Expectancy: The One Formula That Connects Win Rate, Payoff, and Costs

Abstract

Expectancy is the mathematical formula that determines the average profit or loss per trade by combining three elements: the probability of winning, the average size of winners, and the average size of losers. ^[1] The formula reveals a counterintuitive truth: a trading strategy can be profitable with a win rate below 50% if winners are large enough, and equally, a high win rate can destroy capital if losses are larger than wins. ^[2] Understanding expectancy is essential for evaluating any systematic approach to trading, regardless of market or instrument.

Core Concept

Expectancy answers a single question: on average, how much profit or loss should a trader expect per trade if the system is executed repeatedly over time? The answer rests on three pillars that are often discussed separately but must be evaluated together.

First, the win rate (also called hit rate or probability) is the percentage of trades that close at a profit. A trader with a 40% win rate wins 4 out of every 10 trades; a trader with 70% win rate wins 7 out of every 10.

Second, the payoff ratio is the relationship between the average size of winning trades and the average size of losing trades. A payoff ratio of 2:1 means winning trades are twice as large as losing trades. A 1:3 ratio means every win is one-third the size of every loss.

Third, costs include commissions, exchange fees, bid-ask spreads, and slippage from order execution. These reduce the gross profit from every trade.

The insight is that these three elements interact. None can be evaluated in isolation. A 70% win rate paired with a payoff ratio of 1:5 (wins five times smaller than losses) will lose money. A 30% win rate paired with a 5:1 payoff ratio (wins five times larger than losses) will make money. The formula makes this relationship explicit.

Mechanical Formula

The basic expectancy formula is:

Expectancy (per trade) = (Win Rate × Average Win) − (Loss Rate × Average Loss) − Costs

Expressed algebraically:

E = (p × W) − ((1 − p) × L) − C

Where:

• p = probability of a winning trade (as a decimal; 60% = 0.60)
• W = average profit on a winning trade (in dollars or points)
• L = average loss on a losing trade (in dollars or points, stated as a positive number)
• C = average costs per trade (commissions, fees, slippage)

The result is the expected profit (or loss, if negative) per trade. If a system produces an expectancy of $15 per trade on average, and a trader executes 100 trades, the long-run expected total profit is $1,500 before capital depletion effects.

The formula demonstrates why win rate alone is misleading. Suppose Strategy A has a 70% win rate with wins averaging $100 and losses averaging $200. Expectancy = (0.70 × 100) − (0.30 × 200) − 0 = $70 − $60 − 0 = $10 per trade. Now suppose Strategy B has a 40% win rate with wins averaging $400 and losses averaging $100. Expectancy = (0.40 × 400) − (0.60 × 100) − 0 = $160 − $60 − 0 = $100 per trade. Strategy B, despite winning less than half the time, has an expectancy ten times better than Strategy A.

Worked Example: Historical Context

Consider a simplified illustration using historical volatility data. During the 2008-2009 financial crisis, many systematic trend-following traders (who profit from sustained price moves in one direction) operated with relatively low win rates, typically 40-50%. However, their payoff ratios were often 3:1 or higher because losing trades were exited quickly on reversal signals, while winning trends were held. ^[3]

Suppose a trend-following system on the S&P 500 E-mini futures contract (ES) executed during 2009 produced the following characteristics:

• Win rate: 45% (9 winners out of 20 trades)
• Average winning trade: 80 points
• Average losing trade: 40 points
• Costs per trade: 2 points (combined commissions and slippage)

Expectancy = (0.45 × 80) − (0.55 × 40) − 2 = 36 − 22 − 2 = 12 points per trade

At that time, each point on the ES was worth $50, so expectancy = 12 × $50 = $600 per trade. Over 100 trades, the expected gross profit would be $60,000 (before accounting for drawdown and position sizing). The trader can be systematically profitable despite failing to predict the direction of direction correctly even half the time, because the payoff structure and cost efficiency make up for the low win rate.

This illustrates why low-win-rate systems are viable in practice. They are common in trend-following, breakout strategies, and certain statistical arbitrage approaches, where capturing large infrequent moves is the goal, not batting average.

Limitations

Several critical limitations constrain the utility of the expectancy formula.

First, expectancy assumes statistical stability. The formula uses historical averages for win rate, average win, and average loss. If market conditions, volatility, or the trader's execution change, these averages change, and forward expectancy diverges from historical expectancy. A system that worked perfectly in 2019 may fail entirely in a different volatility regime. Expectancy is backward-looking; forward expectancy is unobservable. ^[4]

Second, expectancy does not account for drawdown or capital depletion. A system with positive expectancy of $10 per trade is still dangerous if individual losses of $10,000 occur before 100 trades accumulate. The formula treats all trades as if capital is unlimited. Position sizing and risk management, which limit the size of each loss, are essential to survival and must be applied separately. ^[5] Expectancy alone cannot answer the question "Will I run out of capital before the strategy converges to its long-run average?"

Third, the formula assumes independence of trades. In reality, trading outcomes are often correlated. A losing trade in a trending market may be followed by further losses as the trend accelerates against the trader. Win/loss sequences are not random; they cluster. This clustering can produce drawdowns far worse than expectancy would suggest.

Fourth, costs are often underestimated. The formula requires honest estimation of slippage, bid-ask spreads, and market impact. Retail traders often undercount slippage and market impact, inflating their expected results. ^[6] Bid-ask spreads and commissions are measurable, but slippage depends on liquidity, order size, and market conditions and varies significantly.

Fifth, expectancy conflates two different problems: strategy design and capital preservation. A strategy can have positive expectancy but fail because of inappropriate position sizing or insufficient capital. Conversely, a strategy with negative expectancy can avoid ruin if position size is so small that losses are negligible. Expectancy is necessary but not sufficient for profitable trading.

Sixth, the formula assumes transactions occur at consistent prices. It does not model regime change, liquidity collapse, flash crashes, or execution failure. During market stress (March 2020, November 2022), actual slippage and execution prices diverged sharply from normal conditions.

Key Definitions

Expectancy: The average profit or loss per trade, calculated as (Win Rate × Average Win) minus (Loss Rate × Average Loss) minus transaction costs.

Win rate: The percentage of closed trades that end at a profit; also called hit rate or probability of success.

Payoff ratio: The relationship between the size of average winning trades and average losing trades, expressed as a ratio (e.g., 2:1 means wins are twice as large as losses).

Average loss: The mean profit or loss on all losing trades in a sample, stated as a positive number to represent drawdown magnitude.

Average win: The mean profit on all winning trades in a sample.

Slippage: The difference between the expected execution price and the actual execution price, caused by market movement during order transmission and execution.

Position size: The quantity of contracts or shares traded; determines the dollar impact of each point or percentage move in the underlying instrument.

References

1. Pardo, Robert, "The Evaluation and Optimization of Trading Strategies," John Wiley & Sons (2008). ISBN 0-471-59556-7.

2. Vince, Ralph, "The Mathematics of Money Management: Risk Analysis Techniques for Traders," John Wiley & Sons (1992). ISBN 0-471-56725-X.

3. Polge, Courtney et al., "A Study of Trend-Following Performance and Volatility Regimes," working paper, SSRN (2018). Https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3083821

4. Aronson, David, "Evidence-Based Technical Analysis: Applying the Scientific Method and Statistical Inference to Trading Signals," John Wiley & Sons (2007). ISBN 978-0-470-00874-7.

5. de Prado, Marcos López, "Advances in Financial Machine Learning," John Wiley & Sons (2018). ISBN 978-1-119-48208-9.

6. Kissell, Robert, "The Science of Algorithmic Trading and Portfolio Management," Academic Press (2013). ISBN 978-0-124-01691-3.

Educational research on historical data only. Not investment advice, not a signal, and never a performance promise. Past results do not predict future performance. Every reference is link-verified before publication and every paper is re-audited weekly against the library's editorial standard.

Last reviewed by the PropLedger research pipeline: 2026-09-27. Educational research on historical data, not financial advice.