Expected growth per trade at the wrong win rate: full Kelly, half Kelly and a 2% risk, with break-even win rates, two ways
Our own calculation, 8 November 2026 (UTC), for the article "Full Kelly, half Kelly or 2% risk: what a misjudged win rate costs per trade". A trade gains 10% of the position with probability p and loses 10% otherwise; f is the position as a share of capital (f = 2 is full Kelly for an estimated 60% win rate: f* = p/l - q/g = 0.6/0.1 - 0.4/0.1 = 2; f = 1 is half Kelly; f = 0.2 risks 2% of capital). Expected log growth per trade: p ln(1 + 0.1 f) + (1 - p) ln(1 - 0.1 f). Way 1: that formula. Way 2: the average log growth over an exact sequence of 10,000 trades with p x 10,000 wins (the same to rounding). Break-even win rates: way 1 a numeric search for the p where growth is zero; way 2 the closed form p* = ln(1/(1 - 0.1 f)) / ln((1 + 0.1 f)/(1 - 0.1 f)); the two agree to nine decimals.
What we found
- Full Kelly at an estimated 60% (f = 2) grows +2.01% per trade if 60% is right, -0.01% at 55%, -2.04% at 50% and -6.10% at 40%.
- Half Kelly (f = 1): +1.50% at 60%, +0.50% at 55%, -0.50% at 50%, -2.51% at 40%.
- A 2% risk (f = 0.2): +0.38% at 60%, +0.18% at 55%, -0.02% at 50%, -0.42% at 40%.
- A position of 20% of capital (f = 0.2), which risks 2% of capital per trade, loses 0.42% per trade at 40% and still grows at 55% (+0.18%).
- Break-even win rates: 50.50% at f = 0.2, 52.50% at f = 1 and 55.03% at f = 2.
| True win rate | f = 0.2 (risk 2%) | f = 1 (half Kelly) | f = 2 (full Kelly at 60%) |
|---|---|---|---|
| 40% | -0.42% | -2.51% | -6.10% |
| 45% | -0.22% | -1.51% | -4.07% |
| 50% | -0.02% | -0.50% | -2.04% |
| 55% | +0.18% | +0.50% | -0.01% |
| 60% | +0.38% | +1.50% | +2.01% |
| 65% | +0.58% | +2.51% | +4.04% |
What this cannot tell you: real markets, where gains and losses are not a fixed 10% and the win rate drifts.