ROR is the % chance you go broke if you play (in a certain way) forever. The alternative to busting out is to growing your money to infinity.
ROR is exponential/logarithmic. If you have 200 “units” and are playing at a 40% ROR….if your bankroll gets to 400 units, then your ROR is now 40% x 40% = 16%. If you think about it logically, what is the chance you lose 200 units? 40%. What’s the chance of doing that twice in a row? Well, 40% times 40%.
Similarly, if your bankroll goes down to 100 units, then your ROR becomes the square root of 40%, or 0.4^(1/2) = 0.63245553203 = 63.24%. Let’s say you grow your bankroll back up to 200 units, let’s make sure our math checks out. 0.63245553203^2 = 0.4 = 40%, which makes sense, since that was the premise.