Expected value measures the average profit or loss a trading strategy would produce per trade under its assumed outcome probabilities. It combines how often the strategy wins, how much its winners earn, how much its losers lose, and what execution costs.
A high win rate alone cannot answer whether a strategy is profitable. A strategy that loses more often than it wins can still have positive expectancy if its winning trades are large enough. Within trading probability analysis, expected value connects the likelihood of an outcome with its financial consequence.
The Expected Value Formula for Trading
For a strategy with winning and losing trades, the usual formula is:
Gross expected value = (p × W) − ((1 − p) × L)
Here, p is the probability of a winning trade, W is the average profit on winners, and L is the average loss on losers, expressed as a positive amount. Use 0.40 for a 40% win probability, not 40.
When those payoffs exclude trading costs, subtract the expected cost of a complete trade:
Net expected value = (p × W) − ((1 − p) × L) − C
The symbol C represents average round trip costs. All amounts must use the same unit: dollars, percentage returns, or units of initial risk.
This is a probability weighted average, consistent with the mathematical definition in Penn State’s material on mathematical expectation. With several possible outcomes, multiply each outcome by its probability, then add the results. The probabilities must cover all possible outcomes and sum to one.
A positive result means the assumed distribution favors an average gain. A negative result means it favors an average loss. Neither result predicts the next trade. Also, the formula does not supply its own inputs: probabilities and average payoffs must come from a defensible model or an estimate based on trading records.
A Worked Example Using 100 Trades
Consider a hypothetical strategy with the following historical results. The profit and loss figures exclude costs, and every completed trade incurs a $10 round trip cost.
| Measure | Result |
|---|---|
| Completed trades | 100 |
| Winning trades | 40, averaging $200 profit |
| Losing trades | 60, averaging $100 loss |
| Gross profit across all trades | $8,000 − $6,000 = $2,000 |
| Total trading costs | 100 × $10 = $1,000 |
| Net profit | $1,000 |
Using the observed win rate and average payoffs:
Estimated net EV = (0.40 × $200) − (0.60 × $100) − $10 = $10 per trade
The strategy lost on 60% of its trades, yet its historical average remained positive. Its winners were twice the size of its losers before costs. The relationship between these inputs is examined further in win rate, payoff ratio, and the break-even point.
The $1,000 net profit is an observed result. The $10 figure becomes a forecast only if the sample provides a reasonable estimate of future outcomes. Under unchanged assumptions, another 100 trades would have an expected total profit of $1,000, but their actual result could be much higher or lower.
Account for Costs Without Counting Them Twice
Costs can turn a small gross advantage into a net loss. In the example, gross expectancy was $20 per trade. Raising average costs from $10 to $25 changes net expectancy to −$5, even though the win rate and gross payoffs remain unchanged.
Execution prices matter alongside explicit fees. A quoted price is not necessarily the price available when an order reaches the market, particularly during rapid price changes. The SEC’s explanation of order execution identifies how execution timing and routing can affect transaction prices and costs.
For a backtest, estimate commissions, spread effects, slippage, and applicable financing or borrowing charges. Use costs appropriate to the instrument, order size, and trading conditions rather than one convenient assumption for every trade.
For actual trading records, check what the reported profit already includes. Profit calculated from real entry and exit fills already reflects the spread and execution slippage embedded in those prices. Do not subtract those effects again. Deduct explicit charges only where they have not already been included.
Average costs can also conceal an awkward pattern: the most expensive executions may occur on the largest losing trades. Preserve individual trade records so that later risk analysis retains that relationship.
Estimate Expectancy From Complete Trading Records
The simplest historical estimator does not require separating winners and losers:
Estimated net EV = Sum of net trade results ÷ Number of completed trades
For the example, $1,000 divided by 100 equals $10. This calculation also handles trades that finish flat, partial exits, and differing outcomes without forcing everything into two categories.
Define a trade consistently. If one position exits in three parts, either aggregate those exits into one completed position or use another clearly stated convention throughout. Counting each profitable partial exit separately but treating the final loss as one trade distorts the win rate.
Keep actual results separate from planned outcomes. A target of twice the initial risk does not establish an average winner of twice the risk. Early exits, partial fills, gaps, and discretionary changes can produce a different distribution.
Expressing Results in Units of Risk
An R multiple divides a trade’s net result by its initial planned dollar risk:
Trade result in R = Net profit or loss ÷ Initial planned risk
If every trade in the example initially risked $100, its estimated net expectancy would be 0.10R per trade. That separates performance from the dollar size used for this sample. Position size can then be considered through a method such as fixed-fractional position sizing.
When initial risk varies, calculate each trade’s R multiple before averaging. Dividing total profit by average initial risk does not generally give the same answer. Retain both dollar and R results: one measures money earned at the sizes traded, while the other measures results relative to planned risk. Planned risk is a reference amount, not a guaranteed maximum loss.
Measure the Uncertainty Around Your Estimate
A positive sample average is not proof of positive future expectancy. Another sample from the same process could produce a different mean. A confidence interval makes some of that estimation uncertainty visible.
For independent observations under suitable distributional assumptions, a conventional interval for the mean is:
Sample mean ± t critical value × (s ÷ √n)
Here, s is the sample standard deviation and n is the number of observations. The NIST confidence interval method for a mean sets out this calculation and its interpretation.
Consider a separate illustrative sample of 100 trades with a $10 average net profit and a $150 standard deviation. The estimated standard error is $15. Using a 95% two-sided t interval gives approximately:
$10 ± $29.76, or −$19.76 to +$39.76 per trade
That interval spans zero, so the sample does not establish a positive mean at that confidence level under the test assumptions. It does not prove the strategy has no advantage either. A confidence interval for the mean is also not a prediction interval for the next trade.
Check Whether Trades Provide Independent Evidence
Trade count and independent evidence are not interchangeable. Ten positions opened together in closely related instruments may respond to the same market move. Treating them as ten unrelated observations can overstate the precision of an estimate.
Dependence can also occur across time. A strategy might produce runs of similar results during one market condition. Autocorrelation measures the relationship between observations in the same series at different lags; NIST’s autocorrelation guidance describes its use in detecting departures from randomness.
Where dependence is plausible, do not apply the simple standard error formula without checking its assumptions. Consider analysis that preserves time or exposure clusters rather than treating every result as an independent draw. Little measured autocorrelation does not, by itself, prove independence.
Rare, large losses create another concern. If the sample contains no example of a damaging outcome that remains possible, its average and estimated uncertainty may both look too reassuring. More observations help only if they represent the process you intend to trade.
Separate a Trading Advantage From Backtest Selection
A positive expectancy found after testing hundreds of rule combinations deserves more scrutiny than the same result from one prespecified test. Selecting the best historical result also selects favorable noise. This selection problem is examined in the research paper The Probability of Backtest Overfitting.
Record the rules and all configurations tested, not just the winner. Otherwise, a clean looking expectancy figure can hide a long search through unsuccessful alternatives. A spreadsheet multiplies a weak assumption just as efficiently as a sound one.
Freeze the proposed rules before evaluating new data. Keep development results separate from validation results, and do not repeatedly adjust the strategy to repair disappointing validation performance while continuing to call that data untouched.
Then compare expectancy across periods, instruments, and execution assumptions. Treat these comparisons as checks, not invitations to retain only the profitable slices. A strategy whose average depends on one exceptional trade or one brief period needs a different assessment from one with less concentrated results.
A structured process using walk-forward testing and out-of-sample validation helps separate rule development from evaluation. It cannot guarantee that future probabilities and payoffs will match the past.
Positive Expected Value Is Not a Complete Growth Plan
Expected value measures an average outcome, not the safety of the path taken to achieve it. Two strategies can have the same expected profit per trade but very different loss distributions, capital requirements, and drawdowns.
Position size also changes the consequences of a losing sequence. A genuinely favorable strategy can still expose an account to unacceptable losses if each position is too large. Increasing size does not repair uncertainty in the estimated advantage.
Nor should average percentage returns be confused with compounded growth. Suppose an account gains 50% in one period and loses 40% in the next. Its arithmetic average return is positive 5%, but $100 becomes $150 and then $90. The distinction between CAGR and arithmetic average returns matters when translating trade statistics into account growth.
Trade frequency needs similar care. A larger per-trade expectancy does not automatically mean better performance per month or per unit of capital. Holding time, overlapping positions, and capital tied up in each trade affect that comparison.
Use Expected Value as a Decision Input
A useful strategy assessment states both the estimated expectancy and the conditions behind it: the sample period, number of trades, position sizes, cost treatment, and validation method. “$10 per trade after modeled costs” is more informative than “profitable,” but it still needs an uncertainty estimate and a review of loss exposure.
Stress the inputs before acting on the result. Reduce the assumed win rate, shrink average winners, increase average losses, and raise execution costs. If modest changes turn expectancy negative, the estimated margin for error is thin.
The practical question is not just whether the calculation produces a positive number. It is whether that number survives realistic costs, credible testing, and position sizes the account can withstand.