Growth at fractions and multiples of Kelly: 55% and 1.8
Full Kelly is 0.55 − 0.45 ÷ 1.8 = 30.0%; half Kelly keeps about three-quarters of the growth at half the risk.
| Size | Risk per trade | Log growth per trade | Share of full Kelly's growth |
|---|---|---|---|
| Quarter Kelly (0.25×) | 7.5% | 0.0346 | 45% |
| Half Kelly (0.5×) | 15.0% | 0.0583 | 76% |
| Full Kelly (1×) | 30.0% | 0.0770 | 100% |
| 2× Kelly | 60.0% | −0.0095 | none: below zero |
| 3× Kelly | 90.0% | −0.5064 | none: below zero |
Where the example comes from, and how to read it
The example from our guide to how much to risk per trade: a 55% win rate with an average win 1.8 times the average loss. Full Kelly is 0.55 − 0.45 ÷ 1.8 = 30.0% of the account per trade. The growth column is g(f), the expected log growth per trade, if both inputs were exact.
Half Kelly keeps about three-quarters of the growth at half the risk; quarter Kelly keeps a little under half at a quarter. At twice Kelly the growth is close to zero, and at three times it is clearly negative: past the peak, a larger size buys less growth, not more.
Why twice Kelly grows about zero
The curve peaks at f* and falls back through zero near 2 × f*: above about 58.5% in the 55% example.
| Win rate | Payoff R | Full Kelly | Growth at 2× Kelly | Growth negative above |
|---|---|---|---|---|
| 40% | 2 | 10.0% | 0.0007 | 20.4% |
| 50% | 1.5 | 16.7% | 0.0000 | 33.3% |
| 60% | 1 | 20.0% | −0.0024 | 38.9% |
| 55% | 1.8 | 30.0% | −0.0095 | 58.5% |
What negative growth means with a real edge
The growth curve rises to its peak at f* and falls after it. For small edges it is close to symmetric, so it returns to zero near 2 × f*. Exactly where depends on the inputs: sometimes just above zero, sometimes just below, and above about 58.5% in the 55% example. Beyond that point a real edge, bet too large, has negative expected log growth: over many trades the typical account shrinks, even though the average trade is still profitable.
Why estimated inputs make full Kelly overbet
If the measured 55% is really 45%, full Kelly grows −0.0019 a trade; quarter Kelly still grows 0.0141.
The sampling error, worked
Kelly treats the win rate and payoff as known. Yours come from a sample. From 100 independent trades, a measured 55% win rate has a standard error of √(0.55 × 0.45 ÷ 100) = 5.0 percentage points, so the true rate could plausibly sit anywhere from about 45% to 65%. A backtest chosen because it looked good is more likely to sit at the high end.
Suppose the true win rate is 45% with the same 1.8 payoff. The true Kelly fraction is then 14.4%, and the measured full Kelly of 30.0% is about 2.1 times it. Its expected growth per trade is −0.0019, below zero. Quarter Kelly of the measured numbers, 7.50%, still grows 0.0141 per trade under the true ones. That margin for error is the reason to use a fraction.
Even quarter Kelly here is a large size. Ten losses in a row at 7.50% each leave 0.925¹⁰ = 45.9% of the account. Kelly is a ceiling; your own limits, other open trades and any prop-firm loss limit can all call for less.
From a Kelly percentage to a position size
Take that share of the balance as the loss at your stop, then size the trade from it.
Which calculator does that step
The Kelly fraction is the share of the account you lose if a trade hits its stop. To turn it into a size, take that share of the balance as your dollar risk and divide by the loss per lot or contract at your stop: the forex position size calculator and the futures position size calculator do that step. To see how often a fixed risk per trade reaches a drawdown threshold, run the same win rate and payoff through the risk of ruin calculator.