Poker Bankruptcy Probability Calculation and Risk Management Model

Tool Purpose

The Risk of Ruin (RoR) is a core metric in poker bankroll management, used to assess the probability that a player, given a certain win rate, variance, and bankroll size, will eventually lose their entire bankroll over the long run. By calculating the risk of ruin, you can determine whether your current bankroll is sufficiently safe or whether you need to move down in stakes to control risk.

Formula Principle

The classic approximation formula for risk of ruin comes from the first-passage time model of Brownian motion and applies to independent and identically distributed poker sessions (e.g., fixed expectation and variance per hand):

$$\text{RoR} = e^{-2 \times \text{WR} \times \text{BR} / \text{SD}^2}$$

Where:

• WR: Expected win rate, in big blinds per 100 hands (bb/100)
• BR: Bankroll size, in big blinds (bb)
• SD: Standard deviation, in big blinds per 100 hands (bb/100)

Principle: The formula assumes that future bankroll changes follow a Brownian motion with drift. The risk of ruin decays exponentially as bankroll increases. When WR is positive, the bankroll will grow in the long run, but short-term variance can still lead to ruin. The formula gives a theoretical upper bound on the risk of ruin (actual risk is usually slightly lower due to the discrete nature of bankrolls).

How to Use

1. Estimate parameters: Use at least 100,000 hands of historical data to calculate your actual win rate (WR) and standard deviation (SD). If you have no historical data, you can reference typical values for players at the same stakes (e.g., a regular NL50 player might have a WR of about 5 bb/100 and an SD of about 80 bb/100).
2. Determine your current bankroll: Know the total bankroll (BR) you have allocated for the current stakes, in bb.
3. Plug into the formula: Use a scientific calculator or programming tool (e.g., Python) to compute RoR = exp(-2 * WR * BR / (SD^2)).
4. Evaluate risk threshold: Generally, a RoR below 5% is considered safe bankroll management; if it exceeds 5%, you should consider increasing your bankroll or moving down.

Practical Example

Scenario: You are an online cash game player. Over the last 200,000 hands, your win rate is 6 bb/100, your standard deviation is 90 bb/100, and your current bankroll is 1,500 bb. Calculate the risk of ruin.

Calculation:

• WR = 6
• SD = 90 → SD^2 = 8,100
• BR = 1,500
• Exponent = -2 × 6 × 1,500 / 8,100 = -18,000 / 8,100 ≈ -2.2222
• RoR = e^(-2.2222) ≈ 0.1084 (about 10.8%)

Interpretation: The risk of ruin is about 10.8%, which is above the 5% safety threshold. It is recommended to increase the bankroll to at least 2,500 bb. At that level, RoR = e^(-2×6×2,500/8,100) = e^(-3.7037) ≈ 0.0245 (2.45%), which is within the safe range.

Frequently Asked Questions

How do I estimate standard deviation in the formula?

Standard deviation reflects the volatility of your profit. It is usually directly available from poker tracking software (e.g., Hold'em Manager, PokerTracker). If you cannot obtain it, you can use general ranges for the same stakes (e.g., for 6-max cash games, SD is typically 80-100 bb/100).

Is a risk of ruin of zero absolutely safe?

No. The formula assumes a constant positive win rate, but in reality, win rates can change (due to skill improvement or downswings), and the discrete nature of bankrolls can make actual risk slightly higher than the theoretical value. It is advisable to maintain a safety margin, such as targeting a RoR of no more than 1%.

Can I play multiple stakes simultaneously?

Yes, but you should combine your bankroll. Compute a weighted average of WR and SD by hand volume for each stake, then plug those values into the formula. For example, if 50% of your hands are at NL50 (WR=5, SD=80) and 50% at NL100 (WR=4, SD=90), the weighted WR is 4.5, the weighted SD is approximately 85.3, and BR is your total bankroll.

Further Learning

• Kelly Criterion: Determines the optimal bet size to maximize bankroll growth while avoiding ruin.
• Monte Carlo Simulation: Simulates a large number of bankroll sequences via computer to estimate risk of ruin more accurately, especially useful for non-normal distributions.
• Risk Curve: A chart of risk of ruin at different bankroll levels, allowing you to visually see how risk changes with bankroll.

Mastering the risk of ruin model is the first step toward scientific bankroll management. It moves you from "feeling safe" to "data safe," enabling steady profits over a long poker career.