Expectation with possible ruin

From Stanford Wong's BJ21

Posted by Kim Lee on 28 Dec 1998, 7:35 pm, in response to Re: Formula (please doublecheck), posted by Don Schlesinger on 22 Dec 1998, 11:12 am

Without loss of generality assume length of play is 1, beginning bankroll is W, mean is mu, standard deviation is sigma. The probability of terminal bankroll less than x is

N((x-W-mu)/sigma))+exp(-2muW/sigma^2)*N((-x-W+mu)/sigma)),

where N(.) denotes the cumulative normal. Setting x = 0 gives Don's risk of ruin formula. The expected terminal bankroll is

(W+mu)*N((mu+W)/sigma)+(W-mu)*exp(-2muW/sigma^2)*N((mu-W)/sigma).

The expected terminal bankroll is generally smaller than W+mu because the bank doesn't grow after it is ruined. But the formula approaches W+mu as the beginning bank, W, becomes large, as the mean, mu, becomes large, or as the risk, sigma, becomes small.