Your trading P&L depends on two primary variables: win rate and reward-to-risk ratio. Understanding their interplay is fundamental to consistent profitability. This lesson defines these variables and demonstrates their mathematical relationship to expectancy.
Defining Win Rate (WR)
Win rate is the percentage of trades that generate a profit.
Formula: WR = (Number of Winning Trades / Total Number of Trades) * 100%*
Example: A trader executes 100 trades. 55 trades are profitable. WR = (55 / 100) * 100% = 55%*
A 55% win rate means 55 out of every 100 trades are winners.
Defining Reward-to-Risk Ratio (R:R)
The reward-to-risk ratio (R:R) measures the average profit of winning trades against the average loss of losing trades. It is expressed as a ratio.
Formula: R:R = (Average Profit Per Winning Trade) / (Average Loss Per Losing Trade)
Example: Over 100 trades, a trader's total profit from winning trades is $5,500. There were 55 winning trades. Average Profit Per Winning Trade = $5,500 / 55 = $100
Over the same 100 trades, the total loss from losing trades is $4,500. There were 45 losing trades. Average Loss Per Losing Trade = $4,500 / 45 = $100
R:R = $100 / $100 = 1:1
This trader's average winning trade profit equals their average losing trade loss.
The Expectancy Formula
Expectancy quantifies the average profit or loss per trade over many trades. It combines win rate and reward-to-risk.
Formula: Expectancy = (WR * Average Win Size) - (LR * Average Loss Size) Where: WR = Win Rate (as a decimal) LR = Loss Rate (1 - WR) Average Win Size = Average Profit Per Winning Trade Average Loss Size = Average Loss Per Losing Trade
Alternatively, using R:R: Expectancy = [(WR * R:R) - LR] * Average Loss Size
Let's break down the second formula. If R:R = Average Win Size / Average Loss Size, then Average Win Size = R:R * Average Loss Size. Substitute this into the first expectancy formula: Expectancy = (WR * (R:R * Average Loss Size)) - (LR * Average Loss Size) Factor out Average Loss Size: Expectancy = [(WR * R:R) - LR] * Average Loss Size
This formula shows that positive expectancy requires the product of win rate and reward-to-risk to exceed the loss rate.
Numerical Example: Calculating Expectancy
Consider a futures day trader. Instrument: ES futures contract. Risk per trade: $100 (4 points, 1 contract). Target profit per trade: $150 (6 points, 1 contract).
Over 200 trades, the trader achieves the following: Number of winning trades: 110 Number of losing trades: 90
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Calculate Win Rate (WR): WR = (110 / 200) * 100% = 55% As a decimal, WR = 0.55
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Calculate Loss Rate (LR): LR = 1 - WR = 1 - 0.55 = 0.45
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Calculate Average Win Size: The target profit per winning trade is $150. Assume actual average win size matches target. Average Win Size = $150
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Calculate Average Loss Size: The risk per losing trade is $100. Assume actual average loss size matches risk. Average Loss Size = $100
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Calculate Reward-to-Risk (R:R): R:R = Average Win Size / Average Loss Size = $150 / $100 = 1.5
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Calculate Expectancy (Method 1): Expectancy = (WR * Average Win Size) - (LR * Average Loss Size) Expectancy = (0.55 * $150) - (0.45 * $100) Expectancy = $82.50 - $45.00 Expectancy = $37.50
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Calculate Expectancy (Method 2, using R:R): Expectancy = [(WR * R:R) - LR] * Average Loss Size Expectancy = [(0.55 * 1.5) - 0.45] * $100 Expectancy = [0.825 - 0.45] * $100 Expectancy = [0.375] * $100 Expectancy = $37.50*
Both methods yield an expectancy of $37.50. This means, on average, this trader expects to make $37.50 per trade. Over 200 trades, the total expected profit is 200 * $37.50 = $7,500.*
The Interdependence of WR and R:R
A trader needs a positive expectancy to be profitable. This can be achieved with various combinations of win rate and reward-to-risk.
Scenario 1: High Win Rate, Low R:R Trader A: WR = 70% (0.70) LR = 30% (0.30) Average Win Size = $50 Average Loss Size = $100 R:R = $50 / $100 = 0.5
Expectancy = (0.70 * $50) - (0.30 * $100) Expectancy = $35 - $30 Expectancy = $5.00
This trader is profitable with a high win rate but takes losses twice as large as their wins.
Scenario 2: Moderate Win Rate, Moderate R:R Trader B: WR = 50% (0.50) LR = 50% (0.50) Average Win Size = $150 Average Loss Size = $100 R:R = $150 / $100 = 1.5
Expectancy = (0.50 * $150) - (0.50 * $100) Expectancy = $75 - $50 Expectancy = $25.00
This trader is profitable with a break-even win rate, but their average win is 1.5 times their average loss.
Scenario 3: Low Win Rate, High R:R Trader C: WR = 30% (0.30) LR = 70% (0.70) Average Win Size = $400 Average Loss Size = $100 R:R = $400 / $100 = 4.0
Expectancy = (0.30 * $400) - (0.70 * $100) Expectancy = $120 - $70 Expectancy = $50.00
This trader is profitable despite winning only 30% of their trades. Their average win is four times their average loss.
Break-Even Point
The break-even point occurs when expectancy is zero. This happens when (WR * R:R) = LR.*
Rearranging for WR: WR = LR / R:R WR = (1 - WR) / R:R WR * R:R = 1 - WR WR * R:R + WR = 1 WR (R:R + 1) = 1 WR = 1 / (R:R + 1)
Example: If a trader targets an R:R of 2:1 (meaning Average Win Size is twice Average Loss Size). R:R = 2. Break-even WR = 1 / (2 + 1) = 1 / 3 = 0.3333 or 33.33%. If this trader wins 33.33% of their trades, they break even. If they win more than 33.33%, they are profitable.
If a trader targets an R:R of 0.5:1 (meaning Average Win Size is half Average Loss Size). R:R = 0.5. Break-even WR = 1 / (0.5 + 1) = 1 / 1.5 = 0.6667 or 66.67%. This trader needs to win more than 66.67% of their trades to be profitable.
Practical Application for Day Traders
- Track Your Data: Accurately record every trade
