Poker Bankruptcy Probability Calculation and Risk Management Model: A Mathematical Tool to Protect Your Bankroll

Tool Purpose

The Risk of Ruin (RoR) calculation model is the most essential mathematical tool in poker bankroll management. Based on your win rate, standard deviation, and current bankroll, it quantifies the probability of eventually going broke over an infinite number of games. Using this model correctly allows you to:

• Determine safe buy‑in levels to reduce bankruptcy risk
• Assess whether your current bankroll is sufficient for the current stake level
• Adjust your bankroll strategy according to your personal risk tolerance

Formula Principles

The most commonly used risk‑of‑ruin formula in poker (assuming infinite bets and no rake) is:

[ \text{RoR} = e^{\frac{-2 \times B \times \text{Win Rate}}{\text{Standard Deviation}^2}} ]

Where:

• (B) = current bankroll (in big blinds / buy‑ins)
• (\text{Win Rate}) = average profit per unit time (e.g., BB/100 hands)
• (\text{Standard Deviation}) = standard deviation per unit time (e.g., BB/100 hands)
• (e) = natural constant, approx. 2.71828

Derivation logic: The formula is based on a random walk and Brownian motion model, assuming that game results are independent and identically distributed. Practical usage points to note:

• The win rate must be positive; otherwise the risk of ruin is 100%
• The larger the standard deviation, the higher the risk of ruin with the same bankroll
• The larger the bankroll B, the risk of ruin decreases exponentially

Steps for Use

1. Collect personal data: Record results from at least the last 10,000 hands (or more) and calculate the win rate and standard deviation per 100 hands.
   • Example: A player has played 50,000 hands at NL100 (blinds $0.5/$1), with an average win rate of 8 BB/100 hands and a standard deviation of 80 BB/100 hands.
2. Determine the target risk of ruin: Professional players usually aim for RoR ≤ 1%–5%; recreational players may accept a higher value (e.g., 10%). Conservative players choose 0.5%.
3. Plug into the formula to find the required bankroll: Rearranged formula: [ B = -\frac{\text{Standard Deviation}^2 \times \ln(\text{RoR})}{2 \times \text{Win Rate}} ]
   • Using the example: if RoR = 1% (0.01), (\ln(0.01) = -4.605), then [ B = -\frac{80^2 \times (-4.605)}{2 \times 8} = \frac{6400 \times 4.605}{16} = \frac{29472}{16} = 1842 \text{ BB} ]
   • That means at least 1,842 big blinds, or about 18.42 buy‑ins (each buy‑in is 100 BB).
4. Adjust buy‑in level: If your current bankroll is $1,000 (i.e., 1,000 BB), plugging into the formula gives RoR = ? [ \text{RoR}= e^{\frac{-2 \times 1000 \times 0.08}{80^2}} = e^{-0.025} \approx 0.975 ] The risk of ruin is as high as 97.5%, which is clearly unacceptable. In this case, you should move down in stakes.

Practical Example

Scenario: A player at NL200 (blinds $1/$2) has a win rate of 10 BB/100 hands and a standard deviation of 100 BB/100 hands. He wants the risk of ruin to be ≤ 2%. What is the minimum bankroll?

Solution:

• RoR = 0.02 → ln(0.02) ≈ −3.912
• B = −(100² × −3.912) / (2 × 10) = (10,000 × 3.912) / 20 = 39,120 / 20 = 1,956 BB
• This means at least 1,956 big blinds, or about 19.56 buy‑ins (each buy‑in is 200 BB = $400), so the minimum bankroll is approximately $3,912.

If the current bankroll is only $2,000 (1,000 BB), then RoR = e^{(−2 × 1,000 × 0.10) / (100²)} = e^{−0.02} ≈ 0.9802, i.e., a 98% risk of ruin – extremely dangerous.

Frequently Asked Questions

Q1: Must the win rate be in BB/100? A: Yes, and the standard deviation must be in the same unit. If you use a win rate like hourly profit, you must use a standard deviation with the same time unit. The formula is standardized as long as the “unit time” is consistent.

Q2: How do I account for rake in real games? A: Rake reduces your actual win rate. Use your net win rate (after rake) in the formula. If you are unsure, use a conservative estimate after moving down in stakes.

Q3: Does the formula still apply when multi‑tabling? A: Yes, but the standard deviation increases when multi‑tabling. It is recommended to calculate data per table separately, or combine them to compute the total standard deviation. Multi‑tabling usually increases the average win rate, but the standard deviation grows faster, so the required bankroll may be higher.

Q4: What is an acceptable risk of ruin? A: Professional players typically keep it between 1% and 5%; recreational players can loosen it to 5%–10%. However, if your bankroll is tight, it is safer to stay below 2%.

Further Study

• Book recommendations: Poker Bankroll Management by Matthew Hilger, and the bankroll management chapter in Ace on the River.
• Advanced concept: The Kelly Criterion is used to determine bet sizing and is closely related to risk of ruin.
• Online risk‑of‑ruin calculators (e.g., DeucesCracked’s RoR calculator) are available, but it is important to understand the underlying principles.
• Note: The formula assumes an infinite game; in practice you can move down in stakes or add funds to your bankroll, but the calculation provides a baseline for the worst‑case scenario.