Equation 1: Simple Risk of Ruin Formula for Kelly Bettor:
Let A = the starting bankroll (BR). Then, if betting 100% Kelly (f=1),

Prob. {BR reaches yA before xA} = y*(x-1)/(x-y).

For example, this leads to the rather well-known fraction of 67% for the probability that one doubles the bank before losing one-half. To wit: y = 2; x = 1/2, so Prob. = 2*(.5 - 1)/(.5 - 2) = 2/3.




Equation 2: Fractional Kelly Risk of Ruin Formula:
Generally, for any f (fractional Kelly), the

Prob. {BR reaches yA before xA} = [x^((2/f) -1) - 1]/[(x/y)^((2/f) -1) -1].

Don