Poker Risk of Ruin Calculation and Risk Management Model

Tool Purpose

Poker Risk of Ruin (RoR) is a metric that measures the probability of a player losing all their bankroll given their win rate, variance, and current bankroll size. It is a core tool for bankroll management, helping players determine a safe minimum bankroll to avoid going broke due to short-term fluctuations.

Calculation Principle

The most commonly used RoR formula assumes a fixed win rate and normally distributed returns. The classic formula is:

$$ \text{RoR} = e^{-\frac{2 \cdot W \cdot B}{\sigma^2}} $$

Where:

• $W$: Win rate per unit of time (e.g., per 100 hands, in bb)
• $B$: Bankroll (units consistent with $W$, e.g., bb)
• $\sigma$: Standard deviation per unit of time (units consistent with $W$)
• $e$: Euler's number (approx. 2.71828)

This formula assumes unlimited playing time, constant win rate and variance, and infinitely divisible bankroll. In practice, the time unit is usually 100 hands, since win rates and standard deviations in poker are often reported per 100 hands.

Parameter Acquisition

• Win rate $W$: Obtain from historical data as the average profit per 100 hands (bb/100). Beginners can reference typical values for similar games, e.g., at NL50 about 5–10 bb/100.
• Standard deviation $\sigma$: The standard deviation of profit per 100 hands. Typical values range from 60–120 bb/100; for cash games, 100 bb/100 is often used.
• Bankroll $B$: The total amount you currently have available for that stake.

How to Use

1. Choose an acceptable risk of ruin: Professional players typically aim for 1% or lower; recreational players may accept 5%. Lower probability requires a larger bankroll.
2. Estimate your win rate and standard deviation: At least 10,000 hands of data are needed for reasonable accuracy. Without data, use typical values from comparable players.
3. Calculate your current RoR: Plug the values into the formula and check if it falls within your acceptable range.
4. Calculate the required minimum bankroll: If your current RoR is too high, solve the formula for $B$ given your target RoR:

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

5. Re-evaluate periodically: Update the parameters as your playing style or stakes change.

Practical Example

Example: A player at NL100 cash game has historical data showing a win rate of $W=5\text{bb}$ per 100 hands and a standard deviation $\sigma=100\text{bb}$. Current bankroll is 3000bb (30 buy-ins). Calculate the risk of ruin and find the bankroll needed to achieve RoR ≤ 1%.

Step 1: Calculate current RoR

$$ \text{RoR} = e^{-\frac{2 \times 5 \times 3000}{100^2}} = e^{-\frac{30000}{10000}} = e^{-3} \approx 0.0498 $$

This is about a 5% risk of ruin – acceptable for a recreational player but likely too high for a professional.

Step 2: Solve for required bankroll (target RoR = 1%)

$$ B = -\frac{100^2 \times \ln(0.01)}{2 \times 5} = -\frac{10000 \times (-4.60517)}{10} \approx 4605 \text{bb} $$

That means at least 46 buy-ins (4600bb) are needed to keep the risk of ruin below 1%.

Note: This example assumes constant parameters. In reality, you should account for rake, emotional variance, etc.

Frequently Asked Questions

Q: The formula assumptions are too strict – is it useful in practice?
A: The classic formula relies on idealized assumptions (infinite time, constant win rate, normal distribution), but it still provides strong guidance. It's recommended to also use simulation software (e.g., PRR) for stress testing, considering dynamic strategies like moving down in stakes.

Q: What if my win rate estimate is inaccurate?
A: Win rate errors significantly affect RoR. A conservative approach is to use the lower bound of a 95% confidence interval for $W$, or to calculate the required bankroll assuming a win rate of 0 (which gives RoR = 100%, but you can determine a safety margin through simulation).

Q: Can I reduce the required bankroll by playing games with lower variance?
A: Yes. Lowering the standard deviation (e.g., from 100 bb/100 to 80 bb/100) can substantially reduce the needed bankroll. Players can achieve this by adopting a tight-aggressive style.

Further Learning

• Kelly Criterion: Used to determine optimal bet sizing; combining it with RoR can optimize bankroll growth.
• Multi‑stake bankroll management: A common rule is to move up only when your bankroll reaches 20–30 buy‑ins for the next level, and set a move‑down threshold.
• Simulation tools: Software like the Poker Risk of Ruin Calculator (PRR) can simulate different strategies and their paths to ruin.