Trading Expectancy, Sample Size and Sizing Beyond the 1% Rule

The 1% rule exists for a plain reason. An account is very hard to destroy while that rule is obeyed, which makes it an excellent place to begin. As a place to stop, I would argue, it is rather strange.

The arithmetic that follows is not difficult, and most traders never perform it. It answers a question that traders rarely ask aloud, and that few trading educations bother to raise at all: how confident am I that the edge I believe I have is real? Once a trade log exists in sufficient quantity to be analysed, sizing stops being a matter of taste and becomes a question with an answer, although the first step is establishing whether the edge being sized exists at all.

Expectancy, properly stated

Every closed trade converts into an R multiple, which is the result divided by the amount risked. A full stop-out is −1R. A trade that made twice the risk is +2R.

Expectancy = the mean of your R multiples

The resulting figure is your average result per trade, and its virtue is comparability: across instruments, across account sizes and across time periods it carries the same meaning, which is something dollars and percentages cannot manage.

A worked example from a 100-trade log:

	Count	Average R
Winners	38	+2.4R
Losers	62	−0.95R

Expectancy = (0.38 × 2.4) − (0.62 × 0.95) = +0.323R per trade

At $250 of risk per trade that comes to about $81 per trade, or $8,100 over the hundred. The figure looks decisive. It is considerably less decisive than it looks.

The question almost nobody asks: how confident are you?

Expectancy is a sample mean. Sample means carry error bars.

Standard error = standard deviation of R multiples ÷ √n

Suppose those R multiples have a standard deviation of 1.6R, which is entirely typical when winners run to +4R and losers cluster near −1R.

• Standard error = 1.6 ÷ √100 = 0.16R
• 95% confidence interval ≈ 0.323 ± (1.96 × 0.16) = +0.01R to +0.64R

So the edge sits only just clear of zero after a hundred trades. Run the identical arithmetic across a 30-trade sample and the interval swallows zero comfortably, which amounts to saying that the data cannot distinguish the strategy from a coin flip, however convinced the owner of that strategy happens to feel about it.

Halving the width of that interval requires four times the trades, because the error shrinks with the square root of n, which makes evaluation slow for reasons of arithmetic alone.

The Kelly criterion, and why not to use it in full

Kelly gives the fraction of capital that maximises long-run growth. In trading terms:

f = W − (1 − W) ÷ R

where W is the win rate and R is the ratio of the average win to the average loss, both of them taken from your own log.

Using the log above: W = 0.38, R = 2.4 ÷ 0.95 = 2.53

f = 0.38 − (0.62 ÷ 2.53) = 0.38 − 0.245 = 0.135

The formula says risk 13.5% of capital on each trade, thirteen times a 1% habit, and nothing about the arithmetic is wrong. It remains a poor idea. Three reasons, roughly in order of how much damage they do.

1. It assumes your edge estimate is exact, and it is not. You have just calculated a confidence interval running from +0.01R to +0.64R. Kelly sized on the optimistic end of that range is catastrophically oversized.
2. The drawdowns are unexecutable. Full Kelly routinely produces drawdowns above 50% even when the edge is genuine, and almost nobody keeps executing through that.
3. Overbetting is asymmetric. Sizing above optimal reduces growth and raises the risk of ruin, while sizing below optimal costs you growth alone, so the error worth making is the conservative one.

Half Kelly captures roughly three quarters of the growth rate with about half the volatility, and quarter Kelly, 3.4% on the numbers above, is the more cautious setting again, sitting well above most retail practice while remaining a long way below the raw formula.

Volatility-adjusted sizing

A flat percentage stop treats a utility and a biotech as equivalent instruments, which they plainly are not.

Using ATR as the unit of measurement addresses the problem. The rule becomes “stop 2 ATR below entry”:

	Stock A	Stock B
Price	$50	$50
14-day ATR	$0.60	$3.20
2-ATR stop distance	$1.20	$6.40
Shares at $250 risk	208	39

Both positions now carry the same statistical risk, and both are stopped out by roughly the same degree of abnormal movement, which is the entire point of the exercise. A flat 5% stop would have sat far too tight on Stock B, where ordinary daily movement covers most of that distance, and needlessly wide on Stock A.

Everything downstream is unchanged. Feed the result to the position size calculator. Only the derivation of the stop distance has altered.

A framework that survives contact with reality

1. Fixed fractional as the base: 0.5% to 1% while the sample remains under roughly 200 trades.
2. Volatility-adjusted stops, using ATR multiples in place of flat percentages.
3. Scale with measured confidence. Size up only when the lower bound of the expectancy interval rises, and treat a run of three winners as noise.
4. Cap at a fraction of Kelly. Quarter Kelly as a ceiling, and only once n is large.
5. Portfolio heat limit. Total open risk capped at 5% to 6% whatever per-trade sizing suggests, because correlated positions behave as a single position.

Sequencing is the part that gets skipped. A sizing model applied to an edge that nobody has validated will amplify whatever mistake is buried in the estimate, and it will amplify it in proportion to the confidence placed in the model, which is why the order runs sample first, confidence interval second, sizing model third.

Next: backtesting without fooling yourself for why the edge you measured is probably smaller than it appears.

Two companion pieces take the sizing half further. The 1% figure this page works beyond is a convention rather than a measured result, and set risk per trade from your drawdown limit runs the arithmetic in reverse, deriving a percentage from the drawdown you would actually tolerate and the losing streak your win rate produces. And fixed fractional vs fixed dollar position sizing covers the choice underneath that number: whether risk scales with the account or stays flat, why only one of the two has a floor, and which equity figure the percentage should be taken from.