Risk of Ruin: The Formula, a Computed Curve and How to Lower It
The probability that losses reach the level where you stop trading. The gambler's ruin formula explained step by step, a curve and table computed from it, an approximate formula for unequal wins and losses tested by simulation, and the one input that matters most.
On this page
Risk of ruin is the probability that a run of losses takes your account down to the level at which you would stop trading, before your edge has time to pay. Three inputs decide it: your edge (win rate and the size of wins against losses), the amount you risk on each trade, and the loss you call ruin. Risk per trade is the only one you control completely, and it moves the result the most: with the same hypothetical 55% win rate, risking 2% of the account per trade gives a 0.66% chance of losing half the account, and risking 10% gives 36.7%.
Position size from account risk and stop distance
The risk of ruin formula
The basic formula comes from the gambler's ruin problem of probability theory. A player wins one unit with probability p and loses one unit with probability q = 1 − p, and starts z units away from ruin. Against an opponent with unlimited money, the probability of eventual ruin is (q/p)z when p is larger than q, and 1 otherwise (Grinstead and Snell, Introduction to Probability, section 12.2, free PDF from Dartmouth). The result is old: the same book credits De Moivre with a derivation in 1711.
For a trader, one unit is the fixed amount risked on a trade, a win pays the same amount as a loss (a 1:1 payoff), and z is the number of those losses that separate you from your ruin level.
Risk of ruin by risk per trade: the curve
The curve below applies the formula to three hypothetical win rates with a 1:1 payoff, ruin set at a loss of 50% of the starting capital, and a fixed dollar amount risked on each trade, expressed as a share of the starting capital.
Risk of ruin against risk per trade, by win rate
The same numbers as a table, for quick lookup with a 1:1 strategy:
| Win rate | 1% per trade | 2% | 3% | 5% | 10% |
|---|---|---|---|---|---|
| 52% | 1.83% | 13.5% | 26.3% | 44.9% | 67.0% |
| 55% | 0.004% | 0.66% | 3.53% | 13.4% | 36.7% |
| 60% | under 0.001% | 0.004% | 0.12% | 1.73% | 13.2% |
Two things stand out. For the 55% and 60% cases the probability falls below 1% at 2% per trade or less, and above 5% it climbs fast even for the 60% case. And a thin edge is fragile: going from 55% to 52% at 2% per trade multiplies the risk of ruin by about twenty.
The edge form of the formula
The same formula is also written as ((1 − edge) ÷ (1 + edge))u, where edge = win rate minus loss rate and u is the number of units to ruin. With a 1:1 payoff this is exactly (q/p)z, since (1 − p + q) ÷ (1 + p − q) = 2q ÷ 2p.
Unequal wins and losses: an approximate formula
When wins and losses differ in size, the gambler's ruin formula no longer applies directly. The approximation below is attributed to Ralph Vince, but we could not confirm it. His book Portfolio Management Formulas (Wiley, 1990) has a chapter titled Risk of Ruin (publisher's contents page). We could not check the formula against the book, and we found no reference we could verify that ties it to him, so treat the attribution as unconfirmed. What we can do is test it.
A variant of it subtracts the loss term inside the square root of A. Kept with P = 0.5 × (1 + Z ÷ A), it still returns a number, so it can be compared. To see which version holds up, we simulated 200,000 accounts per case, each trading up to 20,000 times with fixed dollar risk, and counted how many lost half of their starting capital.
| Hypothetical case | Formula, plus sign | Minus sign, P = 0.5 × (1 + Z ÷ A) | Simulation |
|---|---|---|---|
| 55% wins, win 5%, lose 5% | 13.4% | under 0.001% | 13.4% |
| 40% wins, win 4%, lose 2% | 1.03% | 0.004% | 0.77% |
| 40% wins, win 10%, lose 5% | 16.1% | 1.73% | 14.4% |
| 50% wins, win 3%, lose 2% | 0.041% | under 0.001% | 0.024% |
The plus sign version lands close to the simulation and errs on the cautious side; the minus sign version, computed with the same P, understates it by a factor of eight or more. Wins and losses in the table are shares of the starting capital. How the simulation is done, so that anyone can rerun it: each case starts a fresh NumPy PCG64 generator, default_rng(20261101), draws the trades in blocks of 500 for the accounts still alive, and stops an account the first time it touches half of its starting capital. With 200,000 accounts the sampling error is small: one standard error is about 0.08 percentage points at 13%, and 0.02 at 0.77%, far less than the gap between the two formulas.
The approximate formula against a simulation
- Formula, plus sign
- Simulation
If the minus sign version is written with a different P, 0.5 × [(1 + Z) ÷ A], the 1 added to Z before dividing by A, the formula gives no probability at all. In the four cases above, P comes out between 10 and 32, far above 1, so (1 − P) ÷ P is negative and cannot be raised to the power M ÷ A. The same happens with the plus sign. If a calculator returns a risk of ruin from that version, check how it computes P.
Fixed fractional sizing changes the answer
Both formulas assume you risk the same dollar amount on every trade. Fixed fractional position sizing risks a fixed share of the current balance instead, so the stake shrinks after losses. We simulated that case too (55% wins, 1:1 payoff, 100,000 accounts, up to 5,000 trades, ruin at half the starting capital, same generator and seed; one standard error is at most 0.16 percentage points):
| Risk per trade | Fixed dollars (formula) | Fixed share of current balance (simulation) |
|---|---|---|
| 2% | 0.66% | 0.18% |
| 5% | 13.4% | 11.7% |
| 10% | 36.7% | 47.5% |
At small risk, shrinking the stake after losses cuts the risk of ruin. At 10% it makes things worse, because large swings in a compounding account drag the balance down even when wins and losses alternate: a 10% gain followed by a 10% loss leaves 99% of the money. The position size calculator guide shows how to set the share risked per trade.
Fixed dollars or a fixed share of the balance
- Fixed dollars (formula)
- Fixed share (simulation)
Risk of ruin is not drawdown
A drawdown is how far an account falls from its peak; risk of ruin is the probability that the fall reaches a level you have fixed in advance. The answer depends on that level, so write it down: for one trader ruin is a 100% loss, for another it is the 30% drop after which they would stop. With a 1:1 payoff and 55% wins, risking 5% per trade gives 13.4% for a 50% ruin level and 1.8% for a 100% level.
How to lower your risk of ruin
- Cut the risk per trade first. It is the input with the largest effect and the only one you set directly. Along each curve above, it is the only input that changes.
- Define ruin honestly. Use the loss at which you would actually stop, not 100%.
- Measure the edge on enough trades, and use the pessimistic end of the estimate. The trading math guide covers expectancy and sample size.
- Check the payoff, not only the win rate. A 40% win rate can be safer than a 55% one if the wins are twice the losses; the risk reward ratio guide shows the break-even win rate for each ratio.
- Watch leverage on small accounts. High leverage makes it easy to risk 10% or more on one trade without noticing; see the best leverage for a small account.
More risk and sizing topics are in trading concepts.
Frequently asked questions
What is risk of ruin in trading?
The probability that your account falls to a loss level you have defined as ruin, for example half of the starting capital, before your edge has time to pay. It depends on the win rate, the payoff and the amount risked per trade.
What is the risk of ruin formula?
For equal wins and losses and a fixed amount risked per trade: (q/p)z, where p is the win rate, q the loss rate and z the number of losses between you and ruin. If p is not larger than q, the risk of ruin is 1.
Is there a risk of ruin formula for unequal wins and losses?
An approximation attributed to Ralph Vince, not confirmed: Z = W × average win − L × average loss, A = the square root of (W × average win² + L × average loss²), P = 0.5 × (1 + Z ÷ A), and risk of ruin = ((1 − P) ÷ P) to the power (ruin level ÷ A), all in shares of starting capital. We could not confirm the attribution or check it against Vince's book; with a plus sign inside the root it matched our simulations. Watch how P is written: with P = 0.5 × (1 + Z) ÷ A, the formula gives no probability.
What is an acceptable risk of ruin?
There is no official threshold; it is a personal choice. The table on this page shows how much it depends on the edge: with a hypothetical 55% win rate and a 1:1 payoff, risking 2% per trade gives 0.66%, but with 52% the same 2% gives 13.5%, and 5% or more puts both in double digits.
Does a 50% win rate always lead to ruin?
With equal wins and losses and no end to trading, yes: when p equals q, the gambler's ruin probability is 1. Over a limited number of trades it is lower, but nothing protects the account in the long run without an edge.
Is risk of ruin the same as maximum drawdown?
No. Maximum drawdown is the largest fall from a peak that actually happened. Risk of ruin is a probability that a fall reaches a level you fix in advance.
Sources
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