Risk of Ruin Formula

The Formula

For cash games with a constant win rate and standard deviation, the closed-form approximation:

RoR = e^(-2 x WR x BR / SD^2)

Where:

• WR = win rate (bb/100)
• BR = bankroll (in bb)
• SD = standard deviation (bb/100)
• e = Euler's number (2.71828...)

Example Calculations

Scenario 1: Micro stakes grinder

• Win rate: 7 bb/100
• Standard deviation: 75 bb/100
• Bankroll: 2,000 bb (20 buy-ins at NL50)

RoR = e^(-2 x 7 x 2000 / 5625) = e^(-4.98) = 0.7%

With a strong 7 bb/100 win rate and 20 buy-ins, risk of ruin is under 1%.

Scenario 2: Mid-stakes regular

• Win rate: 3 bb/100
• Standard deviation: 85 bb/100
• Bankroll: 2,500 bb (25 buy-ins at NL100)

RoR = e^(-2 x 3 x 2500 / 7225) = e^(-2.08) = 12.5%

With a modest 3 bb/100 win rate and 25 buy-ins, there is a 12.5% chance of going broke. This is within many players' risk tolerance but higher than conservative bankroll management would recommend.

Scenario 3: Marginal winner

• Win rate: 1.5 bb/100
• Standard deviation: 90 bb/100
• Bankroll: 2,000 bb (20 buy-ins at NL200)

RoR = e^(-2 x 1.5 x 2000 / 8100) = e^(-0.74) = 47.7%

Nearly a coin flip. A 1.5 bb/100 winner with 20 buy-ins has an unacceptable risk of ruin. This player needs either more bankroll, a higher win rate, or lower stakes.

Bankroll Requirements by Win Rate

These numbers assume a fixed stake. Players who move down when losing (a standard bankroll management practice) have substantially lower risk of ruin than the formula suggests.

Limitations of the Formula

The closed-form formula assumes:

1. Constant win rate. Your actual win rate varies as your skill improves, the player pool changes, and you tilt.
2. Normally distributed results. Poker results are approximately normal over large samples but have slightly fatter tails.
3. No moving down in stakes. The formula assumes you play the same stake until ruin or success. Moving down when your bankroll drops reduces actual risk of ruin significantly.
4. No withdrawal/deposit. The formula assumes a closed system. Adding to your bankroll from external income changes the math.

For more accurate modeling that accounts for these factors, use the cash bankroll simulator, which runs Monte Carlo simulations with customizable parameters.

Tournament Risk of Ruin

Tournament risk of ruin is harder to compute analytically because tournament results do not follow a normal distribution. They are heavily right-skewed (many small losses, occasional large wins).

The practical approach:

• 100 buy-ins is the minimum for tournament players with positive ROI
• 200 buy-ins is conservative and recommended for players with <15% ROI
• 300+ buy-ins is appropriate for players mixing buy-in levels or playing high-variance formats (turbo, hyper, PKO)

Use the MTT bankroll simulator to model tournament bankroll survival with your specific ROI and field sizes.

Practical Bankroll Management

The formula gives you the math. Here are the behavioral guidelines that complement it:

1. Know your win rate. Track at least 30,000 hands before trusting your win rate estimate. Use the variance calculator to see the confidence interval around your sample.

2. Move down before you have to. Losing 5 buy-ins should trigger a stake review. Losing 10 should trigger a move down. Do not wait until risk of ruin becomes real.

3. Separate your poker bankroll from your life bankroll. Money you need for rent, food, or other essentials is not part of your poker bankroll.

4. Use Kelly criterion for stake selection. The cash Kelly calculator computes the optimal fraction of your bankroll to risk at each stake, maximizing long-term growth while controlling ruin risk.

5. Recalculate when your game changes. A move in stakes, a new player pool, or a format change shifts the inputs, and your old risk of ruin calculation goes stale.