Poker Bankroll Risk Management Models

Tool Purpose

Risk of Ruin (RoR) measures the probability that a poker player will eventually lose all their money given a specific bankroll size, skill level, and variance. This model helps players:

• Determine the minimum safe bankroll
• Assess the variance risk of current games
• Develop moving up/down and withdrawal strategies
• Avoid going broke due to short-term negative variance

Formula Principles

The most commonly used Risk of Ruin formula is based on the normal distribution assumption and applies to No-Limit Hold’em cash games. The formula is:

[ R = e^{-2 \cdot B \cdot \frac{WR}{\sigma^2}} ]

Where:

• R = Risk of Ruin (a value between 0 and 1, convertible to percentage)
• B = Current bankroll (in big blinds or buy-ins)
• WR = Expected Win Rate (in big blinds per hand or BB/100)
• σ = Standard Deviation (in big blinds per hand or BB/100)

Derivation Principle: The formula originates from random walk and Brownian motion models. It assumes that a player’s profit follows a normal distribution with independent increments. The formula gives the probability of the bankroll reaching zero over an infinite time horizon.

Another Common Form (for tournaments): Uses number of buy-ins and average ROI, but the above formula is more common for cash games.

Usage Steps

1. Estimate Win Rate (WR): Use historical hand statistics, e.g., over the last 100,000 hands, profit of 0.5 BB per 100 hands (i.e., 5 BB/100). Beginners can refer to the average win rate of players at the same stake.

2. Estimate Standard Deviation (σ): Standard deviation reflects variance magnitude. Typical values for cash games are around 80–100 BB/100 (6-max) or 100–120 BB/100 (full-ring). You can obtain it directly from poker tracking software (e.g., Hold’em Manager, PT4).

3. Determine Current Bankroll (B): Record in the same unit, e.g., B = 2000 BB (2000 big blinds).

4. Plug into Formula: Use a calculator or Excel (enter formula =EXP(-2BWR/σ^2)).

5. Interpret Results:

   • RoR < 0.5%: Very safe, can consider moving up
   • RoR 0.5%–2%: Normal safe level
   • RoR 2%–5%: Proceed with caution, consider adding bankroll or moving down
   • RoR > 5%: High risk, adjust immediately

Practical Example

Case: A player at NL100 ($0.5/$1) with a bankroll of $2000 (i.e., 2000 BB). From 100,000 hands:

• Win Rate: 5 BB/100 (5 big blinds per 100 hands)
• Standard Deviation: 90 BB/100

Calculation: [ R = e^{-2 \times 2000 \times \frac{5}{90^2}} = e^{-2 \times 2000 \times \frac{5}{8100}} = e^{-2 \times 2000 \times 0.00061728} = e^{-2.46912} \approx 0.0847 ]

RoR ≈ 8.47%, which is high risk. To reduce RoR to below 1%, what bankroll is needed?

Find Required Bankroll: Set R = 0.01, solve for B: [ 0.01 = e^{-2 \cdot B \cdot \frac{5}{8100}} ] Take natural log: [ \ln(0.01) = -2 \cdot B \cdot \frac{5}{8100} ] [ -4.60517 = -2 \cdot B \cdot 0.00061728 ] [ B = \frac{4.60517}{2 \times 0.00061728} \approx 3730 \text{ BB} ] Thus, approximately $3730 bankroll, or 37.3 buy-ins (assuming 100 BB buy-in) is needed to keep RoR below 1%.

FAQ

Q: Does the formula require an exact win rate? A: Win rate is the most sensitive parameter in the model. Inaccurate estimates can lead to large RoR deviations. It is recommended to use at least 50,000 hands of data and allow a margin of safety.

Q: How do I obtain standard deviation? A: Poker tracking software can display it directly. Without data, refer to typical values for your game type: cash 6-max ~80–100, full-ring ~100–120.

Q: Is this model suitable for tournaments? A: Not entirely, because tournament profit distributions are more skewed. For tournaments, similar concepts apply but with more complex models, such as the Kelly Criterion or simulations.

Q: Is bankroll safety the same as RoR < 1%? A: Different players have different risk preferences. Professional players typically require RoR < 0.5%, while recreational players may accept 5%. However, it is recommended to maintain at least 20 buy-ins (for standard cash games) or more.

Further Learning

• Kelly Criterion: Used to determine optimal bet sizing to maximize long-term growth.
• Monte Carlo Simulation: Simulate thousands of bankroll paths to obtain more realistic Risk of Ruin probabilities, especially for tournaments.
• Multi-Table Bankroll Management: Playing multiple tables increases variance, requiring adjusted bankroll requirements. See the "Multi-Table Risk of Ruin" extension model.