Poker Risk of Ruin Calculation and Bankroll Management Model: The Core Tool for Bankroll Safety

Tool Purpose

Risk of Ruin is a core metric in poker bankroll management, used to evaluate the probability that a player will go broke due to a series of losses given a specific bankroll size, win rate, and volatility. By calculating the risk of ruin, players can determine whether their current bankroll is sufficiently safe and decide whether to move up, move down, or adjust their game strategy.

Calculation Formula Principles

The standard risk-of-ruin model assumes that the returns per hand or per 100 hands are independent and identically distributed and approximately follow a normal distribution. The commonly used formula is:

$$ R = e^{-2 \cdot B \cdot EV / \sigma^2} $$

where:

• R = Risk of Ruin (between 0 and 1)
• B = Current bankroll (in the same units as EV and standard deviation, e.g., BB or dollars)
• EV = Expected value per unit of hands (e.g., win rate per 100 hands)
• σ = Standard deviation per unit of hands (e.g., standard deviation per 100 hands)

This formula is derived from the first-passage time distribution of Brownian motion and is the most common approximation used in poker bankroll management. More precise calculations need to account for discreteness, but for most players this formula is sufficient.

Usage Steps

1. Collect Data: Extract your win rate per 100 hands (BB/100) and standard deviation (BB/100) from poker tracking software (e.g., Hold'em Manager or PokerTracker). If your sample size is insufficient (at least 50,000 hands recommended), use industry averages (e.g., for 6‑max cash games, win rate around 5–10 BB/100, standard deviation about 80–100 BB/100).
2. Determine Bankroll: Calculate your current available playing bankroll (excluding living expenses).
3. Plug into Formula: Input the data into the risk-of-ruin formula.
4. Set Safety Threshold: A risk of ruin below 5% is generally considered “safe,” but professional players often require less than 1% or even 0.1%.
5. Review Regularly: As your results and bankroll change, recalculate and adjust accordingly.

Worked Example

Suppose you are a NL100 (big blind $1) cash game player. After 100,000 hands you have: win rate 5 BB/100 hands, standard deviation 80 BB/100 hands. Current bankroll is 2000 BB (i.e., 20 buy‑ins).

• EV = 5 BB/100 hands
• σ² = 80² = 6400 (BB²/100 hands)
• B = 2000 BB

Plug into the formula:

$$ R = e^{-2 \times 2000 \times 5 / 6400} = e^{-3.125} \approx 0.044 = 4.4% $$

This means that, assuming you maintain your current performance, you have about a 4.4% chance of going broke at some point. At 4.4% you are on the acceptable borderline; it is recommended to increase to 30 buy‑ins (3000 BB), which lowers the risk of ruin to about 0.9% – much safer.

Frequently Asked Questions

Q1: What is a safe risk of ruin?
A: Generally, below 5% is recommended; professional players should aim below 1%. If the risk exceeds 10%, it is strongly advised to move down or increase your bankroll.

Q2: How do I obtain the standard deviation for the formula?
A: Most poker tracking software directly displays the standard deviation per 100 hands. If you don’t have it, you can use typical values for your game: for 6‑max cash games about 80–100 BB/100; for tournaments, it is much higher due to payout structure differences.

Q3: Does the risk of ruin increase during a downswing?
A: Yes. The formula assumes a long‑term stable win rate, but in reality there is variance. A conservative approach is to use a lowered win rate (e.g., 80% of your actual win rate) or an inflated standard deviation as a buffer.

Q4: Is this formula applicable to tournaments?
A: Not fully, because tournament payout structures are complex (pay jumps, ICM). Bankroll management for tournaments typically requires more conservative buy‑in guidelines, such as having at least 100 buy‑ins, or using a dedicated tournament risk‑of‑ruin model.

Further Learning

• The Mathematics of Poker – detailed derivation of risk of ruin and related models.
• Online bankroll calculators: Tools like BankrollMob, PokerBankrollCalculator offer interactive calculations.
• Advanced topics: Consider multi‑level mixed games or incorporate the Kelly Criterion to optimise bankroll growth.

Mastering risk‑of‑ruin calculation and bankroll management models is essential for any long‑term profitable poker player. By prioritising bankroll safety, you can navigate the game’s variance and play steadily.