Risk of ruin formula: how to calculate it (and what breaks it)

The formula, and the one assumption that matters

For even-money bets, betting a fixed amount each time, starting with N units of capital where each unit is one bet, the probability of eventually losing all of it is:

P(ruin) = ((1 − W) / W) ^ units

where W is your win rate and units is starting capital divided by the amount you risk per trade. Risk 200 per trade with 10,000 of capital and you have 50 units.

That expression is ((1−W)/W), often written q/p, raised to the number of units. It is the classical gambler's ruin result, and it is exact — I verified it against an independent recursion, and the two agree.

Now the assumption. The formula assumes each bet is a **fixed amount**. Not a fixed percentage. If you risk 1% of your account and your account shrinks, the amount you risk shrinks too, and that changes the problem completely. The formula does not describe that strategy, and using it there is not a small approximation error.

What the formula actually says

The table below is the classic formula, for even-money bets at a fixed size.

units = 10: at a 51% win rate, ruin probability is 67.03%. At 52%, 44.91%. At 55%, 13.44%.

units = 20: 51% gives 44.93%, 52% gives 20.17%, 55% gives 1.81%.

units = 50: 51% gives 13.53%, 52% gives 1.83%, 55% is effectively zero.

units = 100: 51% gives 1.83%, 52% gives 0.03%.

Three things fall out of this. First, the sensitivity to win rate is brutal: going from a 51% to a 55% win rate at 10 units takes ruin from about two-thirds to about one in seven. Second, units matter as much as edge — doubling your capital per unit squares the ruin probability, which is why the same strategy is far safer at a larger account. Third, at a 50% win rate the formula returns 100% for every value of units. That is not a bug. With no edge and a fixed bet, a random walk with no drift eventually reaches zero with certainty, given unlimited time. The edge is what stops it.

That last point is worth sitting with: the formula says the only thing standing between you and eventual ruin is a positive edge, and the size of that edge governs how long you last.

Where the formula breaks: percentage-based sizing

Most traders — and nearly all systematic ones — size as a percentage of current equity, not as a fixed amount. That is a different mathematical object, and the classic formula does not apply to it. The honest thing to do is compute it properly rather than force it into the wrong formula.

The calculation for percentage sizing is a recursion rather than a closed form: track the distribution of outcomes where each trade multiplies equity by (1 + f) or (1 − f), and stop when equity first crosses your ruin level. It is small enough to run exactly.

Here is the result for risking a fixed percentage of equity per trade, with even-money outcomes, calling a 50% drawdown ruin, over 1,000 trades. I computed each cell by exact enumeration and cross-checked it against a Monte Carlo simulation; they agree to within about 0.4 percentage points.

Risk 0.5% per trade: ruin is under 0.01% at every win rate in the table.

Risk 1% per trade: 3.82% at a 50% win rate, 0.15% at 52%, effectively zero at 55%.

Risk 2% per trade: 36.89% at 50%, 7.26% at 52%, 0.17% at 55%.

Risk 3%: 61.84% at 50%, 22.14% at 52%, 1.78% at 55%.

Risk 5%: 85.11% at 50%, 50.80% at 52%, 11.51% at 55%.

Risk 10%: 98.40% at 50%, 87.09% at 52%, 46.85% at 55%.

Compare the 2% row against the classic formula. With 10,000 of capital and 2% risked, the classic formula would have you believe you have 50 units — and at a 52% win rate it would report 1.83% ruin. The correct percentage-based answer over 1,000 trades is 7.26%. The formula understates it by about a factor of four, in the direction that costs money.

Touching a drawdown is not the same as ending in one

There is a second distinction that changes the answer by orders of magnitude, and it is the one most risk discussions skip.

The numbers above are the probability of **touching** a 50% drawdown at any point during 1,000 trades. That is the right number if crossing that line ends your trading — a prop-firm limit, a margin call, or your own decision to stop.

It is a different number from the probability of **ending** below 50%, which is what actually happens to your money if you keep trading.

At a 50% win rate and 2% risk per trade, the chance of ever touching a 50% drawdown is 36.89%. The chance of finishing below it after 1,000 trades is 21.46%. Those are the same order of magnitude, so at this position size the distinction is real but not dramatic.

The gap widens sharply when you have a genuine edge. At a 55% win rate and 2% risk, the chance of touching a 50% drawdown is 0.17%, but the chance of ending below it is 0.0037% — roughly 46 times smaller. In other words, with a real edge and modest size, most of the drawdowns that scare you are temporary, and the rule that stops you out of them is doing more damage than the drawdown would have.

At aggressive sizes the two numbers collapse toward each other: at 10% risk with a 52% win rate, touching is 87.09% and ending below is 53.77%. There is no comfort left in the distinction because both outcomes are likely.

How to calculate your own risk of ruin

Step one: decide what ruin means. This is a choice, not a measurement. Losing everything is the literal definition; a 50% drawdown is a common practical one; a prop-firm limit is a contractual one. The probability changes enormously with the definition — for the same strategy above, ruin at −50% was 36.89% at a 50% win rate and 2% risk, but ruin at −80% was under 0.1%.

Step two: decide which sizing scheme you actually use. Fixed amount per trade → use the closed-form formula. Percentage of equity → use the recursion, not the formula.

Step three: put in an honest win rate. Use the out-of-sample figure, and then run the table again with a win rate one or two points lower. The sensitivity is steep enough that the pessimistic version is the one you should size from — this is the same estimation-error argument that makes full Kelly dangerous.

Step four: cross-check with simulation. A Monte Carlo of the same setup should land within a fraction of a percentage point of the exact figure. If it does not, one of the two is wrong, and you want to find out before you trade it.

What you are looking for is not a precise number. It is whether the answer is 2% or 40%. Those two lead to completely different position sizes, and the arithmetic is the only thing that distinguishes them.

What risk of ruin does not tell you

It does not tell you whether your edge is real. It takes the win rate as given, and if the win rate is an in-sample artefact, the output is a precise answer about a strategy that does not exist.

It assumes your trades are independent draws from a fixed distribution. Real strategies have streaks, clustering, and regimes, which makes the tail heavier than the calculation suggests. Treat the output as a floor, not a ceiling.

It says nothing about time. A 5% ruin probability might be over 100 trades or 10,000, and those are very different situations. The number of trades is an input you must state alongside the answer — the tables above use 1,000, and halving the trade count changes the figures substantially.

And it is not a prediction about your account specifically. It is a statement about a process. Two people running the same process can get different outcomes, and one of them being wiped out does not mean the calculation was wrong.