Poker Bankroll Ruin Probability Calculation and Risk Management Model

Tool Purpose

The Poker Bankroll Ruin Probability Calculator is a core tool for bankroll management, used to assess the likelihood of a player going broke given their profitability and variance. Through calculation, players can determine the minimum bankroll requirement, adjust game stakes, or optimize betting strategies to ensure long-term survival.

Formula Principle

The commonly used ruin probability model is based on the normal distribution assumption, with the formula:

$$ \text{RoR} = e^{-2 \cdot B \cdot \text{EV} / \text{SD}^2} $$

Where:

• RoR (Risk of Ruin): Probability of ruin (0 to 1)
• B: Current bankroll (in big blinds bb)
• EV: Expected profit per 100 hands (in bb/100)
• SD: Standard deviation per 100 hands (in bb/100)

This formula assumes that profit follows a random walk and that EV and SD are constant. For more precise calculations, Poisson distribution or Monte Carlo simulation can be used, but this formula provides a reasonable estimate.

How to Use

1. Estimate EV: Use historical data or HUD statistics to calculate net profit per 100 hands (bb). For example, if profit over 6 months is 5000bb and total hands are 200,000, then EV = 5000 ÷ 2000 = 2.5 bb/100.
2. Calculate SD: Standard deviation measures variance per 100 hands. Use Excel's STDEV function on per-100-hand profit samples. For cash game players, SD typically ranges between 60-100 bb/100.
3. Set Target RoR: Generally recommend RoR ≤ 5% (aggressive) or ≤ 1% (conservative).
4. Solve for B: Given EV, SD, and target RoR, the required bankroll can be derived:

$$ B = -\frac{\text{SD}^2}{2 \cdot \text{EV}} \cdot \ln(\text{RoR}) $$

Practical Example

Problem: A cash game player has EV = 2.5 bb/100, SD = 80 bb/100, and wants a ruin probability below 1%. What bankroll is needed?

Calculation:

• Target RoR = 0.01
• ln(0.01) ≈ -4.605
• B = - (80^2) / (2 * 2.5) * (-4.605) = -6400 / 5 * (-4.605) = -1280 * (-4.605) = 5894.4 bb
• Assuming buy-in is 100bb, this equates to about 59 buy-ins, i.e., 5900 bb.

Verification: Plugging B=5900 back into the original formula, RoR = e^(-2 * 5900 * 2.5 / 6400) ≈ e^(-4.609) ≈ 0.01, matching the expectation.

Frequently Asked Questions

• Q: How to estimate EV and SD? A: EV can be obtained directly from tracking software (e.g., Hold'em Manager). Without data, approximate values: small stakes players EV ≈ 5-10 bb/100, high stakes ≈ 1-3 bb/100. For SD, use sample standard deviation or assume a conservative 80 bb/100.
• Q: What if EV is negative? A: The formula does not apply because ruin probability is 100% (unless you get lucky). First, improve your skill to make EV positive.
• Q: Are the formula assumptions reasonable? A: The assumption of independent and identically distributed profits may not hold in reality (e.g., downswings), but the formula still provides a useful reference. Tournament players should consider nonlinear prize structures and use ICM and tournament bankroll models instead.

Further Learning

• Book: The Mathematics of Poker (Bill Chen)
• Article: "Risk of Ruin for Poker Players" (searchable)
• Tool: Online ruin calculators (e.g., variations on pokerdope.com)

Note: The diagram shows the relationship between bankroll and ruin probability for different EV values.