This lesson examines the diminishing returns of win rate increases. Traders often prioritize win rate above all else. This approach overlooks the interplay between win rate, average win, and average loss. Expectancy is the true measure of a trading strategy's long-term profitability.
Expectancy Formula Review
Expectancy quantifies the average profit or loss per trade. The formula for expectancy (E) is:
E = (Win Rate * Average Win) - (Loss Rate * Average Loss)
Where:
- Win Rate = Number of Winning Trades / Total Number of Trades
- Loss Rate = Number of Losing Trades / Total Number of Trades = 1 - Win Rate
- Average Win = Total Profit from Winning Trades / Number of Winning Trades
- Average Loss = Total Loss from Losing Trades / Number of Losing Trades
A positive expectancy indicates a profitable strategy over time. A negative expectancy indicates a losing strategy.
The Relationship Between Win Rate and Expectancy
Consider a strategy with fixed average win and average loss values. We will analyze how expectancy changes as win rate increases.
Assume:
- Average Win = $200
- Average Loss = $100
Let's calculate expectancy for various win rates:
-
Win Rate = 40%
- Loss Rate = 1 - 0.40 = 0.60
- E = (0.40 * $200) - (0.60 * $100)
- E = $80 - $60 = $20
-
Win Rate = 50%
- Loss Rate = 1 - 0.50 = 0.50
- E = (0.50 * $200) - (0.50 * $100)
- E = $100 - $50 = $50
-
Win Rate = 60%
- Loss Rate = 1 - 0.60 = 0.40
- E = (0.60 * $200) - (0.40 * $100)
- E = $120 - $40 = $80
-
Win Rate = 70%
- Loss Rate = 1 - 0.70 = 0.30
- E = (0.70 * $200) - (0.30 * $100)
- E = $140 - $30 = $110
-
Win Rate = 80%
- Loss Rate = 1 - 0.80 = 0.20
- E = (0.80 * $200) - (0.20 * $100)
- E = $160 - $20 = $140
-
Win Rate = 90%
- Loss Rate = 1 - 0.90 = 0.10
- E = (0.90 * $200) - (0.10 * $100)
- E = $180 - $10 = $170
The increase in expectancy for each 10% win rate increment is:
- 40% to 50%: $50 - $20 = $30
- 50% to 60%: $80 - $50 = $30
- 60% to 70%: $110 - $80 = $30
- 70% to 80%: $140 - $110 = $30
- 80% to 90%: $170 - $140 = $30
In this specific scenario, with a fixed average win and average loss, each 10% increase in win rate yields a constant $30 increase in expectancy. This does not immediately demonstrate diminishing returns. The diminishing returns become apparent when considering the effort or cost required to achieve higher win rates.
The Cost of Increasing Win Rate
Achieving higher win rates often requires compromises in other areas. These compromises can include:
- Reduced Average Win: Strategies designed for high win rates often involve taking smaller profits. Traders might exit trades quickly to "book a win," even if the trade has further profit potential.
- Increased Average Loss: To achieve a high win rate, traders might hold losing positions longer, hoping for a reversal. This can lead to larger average losses when the reversal does not occur.
- Fewer Trading Opportunities: High win rate strategies may require stricter entry criteria, reducing the number of valid setups. This lowers trade frequency.
- Increased Transaction Costs: Frequent small wins, especially with tight stops, can lead to higher commission and slippage costs relative to the profit generated.
Concrete Numerical Example: Diminishing Returns
Consider a day trader specializing in futures contracts, specifically ES (E-mini S&P 500 futures). Each point in ES is $50. A 1-tick movement is $12.50 (0.25 points).
Scenario 1: Balanced Strategy
- Average Win: 4 points ($200)
- Average Loss: 2 points ($100)
- Win Rate: 50%
- Expectancy = (0.50 * $200) - (0.50 * $100) = $100 - $50 = $50 per contract
Scenario 2: High Win Rate Strategy (Attempt 1) The trader adjusts the strategy to increase the win rate. This involves taking profits at 2 points ($100) instead of 4 points. To maintain the higher win rate, the trader also widens stops to 3 points ($150) to avoid being stopped out prematurely.
- Average Win: 2 points ($100)
- Average Loss: 3 points ($150)
- Win Rate: 70% (increased due to smaller profit targets and wider stops)
- Expectancy = (0.70 * $100) - (0.30 * $150) = $70 - $45 = $25 per contract
In this attempt, despite a 20% increase in win rate (from 50% to 70%), the expectancy decreased from $50 to $25. The reduction in average win and increase in average loss outweighed the higher win rate.
Scenario 3: High Win Rate Strategy (Attempt 2) The trader tries another adjustment. They aim for a very high win rate by taking profits at 1.5 points ($75) and maintaining a tight stop of 1 point ($50). However, the tight stop means more trades hit the stop before reaching the target.
- Average Win: 1.5 points ($75)
- Average Loss: 1 point ($50)
- Win Rate: 65% (higher than 50% but lower than the 70% from wider stops)
- Expectancy = (0.65 * $75) - (0.35 * $50) = $48.75 - $17.50 = $31.25 per contract
Here, the win rate is 65%, higher than the original 50%. The expectancy is $31.25, which is better than Scenario 2 but still lower than the original $50. The effort to increase the win rate yielded a lower overall expectancy.
Scenario 4: Focus on Average Win (Lower Win Rate) The trader reverts to the original stop of 2 points ($100) but aims for larger wins by letting profitable trades run to 6 points ($300). This naturally reduces the win rate.
- Average Win: 6 points ($300)
- Average Loss: 2 points ($100)
- Win Rate: 40% (lower than original 50%)
- Expectancy = (0.40 * $300) - (0.60 * $100) = $120 - $60 = $60 per contract
In this scenario, a lower win rate (40% vs. 50%) resulted in a higher expectancy ($60 vs. $50). This illustrates that optimizing for win rate alone can be detrimental.
The Mathematical Proof of Diminishing Returns
Let's fix the average win (AW) and average loss (AL). E = WR * AW - (1 - WR) * AL E = WR * AW - AL + WR * AL E = WR * (AW + AL) - AL
This is a linear equation with respect to WR. The slope is (AW + AL). The rate of change of Expectancy with respect to Win Rate is dE/dWR = AW + AL.
This means that for every
