Expectancy Really Means, Lesson 2
Why Most Traders Lose: Negative Expectancy Exposed
Most traders lose money. This is not anecdotal; it is a mathematical certainty for those operating with negative expectancy. Understanding and quantifying expectancy is fundamental to profitable trading. Without positive expectancy, consistent profitability is impossible.
Defining Expectancy
Expectancy quantifies the average profit or loss per trade over a large sample size. It is a statistical measure, not a guarantee for any single trade. A positive expectancy indicates a profitable system over time. A negative expectancy guarantees long-term losses.
The basic formula for expectancy is:
$E = (W \times AP) - (L \times AL)$
Where:
- $E$ = Expectancy
- $W$ = Win Rate (as a decimal)
- $AP$ = Average Profit per winning trade
- $L$ = Loss Rate (as a decimal, $L = 1 - W$)
- $AL$ = Average Loss per losing trade
This formula provides a dollar value for expectancy. A positive value means profit. A negative value means loss.
The Misconception of Win Rate
Many traders focus solely on win rate. A high win rate does not guarantee profitability. A low win rate does not guarantee unprofitability. The relationship between win rate and average profit/loss determines expectancy.
Consider two traders:
Trader A:
- Win Rate ($W$): 70% (0.70)
- Average Profit ($AP$): $100
- Average Loss ($AL$): $300
Trader B:
- Win Rate ($W$): 30% (0.30)
- Average Profit ($AP$): $500
- Average Loss ($AL$): $100
Let's calculate expectancy for both:
Trader A Expectancy: $E_A = (0.70 \times $100) - (0.30 \times $300)$ $E_A = $70 - $90$ $E_A = -$20$
Trader A has a 70% win rate but a negative expectancy of -$20 per trade. For every 100 trades, Trader A expects to lose $2,000.
Trader B Expectancy: $E_B = (0.30 \times $500) - (0.70 \times $100)$ $E_B = $150 - $70$ $E_B = $80$
Trader B has a 30% win rate and a positive expectancy of $80 per trade. For every 100 trades, Trader B expects to gain $8,000.
This example demonstrates that win rate alone is a misleading metric. The average profit relative to the average loss is equally important.
Risk/Reward Ratio and Expectancy
The risk/reward ratio directly impacts expectancy. It is the average profit divided by the average loss ($AP / AL$).
The expectancy formula can be rewritten using the risk/reward ratio ($RR$):
$E = (W \times AP) - ((1-W) \times AL)$ $E = AL \times [(W \times (AP/AL)) - (1-W)]$ $E = AL \times [(W \times RR) - (1-W)]$
Where:
- $RR$ = Risk/Reward Ratio ($AP / AL$)
For positive expectancy, the term $[(W \times RR) - (1-W)]$ must be greater than zero.
This implies: $W \times RR > (1-W)$ $RR > (1-W) / W$ $RR > (1/W) - 1$
This inequality shows the minimum required risk/reward ratio for a given win rate to achieve positive expectancy.
Example: If a system has a 40% win rate ($W = 0.40$), what is the minimum required risk/reward ratio for positive expectancy?
$RR > (1 / 0.40) - 1$ $RR > 2.5 - 1$ $RR > 1.5$
This means if your win rate is 40%, your average winning trade must be at least 1.5 times larger than your average losing trade to break even. Any ratio greater than 1.5 yields positive expectancy.
Practical Application: Futures Day Trading
Consider a futures day trader trading ES contracts. Each point move on ES is $50.
Scenario 1: Negative Expectancy A trader aims for 4-point wins and accepts 2-point losses.
- Average Profit ($AP$): 4 points $\times$ $50/point = $200
- Average Loss ($AL$): 2 points $\times$ $50/point = $100
- Risk/Reward Ratio ($RR$): $200 / $100 = 2.0
This trader has a favorable risk/reward ratio. However, let's assume their win rate is 30% due to aggressive entries.
$W = 0.30$ $L = 0.70$
$E = (0.30 \times $200) - (0.70 \times $100)$ $E = $60 - $70$ $E = -$10$
Despite a 2:1 risk/reward, a 30% win rate results in a negative expectancy of -$10 per contract per trade. Trading 10 contracts, 20 times a day, this trader loses $2,000 daily before commissions.
Scenario 2: Positive Expectancy Another trader also aims for 4-point wins and accepts 2-point losses.
- Average Profit ($AP$): $200
- Average Loss ($AL$): $100
- Risk/Reward Ratio ($RR$): 2.0
This trader, however, has refined their entry criteria, achieving a 40% win rate.
$W = 0.40$ $L = 0.60$
$E = (0.40 \times $200) - (0.60 \times $100)$ $E = $80 - $60$ $E = $20$
This trader has a positive expectancy of $20 per contract per trade. Trading 10 contracts, 20 times a day, this trader gains $4,000 daily before commissions. The difference is a 10% increase in win rate.
Options Day Trading Example
An options day trader buys a call option for $2.50 (250 per contract). Their strategy involves selling for a 20% profit or stopping out at a 10% loss.
- Average Profit ($AP$): $2.50 $\times$ 0.20 = $0.50 per share (or $50 per contract)
- Average Loss ($AL$): $2.50 $\times$ 0.10 = $0.25 per share (or $25 per contract)
- Risk/Reward Ratio ($RR$): $50 / $25 = 2.0
Suppose this trader has a win rate of 45%.
$W = 0.45$ $L = 0.55$
$E = (0.45 \times $50) - (0.55 \times $25)$ $E = $22.50 - $13.75$ $E = $8.75$
This trader has a positive expectancy of $8.75 per contract per trade. If they trade 50 contracts per day, their daily expected profit is $437.50 before commissions.
The Impact of Commissions and Slippage
Commissions and slippage reduce expectancy. They are direct costs per trade. Let $C$ be the total cost per trade (commissions + slippage).
The adjusted expectancy formula is:
$E_{adj} = (W \times AP) - (L \times AL) - C$
Or, if $C$ is a fixed amount per trade:
$E_{adj} = E - C$
Continuing the Options Example: Assume a commission of $1.00 per contract round trip and slippage averaging $0.50 per contract. Total cost per contract ($C$) = $1.00 + $0.50 = $1.50
$E_{adj} = $8.75 - $1.50$ $E_{adj} = $7.25$
The expectancy per trade reduces from $8.75 to $7.25 after costs. This highlights the importance of minimizing trading costs, especially for high-frequency strategies. A system with a small positive expectancy can become negative after costs.
Why Most Traders Lose: The Numbers
Most retail traders operate with negative expectancy for several reasons:
- Poor Risk/
