Suppose avg(A) and avg(B) are normally distributed. This will be exactly true if A and B are normally distributed, because a sum of normally distributed deviates is itself normally distributed. It will be approximately true for large samples however A and B are distributed, by the central limit theorem.
Then what you are asking is how the ratio of two independent normal deviates is distributed. The answer is the Cauchy distribution. (http://en.wikipedia.org/wiki/Normal_..._distributions)
The Cauchy distribution has infinite variance, so your ratio's standard deviation is infinite.