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Expectancy

Lesson 11 · about 10 min

Expectancy is the average result per trade, in R, that your system produces over many trades. It is the single number that says whether you have an edge, and it can be computed from any trade log with four inputs.

The formula

expectancy (in R) = (win rate × average winner) − (loss rate × average loser)

where win rate and loss rate are fractions that add to 1, and the average winner and average loser are both positive numbers in R.

From the twelve-trade log in the previous lesson:

  • Win rate = 0.417, loss rate = 0.583.
  • Average winner = 2.0R, average loser = 1.03R.
  • Expectancy = (0.417 × 2.0) − (0.583 × 1.03) = 0.834 − 0.600 = +0.234R.

That matches the "+0.23R per trade" we got by dividing total R by the number of trades, as it must.

Reading the number

  • Expectancy above zero: over enough trades, the system makes money before costs.
  • Expectancy at zero: you are paying the spread and commissions to entertain yourself.
  • Expectancy below zero: the more you trade, the more you lose, and sizing cannot save you.

Multiply by trades per year to get expected annual R. A swing trader taking 100 trades a year at +0.234R expects about +23R. At 1% risk that is roughly +23% before compounding (a little more with compounding, a little less after costs). A day trader taking 600 trades a year at +0.08R expects +48R. Expectancy per trade and trade frequency together are what matter, not either alone.

Key idea: Expectancy = (win% × avg win) − (loss% × avg loss). Positive means you have an edge worth sizing; negative means no sizing scheme can help.

Three systems, side by side

System Win rate Avg winner Avg loser Expectancy
A: trend follower 35% 3.0R 1.0R 0.35 × 3.0 − 0.65 × 1.0 = 1.05 − 0.65 = +0.40R
B: swing trader 50% 1.5R 1.0R 0.50 × 1.5 − 0.50 × 1.0 = 0.75 − 0.50 = +0.25R
C: "high accuracy" scalper 75% 0.4R 1.5R 0.75 × 0.4 − 0.25 × 1.5 = 0.30 − 0.375 = −0.075R

System C has the best win rate and loses money. Its winners are small because it takes profits quickly, and its losers are large because it holds losers hoping they come back. This is the most common shape of beginner trading, and it feels great: three wins out of four. The account shrinks anyway.

System A loses almost two trades out of three and is the most profitable of the three. Traders abandon it because it feels terrible.

Breakeven win rate

For any reward-to-risk ratio, there is a win rate below which the system loses. Set expectancy to zero with average loser = 1R and average winner = RR:

w × RR − (1 − w) × 1 = 0, so w = 1 ÷ (1 + RR)

Average winner (in R) Breakeven win rate
0.5R 66.7%
1.0R 50.0%
1.5R 40.0%
2.0R 33.3%
3.0R 25.0%
4.0R 20.0%

This table is worth memorising. If your average winner is 2R, you can lose two out of three trades and break even; anything above a 33% win rate is profit. If your average winner is 0.5R, you must win two out of three just to tread water.

Costs

Every trade pays a spread and usually a commission. Convert costs into R and subtract them from every trade, winners and losers alike.

Say 1R is $100 and a round trip costs $6 in commission plus $4 of spread: $10, or 0.10R per trade.

System B after costs: winners become 1.4R, losers become 1.1R. Expectancy = 0.50 × 1.4 − 0.50 × 1.1 = 0.70 − 0.55 = +0.15R. Costs took 40% of the edge.

System C after costs: winners 0.3R, losers 1.6R. Expectancy = 0.75 × 0.3 − 0.25 × 1.6 = 0.225 − 0.40 = −0.175R. Costs more than doubled the loss.

Small stops make costs a larger fraction of R, which is why very tight-stop, high-frequency approaches are much harder to make work with retail commissions than they look on a chart.

Expectancy per setup

A single expectancy for "my trading" hides the fact that you probably run two or three setups with different numbers. Compute it per setup once each has 50+ trades. The usual discovery is that one setup produces nearly all the profit and another is a slow leak. Remove the leak and the overall expectancy jumps without a single new idea.

Try it: Using your own log (or the twelve-trade sample), compute win rate, average winner, average loser and expectancy. Then subtract a realistic cost in R from every trade and recompute. If the number goes negative, you have learned something worth far more than the trades cost.

Recap

  • Expectancy = (win rate × avg winner) − (loss rate × avg loser), in R per trade.
  • Positive expectancy times trade frequency is your expected annual R.
  • Breakeven win rate = 1 ÷ (1 + reward-to-risk); 2R winners break even at 33%.
  • High win rates with small winners and large losers lose money; low win rates with large winners can be very profitable.
  • Subtract costs in R from every trade; tight stops make costs a large share of R.

See it drawn

Original diagrams for the ideas on this page. Illustrative, not real market data.

The spread of outcomes behind an expectancyA histogram of forty trades: a tall block of small losses on the left, a low spread of larger wins on the right, and a line marking the average outcome.NUMBER OF TRADES051024 LOSSES, AVG −$20016 WINS, AVG +$600EXPECTANCY +$120−$400−$200$0+$200+$400+$600+$800PROFIT OR LOSS PER TRADEexpectancy = (40% × $600) − (60% × $200) = +$120 per trade
Expectancy: the average trade. Forty trades sorted by outcome: 24 small losses and 16 larger wins. Weighting each side by how often it happens gives the average result per trade, marked here by the dashed line at +$120.
The win rate needed to break evenA falling curve: the more a winning trade pays relative to the amount risked, the smaller the share of trades that must win to break even.BREAKEVEN WIN RATE0%20%40%60%80%1:11:21:31:41:5REWARD-TO-RISK RATIO1:1 needs 50%1:2 needs 33.3%1:3 needs 25%breakeven win rate = 1 ÷ (1 + reward-to-risk)above the curve, wins more than cover losses
The win rate needed to break even. How often a method must win just to stay level, for each reward-to-risk ratio. At 1:1 half the trades must win, at 1:2 a third, and at 1:3 a quarter, because each win covers more losses.
Compounding against a flat returnTwo account balances over fifteen years at the same yearly rate: one curve bends upwards as gains are left in, the other rises in a straight line.ACCOUNT VALUE$10k$20k$30k$40k051015YEARSCOMPOUNDED 10% a yearSIMPLE: 10% of the original sumboth start at $10,000 and run 15 years$41,772DIFFERENCE$16,772$25,000
Compounding against a flat return. Two accounts start at $10,000 and earn 10% a year for fifteen years. Leaving the gains in means each year earns on a larger balance, so the curve bends away from the straight line and ends $16,772 higher.