Currently i am trying to derive the volatility of a financial ratio. I have calculated the volatility (standard deviation) of both the denominator and numerator however I am running into trouble whenever trying to derive the standard deviation of the ratio.
Initially I tried normally distributing each financial number (denominator & numerator, 10,000 z-scores), divide them at each individual z-score and take the standard deviation of the derived 10,000 ratios. However i found this to be understating the true standard deviation.
Then I tried to do the inverse; take the maximum of one ratio and the minimum of another ratio and the opposite to come up with the worst possible range. However I found this to be overstating the true standard deviation.
Any help would be incredibly helpful!
This link should be helpful. Note that the ratio of two standard normal variable is Cauchy, with $\sigma = \infty$ so ratios can be much less well behaved than either the numerator or denominator. For your problem, see this paper, which discussed ratios of non-standard normals.