Risk of Ruin in Trading: Formula and Assumptions

Risk of ruin in trading is the probability that an account falls to a loss threshold you define in advance, given a stated edge, a stated distribution of trade outcomes, a stated sizing rule, and a stated time horizon. It is not a property of a strategy. Change any one of those inputs and the number changes, often by a factor of ten or more. For the illustrative model used in this article (45% winners at +2R, losers at −1R), a static floor 20 risk units below the starting balance has a continuous-approximation estimate of about 0.2%. At 10 units it is about 4%. If the true win rate is 35% instead of 45%, the 20-unit estimate rises to about 38%.

Here “ruin” means the loss level at which you would have to stop trading the account: a prop-firm maximum loss, a personal stop-out level, or the point where the method can no longer be traded at its normal size. It rarely means a zero balance. How much to risk on each trade is covered in position sizing in trading. This article covers the probability that follows from that choice. Simulated ranges for drawdown depth and losing streaks from a real trade record are covered in Monte Carlo simulation for trading.

How do you calculate risk of ruin for a trading account?

Define the ruin threshold. Write the loss level that ends the account or the method, and whether it is fixed or trails the equity peak.
Convert the distance to risk units. Divide the distance from current equity to the threshold by the amount risked per trade (1R). A $5,000 buffer at $250 per trade is 20R.
Estimate the edge and variance in R. From a clean record under one rule version, calculate the average outcome per trade (μ) and the variance of outcomes (σ²), both in R and after costs.
Apply the formula. For a static threshold, no time limit, and a positive edge (μ > 0), the Brownian approximation is P(ruin) ≈ exp(−2μB/σ²), where B is the buffer in R. If μ ≤ 0, the corresponding infinite-horizon Brownian model reaches any finite lower threshold with probability 1; the same is true of a non-degenerate iid random walk with finite variance and mean μ ≤ 0. The exponential formula does not apply.
Stress the edge. Recalculate with a smaller μ, because the formula is far more sensitive to the edge than to anything else.
Check the assumptions the formula ignores. Trailing thresholds, time limits, correlated positions, outsized losses, and changes in size after losses all need separate treatment.

What counts as ruin in a trading account?

The word covers several different thresholds. The one you choose decides which calculation applies.

Static loss floor. A fixed balance, such as $20,000 on a $25,000 account. Most formulas assume this case.
Trailing loss floor. A floor that moves up as the account makes new highs. Many evaluation and funded accounts use this or a variant that locks at some level. It needs a different treatment, covered below.
Personal stop-out level. A drawdown at which you have decided to stop and review the method. It is self-imposed, but it works the same way in the calculation.
Literal zero. Relevant mainly for leveraged accounts where a loss can exceed the balance.

Two further choices matter. The first is the sizing rule: a fixed dollar amount per trade, or a fixed percentage of current equity. The second is the horizon: ever, or within a stated number of trades or days. A risk-of-ruin figure given without its threshold, sizing rule, and horizon cannot be compared with any other.

Expressing the buffer in R keeps the calculation independent of account size. A $2,000 maximum loss with $200 risked per trade and a $20,000 maximum loss with $2,000 risked per trade are both 10R buffers. They have the same risk of ruin if the trade outcomes are the same.

What is the risk of ruin formula?

The classical case is the gambler’s ruin problem: a sequence of independent, identical bets, where the question is whether the bankroll loses a set amount before it wins a set amount. Ethier’s treatment derives an explicit formula when every bet pays even money. For other integer payoffs a formula exists only in special cases, and for arbitrary payoffs only upper and lower bounds are available.¹ Trading outcomes are rarely even money, so the exact version is mostly a reference point.

For even-money trades that win with probability p and lose with probability q = 1 − p, with no profit target, the probability of ever losing an integer buffer of B units is (q/p)^B when p is greater than q. A 55% win rate at 1:1 with a 20R buffer gives (0.45/0.55)^20, about 1.8%.

For other payoff shapes, a useful approximation treats cumulative P&L as a Brownian diffusion with drift μ and variance σ² per trade. Thorp gives a related approximation for the probability of ever falling to a fraction of starting capital under fixed-fraction betting. He derives the two-barrier diffusion version from the classical gambler’s ruin formula by letting the step size shrink to zero, and notes that it is the known solution for a Wiener process between two absorbing barriers.² Applied to P&L in R rather than to log equity, the no-target version is:

P(ever losing B) ≈ e^(−2μB/σ²), for μ > 0

This infinite-horizon lower-barrier formula applies only when μ > 0 and σ² > 0. When μ ≤ 0, the corresponding Brownian model—and an iid random walk with finite, nonzero variance under the stated assumptions—reaches any finite lower threshold with probability 1 given unlimited trades. Putting a negative μ into the exponential would return a number above 1, which is not a probability.

For the even-money example above, μ = 0.10R and σ² = 0.99, which gives about 1.8% as well. For even-money bets the two methods agree closely. For skewed payoffs they can differ, as the worked example below shows.

Worked example

The model used through the rest of this article is illustrative, not taken from any trader or strategy. It has 45% winners at +2R and 55% losers at −1R.

μ = 0.45 × 2 − 0.55 × 1 = +0.35R per trade
E[X²] = 0.45 × 4 + 0.55 × 1 = 2.35, so σ² = 2.35 − 0.35² = 2.2275
2μ/σ² ≈ 0.3143

The Monte Carlo column reports a finite-horizon estimate from 20,000 seeded paths of 5,000 trades each (xorshift32, seed 20260923). It is not an infinite-horizon result. At 30R, no path reached the floor, so that simulation does not resolve the probability.

Buffer to static floor	Continuous approximation, ever	Exact discrete, ever	Monte Carlo, within 5,000 trades
10R	4.32%	3.41%	3.54%
20R	0.186%	0.116%	0.12%
30R	0.0080%	0.00397%	not resolved (0/20,000 paths)

The exponential formula is a continuous or diffusion approximation: it replaces the discrete +2R and −1R steps with a smooth process having the same mean and variance. This particular model also has an exact discrete answer because it is skip-free downward: every loss is exactly −1R, so the account reaches an integer-R floor by landing on it rather than jumping past it. Let c be the probability of ever falling 1R below the current level. The recurrence is 0.45c² + 0.55c⁻¹ = 1, or 0.45c³ − c + 0.55 = 0. Its root in (0, 1) is c ≈ 0.71335165, so for an integer buffer B the probability of ever reaching the floor is c^B. This direct calculation from the stated random-walk recurrence gives 3.41%, 0.116%, and 0.00397% for 10R, 20R, and 30R. It is not a result attributed to the cited sources.

As path count and horizon grow, Monte Carlo estimates for this discrete model converge toward the exact discrete result. The finite-sample difference in this table reflects simulation noise and the finite horizon; the difference between the exact discrete result and the higher diffusion estimate is approximation error from replacing discrete trades with a continuous process. Here the diffusion estimate is higher, but that direction is not guaranteed. Occasional −2R or −3R losses can jump past the floor, so neither this exact root nor the direction of the approximation error carries over. That is why variance should be calculated from actual outcomes, not from the planned stop, and why irregular records are usually handled with an approximation or a simulation of the actual outcomes.

The buffer matters a great deal. Each additional 10R multiplies the formula estimate by e^(−3.14), about 1/23, and the exact figure by c^10, about 1/29. At a fixed account size, halving risk per trade doubles the buffer in R, so the effect of that change is large.

Why is risk of ruin so sensitive to the edge?

In the formula, the edge μ multiplies the buffer inside the exponent. Halving the edge roughly takes the square root of the probability, so 0.19% becomes about 4%. Removing the edge entirely makes ruin certain eventually, at any buffer. The table keeps the +2R/−1R payoff and a 20R static buffer, and lowers only the win rate:

Win rate	Edge (μ)	Continuous approximation, ever	Exact discrete, ever	Exact discrete, within 250 trades
45%	+0.35R	0.19%	0.12%	0.12%
40%	+0.20R	2.5%	2.0%	1.9%
35%	+0.05R	37.7%	37.1%	20.8%
33.3%	0R	100% (μ ≤ 0 case)	100%	36.7%

The exact columns use the same recurrence as the worked example. The last column applies it over a fixed 250 trades instead of an unlimited run.

A 10-point drop in win rate turns a small-looking risk into a large one. That makes the edge estimate the input to question first. Over 100 trades, the standard error of a 45% win rate is about 5 percentage points, so a record that size cannot tell 45% from 40% with much confidence, and the reduced-edge rows are not pessimism. They are inside the range the record allows.

The horizon column shows a second effect. With a strong edge, most ruin happens early, so 250 trades and “ever” give almost the same answer. With a thin or zero edge, the probability keeps rising as the horizon grows. At zero edge, the exact figure is 37% within 250 trades, 84% within 5,000, and 100% with no limit. A short test period can therefore make a marginal method look safer than it is.

Same expectancy, different risk of ruin

Expectancy alone does not set risk of ruin. Variance does too. The two models below have the same edge, +0.35R per trade, with a 20R static buffer. The Monte Carlo estimates use 20,000 paths of 5,000 trades each. The +2R/−1R run uses xorshift32 seed 20260923; the +4.4R/−1R run uses seed 20260925. These are finite-horizon estimates, not “ever” probabilities.

Model	σ²	Continuous approximation, ever	Exact discrete, ever	Monte Carlo, within 5,000 trades
45% winners at +2R, losers at −1R	2.23	0.186%	0.116%	0.12%
25% winners at +4.4R, losers at −1R	5.47	7.7%	Not calculated	4.62%

The low-win-rate model reaches the same average by rarer, larger winners. Between those winners it has longer losing runs, so more of its paths reach the floor first. Its payoff process does not have the simple skip-free recurrence used for the +2R/−1R model, so its table entry is a finite-horizon simulation rather than an exact eventual result. For the same account, the low-win-rate method needs a larger buffer in R, which means less risk per trade, to reach the same probability. Win rate and reward-to-risk ratio are not interchangeable when the question is survival rather than long-run average.

Does fixed-fractional sizing remove the risk of ruin?

Within an idealized bounded-loss model, fixed-fraction sizing keeps capital above exact zero at every finite step if the position size is always a fraction below 100% of current capital and every loss is bounded by the modeled amount. Thorp establishes this for fixed-fraction coin-toss betting with a fraction between zero and one; the same multiplicative logic applies to the model here. It does not cover gaps, liquidation, negative balances, or execution events that cause losses beyond the modeled amount. It also does not prevent a practical drawdown or stop-out threshold from being reached. Thorp gives the diffusion approximation for the probability of ever falling to a fraction x of starting capital: x raised to the power 2m/s², where m and s² are the mean and variance of log growth per bet.²

Risking a percentage of current equity shrinks the dollar risk after losses, so the floor is harder to reach than under fixed-dollar risk that starts at the same size. In the fixed-dollar rows, each risk amount is set from starting equity; in the fixed-percentage rows, the same percentage is reapplied to current equity after every trade. For the worked model, with a floor 20% below the starting balance, the Monte Carlo estimates use 1,000,000 seeded paths of 1,000 trades each (xorshift32, seed 20260924):

Risk per trade	Fixed dollars from starting equity, Monte Carlo	Fixed % of current equity, Monte Carlo	Thorp diffusion approximation
1%	0.12%	0.059%	0.106%
2%	3.44%	2.55%	3.55%
5%	25.94%	23.10%	29.19%

The fixed-percentage column is lower in every row, but a 20% decline cuts position size by at most 20%, so the sizing rule matters much less than the risk level. Moving from 1% to 2% per trade raises either column far more than switching between the two rules. At the growth-maximizing Kelly fraction for this model, about 17.5% of equity per trade, the same simulation reached the −20% floor on 71.54% of paths and lost half the account on 45.79%. In Thorp’s continuous model, the probability of ever falling to a fraction x of the start under full Kelly is x itself.² Why a fraction of the Kelly amount is often preferred is covered in prediction-market position sizing.

Why does a trailing drawdown change the answer?

Every formula above assumes a static floor. A trailing floor moves up with each new equity high, so the question changes. Instead of “will the account ever lose B from the start?”, it becomes “will the account ever give back B from any peak?”

Under the iid +2R/−1R worked model with a non-zero loss probability, any fixed trailing drawdown eventually occurs with probability 1: an all-loss run long enough to cross the buffer has positive probability, and such a run eventually occurs in an unlimited iid sequence. The same limiting result holds for a Brownian model with positive drift and non-zero variance. A deterministic positive-return process is a counterexample, so positive expectancy alone is not enough to guarantee it. Separately, Magdon-Ismail and colleagues derive the maximum-drawdown distribution for Brownian motion with drift and show that, under positive drift, its expected maximum drawdown grows logarithmically with the horizon.³ A trailing-floor risk figure therefore needs a stated horizon.

For the worked model, Monte Carlo estimates use 20,000 seeded paths of up to 5,000 trades each (xorshift32, seed 20260923). A displayed 100.00% means all simulated paths reached that drawdown by the listed horizon; it is not an exact probability.

Buffer	Floor type	Within 250 trades	Within 1,000 trades	Within 5,000 trades
10R	Static	3.54%	3.54%	3.54%
10R	Trailing from peak	58.32%	97.12%	100.00%
20R	Static	0.12%	0.12%	0.12%
20R	Trailing from peak	2.12%	9.25%	40.45%

The static rows did not change after the first 250 trades in these simulated paths. The trailing rows keep climbing as the horizon grows. A 10R trailing buffer that sounds comparable to a 10R static one is a different risk. Real programs add details this model leaves out, such as end-of-day recalculation, locks at the starting balance, and separate daily loss limits. Those are usually best handled with a simulation that applies the program’s exact rule.

What is the chance of reaching a target before the floor?

Evaluation accounts ask a two-sided question: will the account reach the profit target before the loss floor? Thorp derives the two-barrier diffusion form for log capital.² Using it here with μ and σ² from R-denominated trade P&L is this article’s application. With θ = 2μ/σ², a buffer B, and a target T, both in R, the Brownian approximation is:

P(floor before target) ≈ (1 − e^(−θT)) / (e^(θB) − e^(−θT))

At zero drift, the continuous approximation reduces to T / (B + T), giving 60% floor-first for a 10R floor and a 15R target. That is not the exact discrete answer for +2R/−1R trades: a +2R win can overshoot the upper boundary, moving from +14R to +16R. Solving the discrete random-walk recurrence with the floor at or below −10R and success at or above +15R gives an exact floor-first probability of about 60.53% when p = 1/3 and q = 2/3. The same recurrence gives the discrete values below; they are direct calculations from this article’s stated model, not values attributed to Thorp.

Win rate (+2R / −1R)	Continuous approximation: floor first	Exact discrete: floor first
45%	4.32%	3.39%
40%	14.86%	13.62%
35%	45.20%	45.33%
33.3% (zero edge)	60.00%	60.53%

At zero edge, the exact discrete target-first probability is about 39.47%, the complement of the 60.53% floor-first result. A method with no edge can still reach the target first. It does so less often than it reaches the floor when the target is further away than the floor. This is a statement about one static-floor setup with no time limit, not a pass rate for any real program.

What the formula leaves out

Every number above assumes things a live account does not guarantee.

Independent trades. Positions that share one driver behave like one larger trade. Three correlated 1R positions can act as a single 3R bet, which cuts the effective buffer. Correlated event exposure covers how to count them.
A stable edge. The formula takes μ as fixed. A method whose edge fades for a few weeks behaves more like a sequence of different strategies than a single one.
Bounded losses. Gaps, slippage past the stop, and news moves produce losses larger than −1R. If they are not in the record, they are not in σ². Add them as a stress case.
Costs. μ must be net of commissions, fees, and slippage. A small gross edge can be close to zero after costs, and the table above shows what that does.
Constant risk per trade. The formula assumes 1R stays 1R through a losing run. Raising size to recover losses shrinks the buffer at the worst possible time. Cutting size by a written rule lengthens it. Adjusting risk during a drawdown covers how to set that rule in advance.
No time limit or daily limit. Evaluation deadlines and daily loss limits add more ways to fail that a single-floor formula does not see.

Most of these push the real risk above the formula’s estimate. Treat the estimate as a way to compare sizing choices under stated assumptions, not as a forecast of the real probability.

Turning the estimate into a written risk decision

Write the threshold and its type. Static or trailing, measured on balance or equity, with any lock or daily limit.
Calculate the buffer in R at your planned risk per trade. Recalculate whenever the account balance or risk per trade changes.
Estimate μ and σ² from realized outcomes. Use one rule version, net of costs, in R.
Calculate the estimate at the observed edge and at a reduced edge. Size from the reduced-edge figure, not the observed one.
Choose the maximum acceptable probability before looking at the result. Then pick the risk per trade whose buffer meets it under the reduced edge.
Record the inputs next to the result. An estimate without its threshold, edge, variance, sizing rule, and horizon cannot be checked later.
Recalculate when an input changes. A new rule version, a different market, a new account rule, or a much larger sample all replace the old estimate.

Where Costante fits

Costante does not calculate risk of ruin, estimate edge, run simulations, calculate position size, connect to brokers, or enforce account rules. The calculation above is done in a spreadsheet or statistical tool.

The formula’s weakest assumption is behavioral: that risk per trade stays where the plan put it, especially during a losing run. Costante supports session planning, self-defined behavioral guardrails, pre-trade and in-session checks, low-friction trade and behavioral logging, and structured review. A record kept against a written plan shows whether size stayed at the planned 1R, which rule version each trade belongs to, and whether a drawdown came with execution drift. Those are the conditions that decide whether a risk-of-ruin estimate describes your trading. The model, the risk decision, and every order remain the trader’s.

Frequently asked questions

What is an acceptable risk of ruin?

There is no universal figure. It depends on what the threshold costs you. Losing an evaluation fee is different from losing trading capital you cannot replace. Choose the acceptable probability before calculating, and apply it to the reduced-edge estimate rather than the observed one. The tables above show why. The same risk per trade can sit anywhere from well under 1% to over one in three, depending only on the win rate assumed.

Can risk of ruin ever be zero?

Not for a fixed floor with no time limit under the iid models described here. Even with a strong edge, a long enough losing run can reach any fixed floor. Fixed-fraction sizing prevents reaching exactly zero at any finite step only within the idealized bounded-loss model; it does not prevent reaching a threshold such as a 20% loss. With no edge, reaching the floor is certain given enough trades.

Is risk of ruin the same as maximum drawdown?

No. Maximum drawdown is the largest decline that happened, or that is simulated, over a period. Risk of ruin is the probability that a decline reaches one specific threshold. They are linked: a trailing-floor risk of ruin is the probability that maximum drawdown within the horizon reaches the buffer. The Monte Carlo article reports full drawdown distributions. This article isolates the threshold probability.

Why do risk-of-ruin calculators give different numbers?

They make different choices about the threshold, the sizing rule, the horizon, and the formula. Some use the even-money gambler’s ruin formula, some use an approximation, and some simulate. A calculator that asks only for win rate and reward-to-risk ratio assumes every win and every loss is the same size, which removes variance the real record contains. Before comparing two figures, check that the threshold, buffer, sizing rule, and horizon match.

Sources

Costante provides educational workflow tools, not financial advice. Trading involves risk.

Ethier, S. N. (2010). Gambler’s Ruin. In The Doctrine of Chances: Probabilistic Aspects of Gambling, Chapter 7, pp. 241–274. Springer, Probability and its Applications. Treats the probability of losing L units before winning W units in independent, identical wagers: an explicit formula for even-money payoffs, special-case formulas for integer payoffs, and bounds for arbitrary payoffs. ↩
Thorp, E. O. (2008). The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market. In S. A. Zenios & W. T. Ziemba (Eds.), Handbook of Asset and Liability Management, Volume 1, Chapter 9, pp. 385–428. North-Holland. Section 2 notes that fixed-fraction betting cannot reach zero for a fraction between zero and one. Section 3.2 gives the continuous approximation for the probability of ever falling to a fraction x of starting capital, with m and s² the mean and variance of log growth per bet. Sections 7.3–7.4 give the result that under full Kelly this probability is x, and the two-barrier diffusion form derived from the classical gambler’s ruin formula. Applying the formula to P&L in R units is this article’s use, not Thorp’s. ↩ ↩² ↩³ ↩⁴
Magdon-Ismail, M., Atiya, A. F., Pratap, A., & Abu-Mostafa, Y. S. (2004). On the Maximum Drawdown of a Brownian Motion. Journal of Applied Probability, 41(1), 147–161. Derives the distribution and expected value of maximum drawdown for Brownian motion with drift. As the horizon grows, expected maximum drawdown grows logarithmically for positive drift. The result concerns an idealized process, not a trading record. ↩