Expectancy Really Means, Lesson 6

Expectancy Per Share vs Expectancy Per Trade vs Expectancy Per Dollar Risked

Expectancy quantifies the average profit or loss per unit of trading activity. Traders define "unit of activity" differently. Common units include shares, trades, and dollars risked. Each definition offers a distinct perspective on a trading strategy's profitability. Understanding these distinctions is essential for accurate performance analysis and strategy optimization.

Expectancy Per Share (EPS)

Expectancy Per Share (EPS) measures the average profit or loss generated for each share traded. This metric is useful for strategies that involve varying position sizes but consistent profit targets or stop losses on a per-share basis.

The formula for EPS is:

$EPS = (P_{win} \times AvgGain_{share}) - (P_{loss} \times AvgLoss_{share})$

Where:

• $P_{win}$ = Probability of a winning trade
• $AvgGain_{share}$ = Average gain per share on winning trades
• $P_{loss}$ = Probability of a losing trade
• $AvgLoss_{share}$ = Average loss per share on losing trades

EPS Example

Consider a strategy trading XYZ stock.

• $P_{win} = 0.55$ (55% win rate)
• $P_{loss} = 0.45$ (45% loss rate)
• $AvgGain_{share} = $0.75$ (average gain of $0.75 per share on winners)
• $AvgLoss_{share} = $0.50$ (average loss of $0.50 per share on losers)_

$EPS = (0.55 \times $0.75) - (0.45 \times $0.50)$ $EPS = $0.4125 - $0.225$ $EPS = $0.1875$

This strategy yields an average profit of $0.1875 for every share traded. If a trader executes 100 trades, each with 200 shares, the total shares traded are $100 \times 200 = 20,000$ shares. The expected profit from these 100 trades is $20,000 \times $0.1875 = $3,750$.

EPS is particularly relevant for strategies where per-share profit targets and stop losses are fixed, such as scalping strategies targeting specific tick increments.

Expectancy Per Trade (EPT)

Expectancy Per Trade (EPT) measures the average profit or loss generated for each trade executed, regardless of position size. This metric provides a direct measure of the average outcome of a single trading event.

The formula for EPT is:

$EPT = (P_{win} \times AvgGain_{trade}) - (P_{loss} \times AvgLoss_{trade})$

Where:

• $P_{win}$ = Probability of a winning trade
• $AvgGain_{trade}$ = Average dollar gain on winning trades
• $P_{loss}$ = Probability of a losing trade
• $AvgLoss_{trade}$ = Average dollar loss on losing trades

EPT Example

Consider a futures trading strategy.

• $P_{win} = 0.48$ (48% win rate)
• $P_{loss} = 0.52$ (52% loss rate)
• $AvgGain_{trade} = $300$ (average gain of $300 per winning trade)
• $AvgLoss_{trade} = $150$ (average loss of $150 per losing trade)_

$EPT = (0.48 \times $300) - (0.52 \times $150)$ $EPT = $144 - $78$ $EPT = $66$

This strategy yields an average profit of $66 for each trade. If a trader executes 500 trades, the expected profit is $500 \times $66 = $33,000$.

EPT is useful for comparing strategies where position sizing varies, but the primary concern is the average dollar outcome of each trade. It simplifies performance tracking when a trader uses dynamic position sizing based on volatility or account size.

Expectancy Per Dollar Risked (EPDR)

Expectancy Per Dollar Risked (EPDR), also known as the R-Multiple Expectancy, measures the average profit or loss generated for each dollar risked on a trade. This metric normalizes expectancy by the initial risk, making it ideal for comparing strategies with different risk profiles and position sizing methodologies.

The formula for EPDR is:

$EPDR = (P_{win} \times AvgR_{win}) - (P_{loss} \times AvgR_{loss})$

Where:

• $P_{win}$ = Probability of a winning trade
• $AvgR_{win}$ = Average R-multiple of winning trades (Average Gain / Average Risk)
• $P_{loss}$ = Probability of a losing trade
• $AvgR_{loss}$ = Average R-multiple of losing trades (Average Loss / Average Risk)

Note: By definition, $AvgR_{loss}$ is typically 1.0 if the average loss equals the initial risk unit. If losses are sometimes less than the full initial risk, $AvgR_{loss}$ will be less than 1.0. If losses exceed the initial risk (e.g., slippage), $AvgR_{loss}$ will be greater than 1.0. For simplicity in many contexts, $AvgR_{loss}$ is often assumed to be 1.0.

A more direct calculation for EPDR is:

$EPDR = \frac{EPT}{AvgRiskPerTrade}$

Where:

• $EPT$ = Expectancy Per Trade
• $AvgRiskPerTrade$ = Average dollar amount risked per trade

EPDR Example

Consider an options trading strategy.

• $P_{win} = 0.60$ (60% win rate)
• $P_{loss} = 0.40$ (40% loss rate)
• Average winning trade profit = $250
• Average losing trade loss = $100
• Average dollar amount risked per trade = $100 (This is the defined 1R unit)

First, calculate $AvgR_{win}$ and $AvgR_{loss}$: $AvgR_{win} = \frac{$250}{$100} = 2.5$ $AvgR_{loss} = \frac{$100}{$100} = 1.0$

Now, calculate EPDR: $EPDR = (0.60 \times 2.5) - (0.40 \times 1.0)$ $EPDR = 1.5 - 0.4$ $EPDR = 1.1$

This strategy yields an average profit of $1.1 for every dollar risked. If a trader risks $500 on a trade, the expected profit from that trade is $500 \times 1.1 = $550$. If a trader executes 200 trades, with an average risk of $100 per trade, the total expected profit is $200 \times ($100 \times 1.1) = $22,000$.

Alternatively, using the $EPT / AvgRiskPerTrade$ formula: First, calculate EPT: $EPT = (0.60 \times $250) - (0.40 \times $100)$ $EPT = $150 - $40$ $EPT = $110$

Then, calculate EPDR: $EPDR = \frac{$110}{$100} = 1.1$

EPDR is a powerful metric for position sizing and risk management. A positive EPDR indicates a profitable strategy. An EPDR of 1.1 means for every $1 risked, the strategy expects to return $1.10. This is a net profit of $0.10 for every $1 risked.

Comparative Analysis and Application

Each expectancy metric serves a specific purpose:

• EPS: Useful for analyzing strategies with consistent per-share profit/loss targets, such as high-frequency trading or scalping fixed-spread instruments. It helps evaluate the efficiency of capturing price movements.
• EPT: Provides a straightforward measure of average trade profitability. It is beneficial when comparing strategies that may have different position sizing rules but where the ultimate goal is maximizing dollar profit per trade. It is less useful for comparing strategies with vastly different risk profiles unless normalized.
• EPDR: The most versatile and robust metric for strategy comparison and risk management. By normalizing expectancy to the dollar risked, it allows traders to compare the relative efficiency of capital deployment across diverse strategies, asset classes, and risk tolerances. A strategy with a higher EPDR is generally more efficient at generating returns per unit of risk.

Step-by-Step Numerical Example: Comparing Two Strategies

A trader evaluates two day trading strategies, Strategy A (stocks) and Strategy B (futures).

Strategy A (Stocks):

• Win Rate ($P_{win}$) = 0.58
• Loss Rate ($P_{loss}$)