The Ruin Probability Formula
The Ruin Probability Formula
The ruin probability formula is the quantitative heart of bankroll management. It converts three inputs—starting capital, profit target, and trading edge—into a single probability between 0 and 1 representing the likelihood of account ruin before reaching the profit goal. Unlike vague guidelines ("never risk more than 2% per trade"), the formula gives an exact, auditable answer.
There are multiple versions of the ruin formula, depending on whether position sizing is fixed, proportional to bankroll (Kelly-type), or dynamic. This article presents the two most practical versions: the classical fixed-bet formula (simplest) and the Kelly-based proportional-bet formula (more realistic for professional traders).
Key takeaways
- The ruin probability formula depends on four inputs: starting bankroll B, target capital T, win probability p, and payoff ratio r.
- Fixed-bet and proportional-bet versions give different results: A trader risking a fixed $1,000 per trade faces different ruin odds than one risking a fixed 2% of bankroll.
- The formula is exact for independent identically distributed bets: No simulation required; a calculator or spreadsheet suffices.
- Small edges compound slowly but ruin probability is real: A 52% win rate (0.04 expected value per trade) with 10% bet size faces ~30% ruin probability.
- The formula breaks down if assumptions are violated: Correlated trades, non-constant payoff ratios, or position sizing changes require simulation or adjustment.
The Inputs and Their Definitions
Before presenting the formula, define each input precisely:
B: Starting Bankroll The total capital at the beginning. Example: $30,000. This is the amount the trader begins with, not the amount at risk per trade.
T: Target Capital The desired end capital (profit goal). Example: $40,000. This is a level the trader wants to reach. The ruin probability is the chance of hitting $0 (or a ruin threshold) before reaching T.
p: Win Probability The fraction of trades expected to be winners. Example: 0.55 (55% win rate). This assumes the trader has consistent, documented statistics.
r: Payoff Ratio The ratio of average winning trade size to average losing trade size. Example: 1.2 (average win is $120, average loss is $100). This is crucial—it accounts for the trader's risk management behavior (whether losses are capped smaller than wins).
Ruin Threshold R In many practical cases, this is zero. In regulated trading, it might be the minimum account value required by law (e.g., $25,000 for US pattern day traders). The formula can be adjusted for any threshold by replacing "0" with the threshold value.
The Fixed-Bet Formula
The fixed-bet ruin formula applies when the trader bets a constant dollar amount per trade (e.g., always risking $1,000). This is the simplest model and the one originally solved by Laplace.
For non-fair games (p ≠ 0.5):
Define: q = (1 - p) / (p × r)
If q = 1 (fair game in the payoff sense):
Ruin probability = (T - B) / T
If q ≠ 1:
Ruin probability = (q^B - q^T) / (1 - q^T)
where ^ denotes exponentiation
Alternatively, if B is much smaller than T (which is often true), the formula simplifies:
Ruin probability ≈ q^B when q < 1 (positive edge)
Ruin probability ≈ 1 when q > 1 (negative edge)
Interpreting q
The parameter q is the odds ratio. If q < 1, the game has positive expected value and ruin is possible but not certain. If q > 1, the game has negative expected value and ruin is highly probable. If q = 1, the game is "fair" in the payoff sense.
For a fair game (p = 0.5, r = 1.0):
q = (1 - 0.5) / (0.5 × 1.0) = 0.5 / 0.5 = 1
Ruin probability = (T - B) / T
So a trader starting with $10,000 targeting $20,000 in a fair game faces a ruin probability of ($20,000 - $10,000) / $20,000 = 0.5 (50%).
For a positive-edge game (p = 0.55, r = 1.0):
q = (1 - 0.55) / (0.55 × 1.0) = 0.45 / 0.55 ≈ 0.818
Ruin probability ≈ 0.818^B
With B = $10,000, this is 0.818^10000, which is so close to zero as to be negligible (ruin is extremely unlikely). But this assumes the "unit" of B is a single trade. If B = 10 (meaning 10 trades' worth of capital), then ruin probability = 0.818^10 ≈ 0.14 (14%).
This highlights the critical point: the ruin formula is sensitive to the scale at which B and T are measured. They must be scaled to the bet size.
The Proportional-Bet Formula (Kelly-Like)
In practice, traders do not bet a fixed dollar amount forever. As their bankroll grows, they increase bet size. As it shrinks, they reduce it. A common approach is to risk a constant fraction of bankroll per trade.
For proportional betting, the formula is more complex. Let f be the fraction of bankroll risked per trade. The ruin probability depends on the compound growth factor per trade:
Compound growth per trade G = (1 + r × f)^p × (1 - f)^(1 - p)
The ruin probability is then:
Ruin probability ≈ ((1 - G) / G)^(B / f_unit)
where f_unit is one unit of capital (e.g., $1 or 0.01 of bankroll).
This is more complex to calculate but more realistic. It accounts for the fact that as bankroll grows, bet size grows, amplifying both wins and losses.
Worked Example 1: Fixed-Bet Formula
Setup:
- Starting bankroll B = $10,000 (this is capital in terms of bet units; 10 units of $1,000 per trade)
- Target capital T = $15,000 (15 units)
- Win probability p = 0.55
- Payoff ratio r = 1.0 (1:1 risk/reward)
- Fixed bet size = $1,000 per trade
Calculation:
q = (1 - 0.55) / (0.55 × 1.0) = 0.45 / 0.55 ≈ 0.8182
Ruin probability = (q^B - q^T) / (1 - q^T)
= (0.8182^10 - 0.8182^15) / (1 - 0.8182^15)
= (0.1387 - 0.0345) / (1 - 0.0345)
= 0.1042 / 0.9655
≈ 0.108 or 10.8%
Interpretation: Starting with $10,000 and targeting $15,000 with a 55% win rate, the trader faces approximately a 10.8% probability of the account dropping to zero before reaching $15,000. This is non-trivial. Many traders would find a 10.8% ruin risk unacceptable and would either increase the starting bankroll or reduce bet size.
Worked Example 2: Comparing Edges
Setup: Same trader, but win probability varies.
Win probability = 0.52 (2% edge instead of 5%):
q = (1 - 0.52) / (0.52 × 1.0) = 0.48 / 0.52 ≈ 0.9231
q^10 ≈ 0.3717, q^15 ≈ 0.2120
Ruin probability = (0.3717 - 0.2120) / (1 - 0.2120) ≈ 0.202 or 20.2%
Win probability = 0.50 (fair game):
q = 1
Ruin probability = (T - B) / T = (15 - 10) / 15 ≈ 0.333 or 33.3%
Win probability = 0.48 (negative edge, 2% loss expectation):
q = (1 - 0.48) / (0.48 × 1.0) = 0.52 / 0.48 ≈ 1.0833
q > 1, so Ruin probability ≈ 1 (nearly certain)
Key insight: A 3% edge reduction (55% to 52%) increases ruin probability from 10.8% to 20.2%—roughly doubling it. This shows how sensitive ruin probability is to small changes in edge. A trader's confidence in a 52% win rate must be very high, because even a small margin of estimation error materially changes the ruin odds.
Worked Example 3: Impact of Payoff Ratio
Setup: Win probability fixed at p = 0.52, but payoff ratio r varies.
Payoff ratio r = 1.0 (1:1):
q = 0.48 / 0.52 ≈ 0.9231
Ruin probability ≈ 20.2% (as calculated above)
Payoff ratio r = 1.5 (win $150 for every $100 lost):
q = 0.48 / (0.52 × 1.5) = 0.48 / 0.78 ≈ 0.6154
q^10 ≈ 0.00677, q^15 ≈ 0.000142
Ruin probability ≈ 0.00677 / (1 - 0.000142) ≈ 0.0068 or 0.68%
Payoff ratio r = 0.5 (win $50 for every $100 lost):
q = 0.48 / (0.52 × 0.5) = 0.48 / 0.26 ≈ 1.846
q > 1, so Ruin probability ≈ 1 (nearly certain)
Key insight: Improving the payoff ratio from 1:1 to 1.5:1 drops ruin probability from 20% to less than 1%. Conversely, a 1:0.5 payoff ratio (losing more than winning) leads to near-certain ruin despite a 52% win rate. The payoff ratio is equally important as the win probability for bankroll survival.
Decision Tree for Ruin Probability
Practical Spreadsheet Implementation
Most traders use a spreadsheet to calculate ruin probability. The steps are:
-
Input cells:
- B = starting bankroll (in bet-size units)
- T = target capital (in bet-size units)
- p = win probability (0.5 to 0.6 typical)
- r = payoff ratio (0.5 to 2.0 typical)
-
Intermediate calculation:
- q = (1 - p) / (p * r)
-
Ruin probability (if q ≠ 1):
- Ruin = (q^B - q^T) / (1 - q^T)
-
Sensitivity analysis: Vary p and r to see how ruin probability changes.
Most professional trading platforms (MetaTrader, NinjaTrader, etc.) include ruin probability calculators, but understanding the formula is essential for correct interpretation.
Assumptions and Limitations
The ruin formula assumes:
- Independent trades: The outcome of trade N does not affect trade N+1's probability distribution.
- Constant payoff ratio: All wins are the same size relative to all losses.
- Constant win probability: The p value is stable over time (no skill degradation, no regime change).
- Fixed or proportional bet sizing: No arbitrary changes in position size mid-sequence.
Real trading violates some of these (e.g., correlated trades during flash crashes, dynamic position sizing). The formula is an approximation. If violations are severe (e.g., trades are highly correlated), Monte Carlo simulation is needed instead.
Connecting to Position Sizing
The ruin formula motivates position-sizing rules. If a trader wants to keep ruin probability below 5%, and the trader has a documented 54% win rate with a 1:1 payoff, the formula determines the maximum position size relative to bankroll. For example:
- If target capital T = 2 × B (double the starting bankroll), then the formula solves for the maximum bet size that keeps ruin probability < 5%.
- Professional traders often use this relationship: given their edge (p, r) and their acceptable ruin probability (e.g., 2%), they solve backward for the position size.
This is the bridge between the abstract Gambler's Ruin problem and practical trading rules.
Summary
The ruin probability formula is an exact, non-simulation tool for calculating the likelihood of account bankruptcy before reaching a profit goal. It depends sensitively on the win probability, payoff ratio, and starting capital relative to target. A 52% win rate with a 1:1 payoff faces ~20% ruin risk over a 50% bankroll gain, while a 55% win rate faces ~11% risk—demonstrating the importance of accurate edge estimation. Professional traders use this formula to size positions and choose bankroll levels that keep ruin risk acceptably low (typically < 5%).