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

Abstract

Expectancy, or expected value, is the average profit or loss per trade when win probability, average payoff, average loss, and costs are combined into a single metric. The formula provides a mathematical framework for assessing whether a trading system can be profitable at all, regardless of how much capital is deployed. This paper explains the mechanic, shows a worked example, and discusses why expectancy alone is not sufficient for trading success.

Core Concept

Expectancy answers a fundamental question: on average, how much does each trade make or lose? ^[1] The formula ties together the three elements that determine whether a trader or system will be profitable over time: the probability of winning, the size of wins, and the size of losses. Without expectancy, traders may focus on only one dimension, such as win rate alone, and miss whether their winning trades are actually large enough to compensate for losing trades and the costs of trading.

In mathematical form, expectancy is:

E = (Win Probability × Average Win) − (Loss Probability × Average Loss) − Costs per Trade

where Win Probability and Loss Probability sum to 100 percent (or 1.0), or equivalently, Average Loss is expressed as a negative number:

E = (Win Probability × Average Win) + (Loss Probability × Average Loss) − Costs per Trade

A positive expectancy indicates that, on average, each trade generates a profit; a negative expectancy indicates a loss. An expectancy of zero means the strategy breaks even on a cost-free basis.

This formula is a direct application of expected value, a statistical concept used across insurance, gaming, and finance to measure the long-run average outcome of a probabilistic event. ^[2] Expectancy is not the same as a single trade's profit or loss; it is the center of gravity of many trades.

How It Works Mechanically

The formula operates in three steps: measuring each component, calculating their contribution, and interpreting the result.

Measuring the components. Win probability is the fraction of trades that close profitably, calculated as (number of winning trades) / (total trades). Average win is the mean profit of all winning trades (including commissions and slippage). Average loss is the mean loss of all losing trades, expressed as a negative number or a magnitude. Trading costs per trade include commissions, fees, and an estimate of slippage, the difference between expected execution price and actual price. ^[3] These components must be measured consistently: all in the same currency, all in terms of profit/loss from entry to exit, all including costs if costs are being measured.

Calculating expectancy. Once the win probability is known (say, 55 percent), and the average win and loss are measured, the formula isolates the contribution of each side. A 55 percent win rate contributes (0.55 × average win). The loss side contributes (0.45 × average loss). Costs are subtracted in full. The result is the net expected value per trade.

Interpreting the result. If expectancy is positive, the system is theoretically profitable on a large sample of trades; if negative, it is not. Crucially, expectancy tells the trader nothing about when profit will materialize or how much capital is needed to survive drawdowns. A system with positive expectancy can still go bust if the trader is under-capitalized or if a run of unlucky trades depletes the account before the long-run average is reached.

Worked Example

Consider a trader using a simple mean-reversion system on daily closes of a liquid stock index futures contract. Over 100 trades, recorded with full transaction costs:

• Winning trades: 58 out of 100 (win probability = 0.58)
• Average win: $650 per contract
• Average loss: $400 per contract
• Commission and fees: $25 per round-trip trade
• Estimated average slippage (bid-ask crossing): $15 per round-trip trade
• Total cost per trade: $40

Expectancy = (0.58 × $650) + (0.42 × −$400) − $40 = $377 − $168 − $40 = $169 per trade

This system has a positive expectancy of $169. On a $50,000 account trading one contract per trade, this expectancy translates to approximately $169 / $50,000 = 0.34 percent of account per trade, or about 34 basis points. (Note: this is a simplified illustration; real position sizing is more complex.)

If the trader increases contract size to two per trade (assuming the system continues to behave the same), expectancy doubles to $338 per trade, but so does the risk on each losing trade. If the trader operates this system consistently across 250 trading days, earning 250 × $169 = $42,250 in gross expectancy, the trader still requires adequate capital to absorb drawdowns. A string of 10 losses in a row would lose $4,000 (plus opportunity cost during the drawdown). If the account is $50,000, that is an 8 percent drawdown, which is material but potentially tolerable depending on the trader's risk appetite.

Limitations

Expectancy assumes the past predicts the future. The formula is calculated from historical data: a sample of trades already executed. There is no guarantee that future win rate, average win, or average loss will match the past. Market regimes change; volatility shifts; correlations break. A system with a 58 percent win rate over the past 100 trades may have a 45 percent win rate in the next 100 trades if market conditions are different. The longer the sample used to estimate expectancy, the more solid the estimate, but even long samples can become obsolete. ^[4]

Expectancy does not account for variance or drawdown. A system with an expectancy of +$169 per trade could experience a run of 10 or 20 consecutive losses, even if the long-run win rate is 58 percent. Expectancy tells the trader the average outcome, not the worst-case intermediate outcome. Two systems with the same expectancy but different win rates or payoff ratios will have very different drawdown profiles. A system with a 70 percent win rate and small average wins will feel different from a system with a 40 percent win rate and large average wins, even if both have the same expectancy. Risk of ruin, which measures the probability of losing the entire account before breaking even, depends on capital, expectancy, and variance, not on expectancy alone. ^[5]

Expectancy is blind to costs that aren't per-trade. The formula assumes costs are proportional to trade frequency. It includes commission and slippage but may not fully capture the cost of holding inventory overnight (margin interest), the cost of monitoring the market and executing trades (labor, technology), or the spread between bid and ask when the trader needs to exit quickly. These costs may be material to profitability, especially for high-frequency or discretionary trading.

Expectancy requires a large sample. A sample of 100 trades is often insufficient to distinguish a system with genuine positive expectancy from one that simply got lucky. Statistical tests require much larger samples, often 200 to 500 trades or more, to be confident that a result is not due to chance. A trader who measures expectancy on only 20 trades and finds it positive may be looking at noise.

Expectancy ignores market impact and correlation. The formula assumes that trading one contract (or 100 shares) at a time does not affect market prices and that trades are independent. In reality, large positions may move the market; correlated markets may move together, amplifying drawdowns in opposite directions; and the cost of unwinding a large position may be far higher than the cost of a small one. Expectancy calculated on small position sizes may not hold at large position sizes.

Key Definitions

Expectancy: The average profit or loss per trade, calculated as (Win Probability × Average Win) + (Loss Probability × Average Loss) − Trading Costs per Trade.

Win Probability: The fraction of trades that close with a profit, expressed as a decimal between 0 and 1.

Average Win: The mean profit of all profitable trades, measured from entry to exit and including commissions and slippage.

Average Loss: The mean loss of all unprofitable trades, expressed as a positive magnitude or a negative number, including commissions and slippage.

Trading Costs: Commissions, exchange fees, and bid-ask slippage incurred on each trade.

Drawdown: The peak-to-trough decline in account equity during a period, expressed as a percentage or dollar amount.

Expected Value: A statistical concept: the long-run average outcome of a probabilistic event, equal to the sum of each outcome weighted by its probability.

References

1. Investopedia, "Expected Value," Investopedia. Https://www.investopedia.com/terms/e/expected_value.asp
2. Wikipedia, "Expected Value," Wikipedia. Https://en.wikipedia.org/wiki/Expected_value
3. CME Group, "Risk Management in Futures Trading," CME Group Education. Https://www.cmegroup.com/education/
4. Securities and Exchange Commission, "Risk," SEC Office of Investor Education and Advocacy. Https://www.investor.gov/
5. Ralph Vince, "The Leverage Space Trading Model," John Wiley & Sons (2009).

Summary. Expectancy is a single number that combines win rate, payoff size, and costs to show whether a strategy is profitable on average. The formula is simple, but using it correctly requires measuring the component parts accurately and over a large enough sample to avoid noise. Critically, positive expectancy is a necessary condition for profitable trading, not a sufficient one; capital, risk management, and consistency matter equally. The formula should be the starting point of strategy evaluation, not the end.

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: 2026-10-05. Educational research on historical data, not financial advice.