This lesson is Module 11: Trading Math: Expectancy, Chapter 2: Maximizing Positive Expectancy, Lesson 7.
Expectancy quantifies the average profit or loss per trade. A positive expectancy is necessary for long-term profitability. Ticker selection directly impacts expectancy. Different assets exhibit varying volatility, liquidity, and cost structures. These differences alter win rates, average gains, average losses, and ultimately, expectancy.
Expectancy Formula Review
Expectancy is calculated as:
$E = (W \times AP) - (L \times AL)$
Where:
- $E$ = Expectancy
- $W$ = Win Rate (as a decimal)
- $AP$ = Average Profit per winning trade
- $L$ = Loss Rate (as a decimal, $1 - W$)
- $AL$ = Average Loss per losing trade
Alternatively, using R-multiples:
$E = (W \times AR) - (L \times 1)$
Where:
- $AR$ = Average R-multiple of winning trades
- $1$ = R-multiple of losing trades (representing 1 unit of risk)
Volatility and Expectancy
Volatility, measured by metrics like Average True Range (ATR) or standard deviation, influences potential profit and loss per trade. High volatility stocks offer larger price movements, potentially increasing average profit per win. They also carry higher risk, potentially increasing average loss per trade.
Consider two hypothetical stocks, Stock A and Stock B, traded with the same strategy.
Stock A (Low Volatility):
- Average Daily Range: $1.50
- Strategy parameters: Target $0.50 profit, Stop $0.25 loss
- Observed Win Rate: 60%
- Average Profit (AP): $0.45 (allowing for slippage)
- Average Loss (AL): $0.28 (allowing for slippage)
Expectancy for Stock A: $E_A = (0.60 \times $0.45) - (0.40 \times $0.28)$ $E_A = $0.27 - $0.112$ $E_A = $0.158$ per share
Stock B (High Volatility):
- Average Daily Range: $5.00
- Strategy parameters: Target $1.50 profit, Stop $0.75 loss
- Observed Win Rate: 55% (higher volatility can reduce win rate due to increased stop-outs)
- Average Profit (AP): $1.35
- Average Loss (AL): $0.85
Expectancy for Stock B: $E_B = (0.55 \times $1.35) - (0.45 \times $0.85)$ $E_B = $0.7425 - $0.3825$ $E_B = $0.36$ per share
In this example, Stock B, despite a lower win rate, yields a higher expectancy due to its increased volatility allowing for larger average profits relative to average losses. The trader must ensure their strategy adapts to the volatility.
Liquidity and Expectancy
Liquidity refers to the ease with which an asset can be bought or sold without significantly affecting its price. High liquidity means tighter spreads and less slippage. Low liquidity results in wider spreads and greater slippage, directly impacting average profit and average loss.
Scenario: Trading 1,000 shares
High Liquidity Stock (Stock C):
- Bid-Ask Spread: $0.01
- Average Slippage (entry/exit combined): $0.02 per share
- Strategy: 60% Win Rate, $1.00 AP, $0.50 AL
- Effective AP: $1.00 - $0.02 = $0.98
- Effective AL: $0.50 + $0.02 = $0.52
Expectancy for Stock C: $E_C = (0.60 \times $0.98) - (0.40 \times $0.52)$ $E_C = $0.588 - $0.208$ $E_C = $0.38$ per share
Low Liquidity Stock (Stock D):
- Bid-Ask Spread: $0.10
- Average Slippage (entry/exit combined): $0.15 per share
- Strategy: 60% Win Rate, $1.00 AP, $0.50 AL (theoretical, before slippage)
- Effective AP: $1.00 - $0.15 = $0.85
- Effective AL: $0.50 + $0.15 = $0.65
Expectancy for Stock D: $E_D = (0.60 \times $0.85) - (0.40 \times $0.65)$ $E_D = $0.51 - $0.26$ $E_D = $0.25$ per share
Low liquidity significantly reduces expectancy by eroding profits and increasing losses. This effect is amplified with larger position sizes.
Trading Costs and Expectancy
Trading costs include commissions, exchange fees, and ECN fees. While often small per share, they accumulate and impact expectancy, especially for high-frequency strategies or small profit targets.
Example: Futures Contract (ES Micro)
Consider trading one MES contract.
- Round-trip commission: $1.00
- Exchange fees: $0.50
- Total cost per round trip: $1.50
Strategy Parameters:
- Win Rate: 55%
- Average Profit: 4 ticks ($5.00 per tick * 4 = $20.00)
- Average Loss: 3 ticks ($5.00 per tick * 3 = $15.00)
Before Costs: $E_{raw} = (0.55 \times $20.00) - (0.45 \times $15.00)$ $E_{raw} = $11.00 - $6.75$ $E_{raw} = $4.25$ per contract
After Costs: The $1.50 cost applies to every trade (win or loss). Effective Average Profit: $20.00 - $1.50 = $18.50 Effective Average Loss: $15.00 + $1.50 = $16.50
$E_{net} = (0.55 \times $18.50) - (0.45 \times $16.50)$ $E_{net} = $10.175 - $7.425$ $E_{net} = $2.75$ per contract
Trading costs reduced expectancy by 35.3% ($1.50 / $4.25). For strategies with smaller profit targets, costs can turn a positive expectancy into a negative one.
Ticker-Specific Expectancy Calculation
Traders must calculate expectancy for each ticker or asset class they trade. A strategy profitable on one stock may be unprofitable on another due to differing characteristics.
Step-by-step example: Comparing two stocks for a specific strategy
A day trader employs a breakout strategy. They want to compare its effectiveness on XYZ and ABC.
Stock XYZ:
- Average entry price: $50.00
- Average exit price (win): $50.75
- Average exit price (loss): $49.60
- Observed Win Rate: 58%
- Commission per share: $0.005 (round trip)
- Slippage (estimated, per share, combined entry/exit): $0.03
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Calculate Average Profit (AP) for XYZ: Theoretical profit: $50.75 - $50.00 = $0.75 Net AP = Theoretical profit - Commission - Slippage Net AP = $0.75 - $0.005 - $0.03 = $0.715
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Calculate Average Loss (AL) for XYZ: Theoretical loss: $50.00 - $49.60 = $0.40 Net AL = Theoretical loss + Commission + Slippage Net AL = $0.40 + $0.005 + $0.03 = $0.435
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Calculate Expectancy for XYZ: $E_{XYZ} = (0.58 \times $0.715) - (0.42 \times $0.435)$ $E_{XYZ} = $0.4147 - $0.1827$ $E_{XYZ} = $0.
