Poker Bankroll and Ruin Probability Calculator: How to Scientifically Manage Risk

Tool Purpose

The risk of ruin calculator is used to estimate the probability that a poker player will go broke (lose all their bankroll) given their bankroll size, win rate, and variance. It is a fundamental tool for risk management, helping players determine a reasonable bankroll size to avoid being forced out of the game due to short-term fluctuations.

Formula Principle

The profit process in poker can be approximated as a random walk. Assuming that per-hundred-hand returns are independent and identically distributed following a normal distribution, the relationship between the risk of ruin (risk tolerance) and bankroll is given by the formula:

$$B = -\frac{\sigma^2}{2 \mu} \ln(\alpha)$$

Where:

• $B$: Required bankroll (units consistent with $\sigma$ and $\mu$)
• $\mu$: Expected win rate (average profit per 100 hands)
• $\sigma$: Standard deviation (volatility of profit per 100 hands)
• $\alpha$: Acceptable risk of ruin (risk tolerance, e.g., 0.05 for 5%)

This formula originates from continuous-time financial models, assuming normally distributed returns and constant volatility. In practical poker applications, $\mu$ and $\sigma$ are typically measured in bb/100 hands (big blinds per 100 hands).

How to Use

1. Determine Parameters: Calculate or estimate your win rate $\mu$ and standard deviation $\sigma$. The win rate must be positive (if negative, you will eventually go broke regardless of bankroll). Standard deviation can be computed from historical data (typically around 80–120 bb/100).
2. Set Risk Tolerance: Choose the acceptable risk of ruin $\alpha$, e.g., 5% (0.05) or 1% (0.01).
3. Plug into Formula: Substitute $\mu$, $\sigma$, and $\alpha$ into the formula to calculate the required bankroll $B$.
4. Adjust and Verify: If your current bankroll is less than $B$, consider moving down in stakes or increasing your bankroll; if it is significantly larger than $B$, consider moving up.

Practical Example

Suppose a player at NL100 (where the big blind is $1) has a win rate $\mu = 10,bb/100$ and a standard deviation $\sigma = 100,bb/100$, and wants a risk of ruin below 5%.

Calculation: $$ B = -\frac{100^2}{2 \times 10} \ln(0.05) = -\frac{10000}{20} \times (-2.9957) \approx 500 \times 2.9957 \approx 1497.85 ,bb $$

This is approximately 1498 big blinds, which converts to $1498 (since 1 bb = $1).

If the player wants a risk of ruin of 1% ($\alpha = 0.01$): $$ B = -500 \times \ln(0.01) = -500 \times (-4.6052) \approx 2302.6 ,bb $$

As you can see, the lower the risk tolerance, the larger the required bankroll.

Frequently Asked Questions

• What if the win rate is negative?

  • The formula requires $\mu > 0$. If your win rate is negative, you will eventually go broke in the long run regardless of bankroll size. It is recommended to improve your skills through study or move down in stakes first.

• Is the formula applicable to all game types?

  • It is suitable for scenarios where profit volatility approximates a normal distribution, such as cash games. Tournaments involve high variance and multi-stage structures, so the formula needs adjustments (refer to ICM models).

• How much more bankroll should I have than the calculated value?

  • The calculated value is the minimum requirement. It is advisable to keep an additional 20%–50% buffer to account for statistical errors or moving down costs.

• How do I estimate my own $\mu$ and $\sigma$?

  • Use tracking software (such as Hold'em Manager or PokerTracker) to view your "win rate" and "standard deviation." If the sample size is insufficient, refer to typical values for your stakes (e.g., for regular stakes below $2/$4, win rates of 5–15 bb/100 and standard deviations of 80–120 bb/100 are common).

Further Learning

• Kelly Criterion: A bankroll management formula that maximizes long-term growth rate. It is suitable for compounding scenarios, but it is recommended to use "fractional Kelly" to reduce risk.
• Value at Risk (VaR): Estimates the maximum loss at a given confidence level and can be used alongside risk of ruin.
• Simulation Methods: Use Monte Carlo simulations for more accurate assessment of multi-table or tournament risk of ruin.

Tip: The risk of ruin formula provides theoretical guidance only. In practice, factors such as rake, moving down in stakes, and multi-tabling must also be considered. Always maintain a safety margin to avoid taking excessive risks.