Trading expectancy measures the average profit or loss a strategy is expected to produce per trade. Risk of ruin measures the probability of losing enough capital to stop trading, under a stated set of assumptions. They answer different questions: does the strategy have an edge, and can the account survive long enough to benefit from it?
A positive answer to the first does not guarantee a positive answer to the second. In speculative trading, an apparently profitable strategy can still exhaust its trading budget through oversized positions, clustered losses or an edge that was never as reliable as the backtest suggested.
How to Calculate Trading Expectancy
For a strategy with winning and losing trades, the basic calculation is:
Expectancy = (win probability × average win) − (loss probability × average loss)
Express the average loss as a positive number because the formula subtracts it. This mathematical expectation framework from CME Group combines the frequency and size of outcomes. Win rate alone cannot establish profitability.
Suppose a hypothetical strategy wins 45% of its trades. Its average winning trade earns £180 before costs, while its average losing trade loses £120 before costs:
Gross expectancy = (0.45 × £180) − (0.55 × £120) = £15 per trade.
If average trading costs are £10 per completed trade, net expectancy falls to £5. Across 100 trades, the expected total would be £500 if those probabilities, payoffs and costs remained unchanged. That is an average across possible outcomes, not a forecast that the next 100 trades will deliver £500.
Include Costs Without Counting Them Twice
Use one consistent approach. Either calculate average wins and losses from results already net of costs, or use gross results and subtract average costs separately. Do not subtract spread or slippage again if execution prices already capture them.
For the example above, the win rate needed to break even after costs is:
Break-even win rate = (£120 + £10) ÷ (£180 + £120) = 43.33%.
The estimated 45% win rate sits only slightly above that threshold. Raising average costs from £10 to £15 would remove the estimated edge entirely. Small positive expectancy deserves careful cost checking, not a victory lap.
Use R Multiples When Trade Sizes Differ
An R multiple expresses a trade’s result relative to its initial planned cash risk. If that risk was £100, a £150 net profit is +1.5R and a £120 net loss is −1.2R. Planned risk is a measuring unit, not a guarantee that losses stop at −1R.
Average the net R results across all trades, including scratches, to estimate expectancy in R. Keep the cash results too: a positive average R can coexist with a cash loss if the largest positions were concentrated in losing trades.
Estimated Expectancy Is Not Proven Expectancy
The formula is straightforward. Establishing trustworthy inputs is harder. A sample average describes the trades recorded; treating it as a forecast requires assumptions about future trading conditions.
Consider 45 winners from 100 trades. Under an independent, constant-probability model, the approximate 95% Wilson confidence interval for the win probability runs from 35.6% to 54.8%. This calculation uses the NIST method for confidence intervals on proportions. The interval covers probabilities both below and above the example’s 43.33% break-even threshold.
That interval concerns win probability only. It says nothing by itself about uncertainty in average winning size, average losing size or future execution costs. Treating the observed £180 and £120 averages as fixed truths would leave much of the uncertainty unexamined.
Before relying on an estimate, check whether a few unusually large winners account for most of the profit. Separate results by strategy version and distinguish trades selected under unchanged rules from trades used to develop those rules. Keep unsuccessful tests in the research record rather than retaining only the winner.
There is no universal trade count that proves an edge. One hundred positions opened around the same market event do not provide the same breadth of evidence as observations spread across different conditions.
Define What “Ruin” Means Before Measuring It
For practical planning, define ruin as a threshold beyond which the strategy must stop, rather than assuming it means an account balance of exactly zero.
A trader starting with £10,000 might set £7,000 as the lowest acceptable account equity. In that case, the relevant question is the probability of touching £7,000, including unrealised losses, before the review period ends. The £3,000 difference is the permitted loss budget; the remaining £7,000 is not available for further experimentation.
State the time horizon too. “Probability of crossing £7,000 within 250 trades” is a different question from “probability of ever crossing £7,000 if trading continues indefinitely”. Neither should be presented simply as “the risk of ruin”.
A fixed capital floor also differs from a drawdown measured against a rising account peak. An account can breach a drawdown limit while remaining above its starting balance. The distinction matters when applying drawdown and recovery calculations to trading rules.
A Simple Risk of Ruin Formula—and Its Limits
The classic gambler’s ruin model developed in Columbia University’s probability notes gives a useful illustration. Assume independent outcomes, constant probabilities, and wins or losses of exactly one fixed cash unit. With no upper stopping target and an indefinite horizon, when the win probability exceeds 50%:
Probability of eventual ruin = (q ÷ p)B
Here, p is win probability, q is loss probability, and B is the starting capital measured in whole betting units. Under these assumptions, eventual ruin has probability one when p is 50% or lower.
With p = 55% and q = 45%, starting with 10 units gives an eventual ruin probability of approximately 13.44%. Starting with 20 units reduces it to approximately 1.81%.
These are model results, not trading forecasts. The formula does not directly handle unequal payoffs, changing stakes, dependent losses or a finite trading horizon. In particular, do not insert a 45% win rate from a strategy with larger winners than losers into this equal-payoff formula.
Position Size Changes the Survival Problem
Using a fixed cash stake and risking a fixed percentage of current equity produce different account paths. Under percentage sizing, the cash amount at risk shrinks after losses. Under fixed cash sizing, the same stake consumes a growing share of the remaining account.
Consider ten consecutive losses, each exactly equal to the chosen percentage of equity immediately before that trade. Assume the stated loss includes all costs, with no other account movements:
| Equity lost per trade | Equity remaining | Total decline |
|---|---|---|
| 1% | 90.44% | 9.56% |
| 5% | 59.87% | 40.13% |
| 10% | 34.87% | 65.13% |
The calculation is remaining equity = starting equity × (1 − risk fraction)10. These percentages are illustrations, not suggested allocations.
The table does not estimate how likely that losing run is. It shows what the run would do if it occurred. The same sequence produces very different damage depending on position size, which is why expectancy should be assessed alongside a defined trading risk budget.
In the ideal percentage model, repeated partial losses never reach exactly zero after a finite number of trades. That mathematical property offers little comfort if equity has already crossed the trader’s stopping threshold. “Not technically bankrupt” is a poor performance objective.
Positive Expectancy Does Not Guarantee Compound Growth
Arithmetic expectancy and compounded wealth are different measures. Suppose an illustrative trade has an equal chance of returning +60% or −50% of current equity. Its arithmetic expected return is positive:
(0.50 × 60%) − (0.50 × 50%) = +5%.
Yet one win and one loss leave only 80% of the starting account: 1.60 × 0.50 = 0.80. A £1,000 account becomes £800. That pair does not describe every possible two-trade outcome; it demonstrates why averaging percentage returns does not describe their compounded result.
The original Kelly paper on capital growth addresses this distinction by maximising expected logarithmic growth rather than expected cash profit alone. Its mathematical objective is long-run growth under the assumed probability model, not keeping short-term drawdowns comfortable.
For practical assessment, examine both average net trade results and the account paths produced by the proposed sizing rule. A positive expectancy figure does not justify increasing stakes without testing the consequences.
Estimate Practical Ruin Risk With Simulated Account Paths
When actual payoffs do not fit the simple formula, build simulated paths that reflect the proposed trading process. This is a modelling exercise, not a way to turn uncertain inputs into certain answers.
Start with a defined account balance, stopping threshold and horizon. Generate trade sequences, apply the sizing rule after each result, deduct any costs not already included, and record whether equity crosses the threshold at any point. Checking only the final balance would miss paths that failed earlier and later recovered on paper.
Suppose a hypothetical simulation produces 10,000 paths and 600 cross the chosen floor. Its estimated breach probability is 6% under that model. It is not a measured 6% probability for the live account.
Resampling historical trades individually treats their order as interchangeable. Where dependence matters, block bootstrap research on dependent observations provides a basis for resampling consecutive groups instead. This retains some local dependence, although the block length and stability of the underlying process still require judgement.
Stress the inputs as well as the ordering. Test smaller average winners, larger losses, higher costs and periods with no positive edge. Resampling alone cannot create a type of loss absent from the original data.
Model Shared Exposure, Not Just Separate Tickets
If several positions are open together, assess their combined equity path. Treating each ticket as an independent bet can miss a common source of loss. Three positions responding to the same currency move may behave more like one larger exposure than three separate opportunities.
Account for correlation and hidden concentration across trades when constructing scenarios. Include open-position losses rather than relying solely on completed-trade records. Otherwise, the model may report survival even when the account would have crossed its stopping threshold between exits.
Turn the Estimates Into Review Rules
A useful assessment should leave a written decision record, not just an expectancy number. Record the net expectancy estimate, its sample size, the sizing rule, the capital floor and the horizon used to measure breach risk. Keep the stressed results beside the base case so the favourable assumptions do not become the only assumptions anyone remembers.
Choose review triggers before trading. These might include costs exceeding the tested allowance, losses outside the modelled range, or account equity reaching the stated floor. A trigger should prompt a defined response, such as pausing new positions and checking the evidence, rather than automatically increasing size to recover losses.
Keep strategy development separate from assessment. Testing trading rules on data not used to design them is a useful discipline, although it cannot guarantee future profitability. If the rules change after a disappointing result, record a new strategy version rather than treating the revised test as independent confirmation.
Expectancy asks whether the proposed process appears profitable after costs. Risk of ruin asks whether its losses could force an early stop. Assess both with conservative assumptions: an estimated edge has little practical value if the account cannot withstand being wrong about it.