Trade expectancy combines the probability and size of gains and losses into net P&L per trade. Calculate its break-even win rate and distinguish model assumptions from a trading record.
Trade expectancy
Trade expectancy is the expected net profit or loss per trade under a specified distribution of trade outcomes. It combines how often a trade wins with how much it wins or loses. For a quant reviewing a strategy, a high win rate is incomplete evidence: frequent small gains can be outweighed by less frequent larger losses.
Expectancy is an application of expected value to trading P&L. Its units follow the input: dollars per trade for dollar outcomes. A historical average trade is an estimate of that quantity, not a promise about the next trade.
The net payoff ledger
Consider one fixed-size, hypothetical trade with two exhaustive outcomes. Its P&L is already net of commissions, spread and slippage as specified by the model; assume no other costs in this example. The probabilities are stipulated, not estimated from a backtest.
Outcome
Probability
Net P&L
Contribution to expectancy
Win
60%
+10 USD
+6.00 USD
Loss
40%
−12 USD
−4.80 USD
Total
100%
—
+1.20 USD per trade
Let X be this trade's net P&L. Then
E[X]=0.6(10)+0.4(−12)=1.20 USD.
That positive mean coexists with a 40% chance of losing USD 12. Neither possible outcome is USD 1.20: the expectation summarizes the distribution rather than naming an amount the next trade will pay. MIT's expected-value notes explain this weighted-average interpretation.
Net outcome contributions and the break-even win probability for a stipulated +10 or −12 USD trade.
Original two-outcome model, already net of specified costs. The line varies only win probability while holding the +10/−12 USD payoffs fixed; it is not a fitted relationship or historical result.
How good must the win probability be?
With win probability p, a positive winning payoff W, a loss magnitude L>0, and no zero-P&L outcomes,
e(p)=pW−(1−p)L.
Here W and L may be fixed payoffs, as in our example, or the conditional mean gain and loss magnitude within their respective groups. This is the frequency-and-size decomposition used in CME Group's trading education.
Holding those payoffs fixed, the break-even probability solves
pW−(1−p)L=0⇒pBE=W+LL.
For the example,
e(p)=22p−12,pBE=2212=116≈54.55%.
A 60% win probability lies above that threshold. At 50%, expectancy would be −1 USD per trade. At the boundaries, e(0)=−12 and e(1)=10 USD. The calculation isolates the payoff–probability trade-off; changing an entry or exit rule in a real strategy can change all three inputs together.
Keep the measurement basis intact
A backtest produces a record of trades. For n>0 closed trades with comparable net P&Ls xi, the observed expectancy is their sample mean:
e=n1i=1∑nxi.
Include flat trades in n. If they have zero net P&L, they contribute zero to the sum; do not silently discard them and treat the loss rate as one minus the win rate. Classify wins and losses after applying the stated cost convention.
Transaction costs must enter once. Starting from gross P&L requires subtracting expected costs; starting from the net outcomes in the ledger requires no further deduction. Actual fills, holding costs and trade size can change the relevant net distribution.
Dollar expectancy has no account-capital denominator, so USD 1.20 per trade is not a portfolio percentage return or compounded growth rate. Averaging dollar P&L across changing position sizes also mixes exposures. State the size and measurement unit before comparing strategies.
A positive number is only one part of the research case
Computing one trade's expectation does not require independence. For a fixed number of trades, expected P&Ls add by linearity even when trades are dependent. Describing repeated averages as converging to this mean requires additional assumptions, such as independent, identically distributed trades with a finite mean; those assumptions do not follow from the ledger.
Dependence, changing market conditions, omitted losses and selection bias can undermine an estimate or change the risk of a trade sequence. Expectancy alone gives neither loss limits nor the capital needed to withstand a drawdown.
To study the spread around a favorable average, SelfMadeQuant's Expectation, Variance and Covariance develops the probability-weighted mean alongside variance. The public page is a preview; the full Probability and Statistics primer topic is available with a free signed-in account.
Dimitri P. Bertsekas and John N. Tsitsiklis, Introduction to Probability, 2nd ed. (Athena Scientific, 2008), Chapter 2 §2.4, p. 82, and §2.6, pp. 104–105: discrete expectation and decomposition through conditional expectations. Local grounding text; the trade ledger and figure are original.
