Expectancy and Risk of Ruin: The Two Formulas Behind Every Sizing Rule
Expectancy says whether a method makes money per trade. Risk of ruin says whether you survive long enough to collect. Both formulas, one grid of win rate against payoff, and the size at which an edge stops mattering.
The trading expectancy formula is one line: expectancy equals the win rate times the average win, minus the loss rate times the average loss. A method that wins 45% of the time, makes $150 on a winner and loses $100 on a loser has an expectancy of 0.45 × 150 − 0.55 × 100 = $12.50 per trade. Over 200 trades that is $2,500 before commissions, on average, with a wide spread around the average. The formula answers whether the game is worth playing. It says nothing about whether you will still be at the table when the average shows up; that is the second formula's job.
Both formulas need inputs from a journal, not from memory, and both assume you already know what a stop is and where it goes. Reading a chart comes first; this post is about what the numbers from that chart add up to.
Expectancy in R
Dollars hide the shape of a method, so the usual move is to measure every outcome in R, where 1R is the amount risked on the trade: the distance from entry to stop in points, times the point value, times the contracts. A winner that makes 1.5 times the risk is +1.5R; every loser at the stop is −1R. With the average winner written as R̄, expectancy per trade becomes:
W is the win rate as a fraction. The 45% method above has R̄ = 1.5, so E = 0.45 × 1.5 − 0.55 = 0.125R: an eighth of the risk, per trade, on average. Set E to zero and solve for W and you get the break-even win rate for any payoff ratio, 1 / (1 + R̄). The grid below is the whole relationship.
| Win rate | R̄ = 0.5 | R̄ = 1.0 | R̄ = 1.5 | R̄ = 2.0 | R̄ = 3.0 |
|---|---|---|---|---|---|
| 30% | −0.55 | −0.40 | −0.25 | −0.10 | +0.20 |
| 40% | −0.40 | −0.20 | 0.00 | +0.20 | +0.60 |
| 50% | −0.25 | 0.00 | +0.25 | +0.50 | +1.00 |
| 60% | −0.10 | +0.20 | +0.50 | +0.80 | +1.40 |
| Break-even win rate | 66.7% | 50.0% | 40.0% | 33.3% | 25.0% |
Three things fall out of the grid. A method that takes profit at half the risk needs two winners in three to break even, which is why scalping small targets against full-size stops is hard to keep positive after costs. A 50% win rate with winners the size of losers is zero, and commissions make it negative. And the cells that look attractive at the bottom right, 60% with 3R winners, describe methods that rarely exist over more than a few dozen trades, because a wide target lowers the win rate on its own.
The grid also explains why R̄ is set by the stop, not the other way round. The stop belongs to the setup: it goes where the trade idea is wrong. Moving it closer to raise R̄ on paper lowers W by more than it raises R̄, and the expectancy cell you land in is usually worse than the one you left.
One more number before the second formula. A win rate measured over 50 trades carries a standard error of about seven percentage points for a method near 45%, which means the true win rate of the 0.125R method could be anywhere from 38% to 52% on that sample. At 38% the expectancy is −0.05R. At 52%, +0.30R. Fifty trades cannot tell a small positive edge from a small negative one; two hundred start to.
Risk of ruin
Risk of ruin is the probability that a run of losses takes the account to a level from which it cannot continue, before the edge has time to work. On a personal account the ruin level is whatever balance you decide you would stop at. On a prop account it is written down for you: the maximum loss limit.
The formula needs one extra input, the ruin distance in units of risk. If the distance to the ruin level is $1,000 and each trade risks $50, the account has 20 units. Risk $200 per trade and it has 5. With even payoffs (R̄ = 1) and a fixed risk per trade, the probability of ruin with no upper limit on how long you trade is:
For winners larger than losers the base is the root of p × r^(R̄+1) − r + (1 − p) = 0 instead, which is a line of code rather than a mental calculation; the table below is that calculation for the 45%, 1.5R method.
| Risk per trade, as a share of the ruin distance | Units | Risk of ruin |
|---|---|---|
| 20% | 5 | 44% |
| 10% | 10 | 19% |
| 5% | 20 | 3.7% |
| 2% | 50 | 0.03% |
| 1% | 100 | under 0.001% |
The same method, the same edge, and the difference between a coin flip and a rounding error is the size. At 5 units the trader is ruined 44 times out of 100 with a positive expectancy in hand. The reason is that a positive expectancy is a statement about averages, and five units is not enough trades for an average to form before an ordinary losing streak, which for a 45% method includes five losses in a row about once every 40 trades, uses up the distance.
An evaluation adds a target, which shortens the game and changes the arithmetic. Take a 25K account with a $1,000 drawdown and a $1,500 target, a fixed floor for simplicity (a trailing floor makes every number below worse), and the same 45%, 1.5R method with independent trades:
| Risk per trade | Probability of hitting the drawdown before the target |
|---|---|
| $50 | 4% |
| $100 | 17% |
| $200 | 35% |
| $250 | 40% |
| $500 | 50% |
A method with zero expectancy, 40% with 1.5R winners, fails the same evaluation 60% of the time at every size in the table, because with no edge the outcome is the ratio of the two distances, 1,500 / (1,000 + 1,500), and size only changes how fast it arrives. The edge shows up in the gap between 60% and 4%, and only at the small sizes.
What the two formulas say together
Expectancy decides whether the method deserves size. Risk of ruin decides how much size it can carry before the method's own losing streaks end the account. Every sizing rule you have read, one percent, two percent, a fixed number of micros on an evaluation, is a way of setting the number of units: risking 2% of the ruin distance is 50 units, risking 1% is 100. The percentages are not moral values. They are the unit counts at which the ruin column turns to zero for methods with small, realistic edges.
Two consequences are easy to miss. First, the ruin distance on a prop account is the drawdown, not the balance: a 50K account with a $2,000 max loss limit and a $100 risk per trade has 20 units, not 500, and belongs in the 3.7% row. Second, the formulas are only as good as the win rate and R̄ you feed them, and those two numbers need a couple of hundred journal entries before they mean anything, which is a reason to trade small while the sample is still being built, independent of the ruin table.
I size from the drawdown rather than the balance for exactly this reason, and I keep the unit count in the tens on any account that has a floor. The expectancy number I trust less; it moves with every hundred trades, and the ruin table is the one that has to be right on the first day.