defined and measured elsewhere on this site, and the page on what expectancy measures and the formula behind it sets out how a trade history produces them and how many trades it takes before either figure means anything. This page treats them as given and looks only at what happens to the answer.

Three illustrative sets of inputs show the shape of the output. At a win rate of 0.55 with a payoff ratio of 1.5, the fraction is 0.55 minus 0.45 divided by 1.5, which is 0.25. At a win rate of 0.40 with a payoff ratio of 3, it is 0.40 minus 0.60 divided by 3, which is 0.20. At a win rate of 0.50 with a payoff ratio of 1, it is 0.50 minus 0.50, which is zero.

That third case is the formula working correctly. With no edge it returns nothing to allocate. The first two are the ones that cause trouble, because 0.25 and 0.20 arrive without units and the reader has to supply them.

Four Sources, Four Different Denominators

Four treatments of this formula were read in full for this page. Each performs the same arithmetic and each attaches the result to a different quantity, and none of the four states that the attachment is a choice.

The encyclopedic treatment is the original gambling frame, where the fraction is the share of a bankroll placed on a single bet. The portfolio-management treatment allocates it as a weight across holdings in a long equity book. The options treatment reserves it as a share of buying-power reduction on a defined-risk position.

The trading-course treatment reads it as a percentage of account capital, applies a further cap of its own choosing, and then converts the survivor into a lot count through the margin required per lot.

Those are four different objects. A bankroll stake is money that disappears if the bet loses. A portfolio weight is money deployed that mostly stays deployed. A buying-power reduction is collateral held against a position whose maximum loss is defined by its structure. Margin is collateral against a position whose loss is not defined at all.

A trader who carries a fraction of 0.25 from one frame into another does not get a slightly different position. They get a position whose worst outcome differs by an order of magnitude. The two variables the formula rests on are shared across all four frames; the denominator is not.

A Leveraged Position Has No Stake Until the Stop Is Placed

The gambling frame works because a stake is simultaneously the amount committed and the amount at risk. Those two numbers are the same number, so a fraction of the bankroll is unambiguous.

A leveraged currency position separates them. The amount committed is the margin held against the position. The amount at risk is whatever the price does before the position is closed. Nothing in the account ties the second figure to the first.

MetaQuotes states the margin calculation for forex plainly: volume in lots multiplied by the contract size and divided by the leverage. One lot of a pair with a contract size of 100,000 units at leverage of 1:100 requires 1,000 units of the base currency as margin. Stop distance is absent from that equation, and so is the win rate, the payoff ratio and every other trading input.

So margin answers a different question. It says what the broker holds to let the position exist, not what the position can cost, and treating it as a Kelly denominator assumes a bound the account never established.

Placing a stop is what creates the bound, which is why the wider discussion of how much to risk on a single trade starts from the stop rather than from the size.

From the Fraction to a Lot Size Without Double-Counting Leverage

Once a stop distance is fixed, the loss on the position is that distance multiplied by the value of one unit of price movement at the chosen size. That product is a stake in the sense the formula requires, and the fraction becomes readable against account equity as a risk percentage for the trade.

The conversion from a risk percentage to a lot size is a separate calculation and it is set out on the page for turning a risk percentage into a lot size, which takes equity, stop distance and the value per pip and returns the size. The Kelly fraction supplies the first of those three and nothing else.

The error worth naming is applying the fraction twice. A trader who reads 0.25 as a quarter of equity and also as a quarter of available margin has produced two different orders from one number, and the larger carries a loss unrelated to the fraction that generated it. Leverage belongs to the lot-size conversion, and to no other step.

What the Formula Assumes That an Estimated Edge Does Not

The derivation treats the win rate and the payoff ratio as known and fixed. A trader supplies neither. Both are sample statistics from a finite record that is usually short, drawn from one market regime, and produced by a method still being adjusted while the trades were taken.

The consequence is arithmetic rather than philosophical. Return to the first illustrative case, a win rate of 0.55 and a payoff ratio of 1.5, which gives 0.25. Move the win rate down by five points to 0.50 and hold the ratio: the fraction becomes 0.50 minus 0.50 divided by 1.5, which is roughly 0.167.

A five-point movement in one estimate has removed about a third of the allocation, and five points is well inside the sampling error of a few dozen trades.

The output is therefore more sensitive to the inputs than the inputs are stable, and the error is asymmetric. Overstating the win rate raises the fraction, which raises the size, which raises the depth of the losing runs the account has to survive. The page on how a drawdown is measured covers what those runs look like once they arrive.

Which Figures This Page Does Not State, and Why

Three of the four treatments read for this page give a recommended reduction to apply to the full fraction, and each gives a different one. None of the three attributes its figure to any source, and no regulator, exchange or platform vendor publishes a figure of that kind, because it is a preference rather than a measurement.

Under the evidence rule this site follows, a number that several publications state and no official source confirms does not appear here at all. The same applies to the historical performance claims in the portfolio-management treatment, every one of which traces back to that firm running its own unpublished simulation.

Two of the four treatments also carry no modified date at all, and one dates its worked example to 2017, so a reader cannot tell whether the figures were ever revisited.

Who This Page Is Not For

A trader without a measured trade history has nothing to put into the formula. Estimating a win rate from memory or from a demo run produces a fraction with the same appearance of precision and none of the content, and the output will be wrong in the direction that increases size.

The formula also has nothing to say to anyone whose entries and exits are decided case by case, because there is no repeated bet for a long-run growth rate to apply to.

Five Checks Before the Number Reaches an Order Ticket

Name the denominator first, before computing anything, and write down which of the four quantities in the table above the fraction will apply to. Confirm the win rate and payoff ratio come from a recorded trade history rather than an impression. Place the stop before the size, so that a bounded loss exists for the fraction to describe.

Recompute the fraction with the win rate moved five points in the unfavourable direction and note how far the answer travels. Then convert the surviving percentage into lots through the position-size calculation, once, and check that leverage entered that step and no other.