Part 4 · Lesson 20 of 25 · Free
Poker risk of ruin: bankroll and variance
Win 5 bb per 100 hands with a standard deviation of 100, and a 1,000 bb bankroll still goes broke 37% of the time. With 3,000 bb, the chance falls to 5%. A winning player can go broke when the bankroll is too small for the swings; risk of ruin is the chance that a bankroll ever hits zero.
The hand
You win 5 bb per 100 hands with a standard deviation of 100 bb per 100 hands, and you have set aside 1,000 bb for poker.
Predict first
What is the chance that you ever lose all 1,000 bb?
- Under 1%
- About 10%
- About 37%
Show the answer
Answer: About 37%
e^(−2 × 5 × 1,000 ÷ 100²) = e^(−1) ≈ 37%. Swings of 100 bb per 100 hands dwarf an edge of 5.
Working it out
For a win rate w and a standard deviation σ in the same units, the risk of ruin with a bankroll B is about e^(−2wB ÷ σ²). Here 2 × 5 ÷ 100² = 0.001 per bb: 1,000 bb gives e^(−1) ≈ 37%, 2,000 bb gives e^(−2) ≈ 14% and 3,000 bb gives e^(−3) ≈ 5%. Each extra 1,000 bb divides the risk by e ≈ 2.7.
The risk falls exponentially as the bankroll grows, and rises sharply as the win rate shrinks or the swings grow.
Formula
Risk of ruin ≈ e^(−2 w B ÷ σ²); bankroll for a target risk r: B = −σ² × ln(r) ÷ (2w)
Check your understanding
Win rate 4 bb/100, SD 90 bb/100. What bankroll gives a 5% risk of ruin, in bb?
Show the answer
Answer: 3,033 bb
B = −90² × ln(0.05) ÷ (2 × 4) = 8,100 × 2.996 ÷ 8 ≈ 3,033 bb.
Your win rate halves and your swings stay the same. For the same risk of ruin you need:
- Twice the bankroll
- The same bankroll
- Four times the bankroll
Show the answer
Answer: Twice the bankroll
B = −σ² × ln(r) ÷ (2w): halving w doubles B. Doubling σ would quadruple it.