Expectancy is the average result of one trade, repeated. It answers whether a method makes money at all, and it says nothing about the order the money arrives in. For an account of fixed size, the order is what decides whether you are still trading when the average has had time to show up.
Both questions need an answer before a setup earns a full-size position. The arithmetic fits on one line and it separates a method you can trade for years from one that throws a career at eight bad trades.
Expectancy in one number
Express every trade in R before averaging anything. One R is what you lose when the stop is hit, so a trade risking $250 and a trade risking $600 both produce 1R losses, and a 2R winner means twice the risk regardless of which account it came from. R makes trades from different accounts and months comparable.
E = (win rate × average win) − (loss rate × average loss)
Take a method that wins 45 percent of the time, with the average winner at 2.1R and the average loser at 1R. Winners contribute 0.45 × 2.1 = 0.945R per trade. Losers take back 0.55 × 1 = 0.55R. Expectancy is 0.945 − 0.55 = 0.395R, about 0.4R per trade before costs.
In dollars: a $10,000 account risking 1 percent, or $100, has an average winner of $210 and an average loser of $100. Then 0.45 × $210 = $94.50 and 0.55 × $100 = $55, leaving $39.50 per trade. On a flat $100 risk with no compounding, that is 253 trades to double the account — 39.5 cents of profit for each dollar of risk carried.
The break-even win rate tells you how much room the method has. Solve p × 2.1 = (1 − p) × 1, so 3.1p = 1 and p = 32.3 percent. A 45 percent hit rate clears break-even by 12.7 points. Tighten the average winner to 1.6R and the same 45 percent becomes 0.45 × 1.6 − 0.55 = 0.17R: less than half the edge, and 2.3 times the trades for the same money.
Costs come off each trade: commission, spread and slippage belong inside the average win and the average loss. A method paying 0.2R gross can be flat or negative once a spread is paid on every entry.
Why a positive expectancy still loses money
A 45 percent win rate implies a 55 percent loss rate, and losing streaks are a property of that number. Eight losses in a row has probability 0.55⁸, about 0.0084, or one in 119. That sounds rare until you count the chances to have one: 200 trades hold 193 overlapping eight-trade windows, so the expected number of such windows is about 1.6. Eight straight losses belong in the plan.
What the streak costs depends on size. At 2 percent risk per trade, eight losses leave 0.98⁸ = 0.851, a 14.9 percent drawdown. At 10 percent the same eight losses leave 0.9⁸ = 0.431, a 56.9 percent drawdown. The trades and the stop distances are the same; only the multiplier on each loss changes.
Inside the streak there is a second problem. You cannot tell it from the edge disappearing. Trades five through eight of a losing run look exactly like a method that stopped working, and the decision to continue is made on eight data points, a sample that cannot answer the question either way. The account therefore has to be sized to survive the streak. The average only pays out to accounts still open.
Risk of ruin is a function of bet size
Fix the fraction risked per trade and the account becomes a product of (1 + f × R) terms. Losses shrink the base that every later trade works from, so one edge produces different curves at different f. With E = 0.395R, a 2 percent stake adds 0.79 percent of equity per average trade; a 10 percent stake adds 3.95 percent.
Growth favors the larger stake until you price the bad run. A 40 percent drawdown takes equity from 100 to 60, and the gain needed to return to 100 is 1/0.6 − 1 = 66.7 percent. At 2 percent risk the average trade repairs 0.79 percent, so the hole closes in ln(1.667)/ln(1.0079) ≈ 65 trades. At 10 percent risk the average trade repairs 3.95 percent, and ln(1.667)/ln(1.0395) ≈ 13 trades.
Compare what it takes to fall in. At 10 percent per trade, 0.9⁵ = 0.590, so five consecutive losses put the account 41 percent down. At 2 percent, reaching the same depth takes 26 consecutive losses (0.98²⁶ = 0.591). A three-loss run is the routine case: 0.55³ = 0.166, one start in six. It costs the 2 percent account 5.9 percent and the 10 percent account 27.1 percent.
The 10 percent line also needs ten times the position for the same 1R stop. On a $50,000 account that is $5,000 at risk, which on a 2-dollar stop is 2,500 shares. The position size calculator does that conversion from account, entry and stop in one step.
Set f from the worst streak you can absorb without changing your process, then hold it. Risk management starts from that number, and the fixed-fraction version of the same decision is worked out in the 1 percent rule.
Reading a distribution of outcomes
Run the method a few thousand times over 200 trades, shuffling winners and losses, and plot the 5th, 25th, 50th, 75th and 95th percentile of equity at each trade. Here is that fan for one edge.
Scroll sideways to read the full diagram.
The levels in the drawing are illustrative; the shape is what transfers. The slope of the 50th band is expectancy per trade, the gap between the 5th and the 95th is dispersion, and that width decides how much size the account can carry.
Read the lower band as a funding requirement. In the drawing the 5th band finishes at 74 against a median of 136, a 45 percent gap between two accounts running the same method for the same 200 trades. If a path like that would force you to cut size halfway through, your stake is too large for the dispersion, whatever the expectancy says.
Sample size
Twenty trades is an anecdote with a spreadsheet. On the worked method, five winners out of 20 pay 5 × 2.1 − 15 = −4.5R. Eleven winners pay 23.1 − 9 = +14.1R. Both come from the same 0.395R edge. The chance of five or fewer winners in 20 trades at a true 45 percent rate is about 5.5 percent, one run in eighteen.
Per-trade dispersion causes that. The standard deviation of a single trade's R is √(0.45 × 1.705² + 0.55 × 1.395²) ≈ 1.54R, where 1.705R and 1.395R are the distances from each outcome to the 0.395R mean. Measured expectancy has standard error 1.54/√n: about 0.34R at 20 trades and 0.11R at 200. A 0.2R result from 20 trades sits inside one standard error of zero.
Win rate needs its own count, with standard deviation √(p(1−p)/n). At p = 0.45, n = 100 gives 5.0 points and n = 275 gives 3.0. Claiming a 45 percent hit rate within three points takes roughly 275 trades.
While the sample is small: hold risk at one fixed fraction so every R means the same thing, log the planned stop, the fill, the exit and the closing R, and review adherence to the plan ahead of the P&L. Keep size at the bottom of the range you would run at full size. The daily chart checklist is where the stop and the R get fixed before the entry, which is what makes the log worth reading later.
Split the log by setup only once each bucket holds about 50 trades. Before that, comparing a 20-trade breakout subset with a 25-trade pullback subset measures noise.
Where the number lies
Curve fitting comes first. Choose the stop, the target and the entry filter on the same 200 trades you then measure, and the expectancy you report is the best fit to that sequence including its noise. The test is whether the rules were fixed before the first trade in the sample and left alone afterwards. Change one parameter after seeing the result and the sample restarts.
Exclusions are easier to justify to yourself. A filter applied after the outcome is known removes losses using information the trade did not have. Valid exclusions are defined before the entry and applied to every trade: no earnings, no half session, spread narrower than a stated width. A skip rule that cannot be written as a condition checked at order entry is a preference.
Regime change hides because nothing in the log looks wrong. An expectancy measured through a low-volatility grind on the index, with 0.5 percent daily ranges and orderly pullbacks, describes that regime. Once the index moves 1.5 percent a day the stops sit at the same distances but the winners run differently, and 0.395R becomes some other number. Quote expectancy with the period and the volatility that produced it.
Variable position size is the most common distortion in real accounts. Size by confidence and the average R is dominated by the trades you were sure about, a different quantity from the average dollar result at fixed size. Log R against planned risk and the dollar result separately, then compute profit per dollar risked; a 0.6R average across trades sized between 1 percent and 5 percent says nothing about a constant-size version.
The denominator leaks too. Cancelled orders, setups that never triggered, scratches booked as 0R when the real exit was −0.2R, partial fills counted as full winners. Each one moves the average. Record the reason for every exclusion when it happens, so someone else could check the sample.
