Risk of Ruin (RoR)
Also known as: risk of ruin, RoR, ruin probability
The probability your bankroll hits zero before you can rebuild it, given your edge, variance, and roll size.
Risk of ruin (RoR) is the probability that variance busts your bankroll before your edge can compound. For a player with a positive win rate, a useful continuous approximation is:
\[ \text{RoR} \approx e^{-\,2 \cdot w \cdot B \,/\, \sigma^2} \]
where \(w\) is win rate per hand, \(\sigma\) is standard deviation per hand, and \(B\) is the bankroll — all in the same units (e.g. bb).
Three levers fall straight out of the formula:
- More bankroll \(B\) → RoR drops exponentially. Doubling the roll squares your survival odds.
- Bigger edge \(w\) → RoR drops exponentially. Game selection is risk management.
- Higher variance \(\sigma\) → RoR rises sharply (it's in the denominator, squared).
The non-negotiable corollary: if \(w \le 0\), RoR \(= 100\%\) — a losing player goes broke eventually regardless of roll size. No amount of bankroll saves a negative edge; it only delays the bust.
This is the math underneath "30 buy-ins for cash, 50–100 for MTTs." Those counts are chosen so RoR is acceptably small for typical edges and variance. The Kelly criterion is the flip side of the same coin: bet sizing that bounds long-run ruin.
Example
Take \(w = 0.05\) bb/hand (5 bb/100), \(\sigma = 100\) bb/100 so \(\sigma^2 = 10000\) bb²/100, i.e. \(\sigma^2 \approx 100\) bb²/hand. With a 3,000 bb roll (30 bi of 100NL): \[ \text{RoR} \approx e^{-2(0.05)(3000)/100} = e^{-3} \approx 0.050. \] Roughly a 5% lifetime ruin estimate. Drop the roll to 1,500 bb and it becomes \(e^{-1.5}\approx22\%\) — halving the roll quadruples-plus the ruin risk.