Chapter 1: What Expectancy Really Means Lesson 9: Expectancy Curves: Mapping Your Edge Across Time
Expectancy is the average profit or loss per trade. A positive expectancy indicates a profitable system over a large number of trades. However, expectancy is not static. It varies with market conditions, time of day, volatility, and other factors. Expectancy curves visualize how your trading edge changes over specific periods. This analysis helps identify optimal trading windows and periods to avoid.
Understanding Expectancy Fluctuation
Expectancy is calculated as:
$E = (P_w \times A_w) - (P_l \times A_l)$
Where: $P_w$ = Probability of a winning trade $A_w$ = Average win amount $P_l$ = Probability of a losing trade $A_l$ = Average loss amount
Each of these variables can shift. For example, during high volatility, $A_w$ and $A_l$ might increase. During low volatility, $P_w$ or $P_l$ might change. Expectancy curves plot this calculation across a chosen time dimension.
Constructing Expectancy Curves
To construct an expectancy curve, you need historical trade data. Each trade requires its entry time, exit time, profit/loss, and instrument.
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Define the Time Dimension: Common dimensions include:
- Time of day (e.g., 9:30 AM to 4:00 PM EST, in 15-minute or 30-minute intervals).
- Day of week (Monday to Friday).
- Volatility levels (e.g., VIX ranges, ATR values).
- Specific market phases (e.g., trend, range, breakout).
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Segment Trade Data: Group trades based on the defined time dimension. For a time-of-day curve, assign each trade to its corresponding time interval based on its entry time.
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Calculate Expectancy for Each Segment: For each segment, calculate $P_w$, $A_w$, $P_l$, and $A_l$ from the trades within that segment. Then compute the expectancy for that segment.
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Plot the Curve: Graph the expectancy values against the time dimension.
Example: Time-of-Day Expectancy for ES Futures
Consider a day trader trading ES futures contracts. The trader has 1,000 historical trades over 6 months. Each trade is 1 ES contract. The commission is $2.50 per round trip.
We want to analyze expectancy in 30-minute intervals from 9:30 AM EST to 4:00 PM EST.
Data Aggregation Steps:
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Extract Trade Data: For each trade, record:
- Entry Time (e.g., 9:45 AM EST)
- Gross Profit/Loss (e.g., +$100, -$50)
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Adjust for Commissions: Subtract the $2.50 commission from each trade's gross profit/loss to get net profit/loss.
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Categorize by Time Interval: Create 30-minute bins:
- 9:30 AM - 10:00 AM
- 10:00 AM - 10:30 AM
- ...
- 3:30 PM - 4:00 PM
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Calculate Metrics per Interval: For each 30-minute interval, count:
- Total trades ($N_t$)
- Winning trades ($N_w$)
- Losing trades ($N_l$)
- Sum of net wins ($\Sigma W$)
- Sum of net losses ($\Sigma L$)
Interval: 9:30 AM - 10:00 AM
Assume the following for this interval from the historical data:
- Total Trades ($N_t$): 150
- Winning Trades ($N_w$): 78
- Losing Trades ($N_l$): 72
- Sum of Net Wins ($\Sigma W$): +$4,290
- Sum of Net Losses ($\Sigma L$): -$3,240
Calculations for 9:30 AM - 10:00 AM:
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$P_w = N_w / N_t = 78 / 150 = 0.52$
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$P_l = N_l / N_t = 72 / 150 = 0.48$
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$A_w = \Sigma W / N_w = $4,290 / 78 = $55.00$
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$A_l = |\Sigma L| / N_l = $3,240 / 72 = $45.00$
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Expectancy ($E$) for 9:30 AM - 10:00 AM: $E = (0.52 \times $55.00) - (0.48 \times $45.00)$ $E = $28.60 - $21.60$ $E = $7.00$
This means, on average, each trade taken between 9:30 AM and 10:00 AM yielded a net profit of $7.00.
Interval: 12:00 PM - 12:30 PM
Assume the following for this interval:
- Total Trades ($N_t$): 80
- Winning Trades ($N_w$): 30
- Losing Trades ($N_l$): 50
- Sum of Net Wins ($\Sigma W$): +$1,200
- Sum of Net Losses ($\Sigma L$): -$1,750
Calculations for 12:00 PM - 12:30 PM:
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$P_w = N_w / N_t = 30 / 80 = 0.375$
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$P_l = N_l / N_t = 50 / 80 = 0.625$
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$A_w = \Sigma W / N_w = $1,200 / 30 = $40.00$
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$A_l = |\Sigma L| / N_l = $1,750 / 50 = $35.00$
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Expectancy ($E$) for 12:00 PM - 12:30 PM: $E = (0.375 \times $40.00) - (0.625 \times $35.00)$ $E = $15.00 - $21.875$ $E = -$6.875$
This indicates an average net loss of $6.88 per trade during this midday period.
Repeat this process for all 30-minute intervals throughout the trading day. Plotting these expectancy values against their respective time intervals creates the expectancy curve.
Interpreting Expectancy Curves
An expectancy curve immediately highlights periods of positive and negative edge.
- Positive Expectancy Peaks: These are optimal trading times. The market conditions, liquidity, or volatility during these periods align best with your strategy. Focus your trading activity here.
- Negative Expectancy Valleys: These periods indicate your strategy loses money on average. Avoid trading during these times, or significantly reduce position size and frequency.
- Flat Expectancy: Near-zero expectancy suggests the market is neutral to your strategy. This might be a period for observation or minimal trading.
For the ES example, if the curve shows a peak from 9:30 AM to 10:30 AM ($7.00 and similar values), a dip from 12:00 PM to 1:00 PM (-$6.88), and another smaller peak from 2:30 PM to 3:30 PM, this provides actionable information. The trader should prioritize trades in the morning and late afternoon, and potentially cease trading during the midday lull.
Beyond Time of Day
Expectancy curves are not limited to time.
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Volatility-Based Expectancy: Categorize trades by ATR (Average True Range) or VIX levels at entry. Plot expectancy against different volatility ranges. A strategy might perform best in moderate volatility (e.g., ATR between 0.5% and 1.5% of price) but fail in very low or very high volatility.
- Example: For SPY options, if VIX is below 15, your expectancy might be -$0.10 per contract. If VIX is between 15-25, expectancy might be +$0.25 per contract. Above 25, it might drop to -$0.05. This suggests avoiding low and high VIX environments.
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Market Phase Expectancy:
