You estimate a hedge ratio as X/Y where X and Y are independent N(0, 1) estimates. Give the distribution of the ratio, say what happens to the average of 1,000 such ratios, and state the practical rule this gives for estimating any ratio.

You estimate a hedge ratio as X/Y where X and Y are independent N(0, 1) estimates. Give the distribution of the ratio, say what happens to the average of 1,000 such ratios, and state the practical rule this gives for estimating any ratio.

Approach: Convert the pair to polar coordinates so the ratio becomes the tangent of a uniform angle, then check whether the resulting law has the moments the law of large numbers requires.

Standard Cauchy. For independent standard normals the joint density is rotationally symmetric, so the angle of the point (X, Y) is uniform on the circle and the ratio X/Y is the tangent of that angle, whose density is 1/(pi*(1 + t^2)). The Cauchy has heavy tails with tail index 1 and no mean, so the law of large numbers does not apply, and the average of 1,000 draws is itself exactly standard Cauchy with the same spread as one observation. Averaging more data buys nothing, and a bootstrap or delta-method standard error computed as though the mean existed will report a figure shrinking like 1/sqrt(n) that means nothing. The practical rule is that a ratio estimator is unstable whenever the denominator has non-negligible probability near zero, so estimate the hedge ratio from a regression of X on Y, which uses the joint information and has finite variance, and report a Fieller interval rather than a symmetric one when the ratio itself is the target.

Follow-up: Y is N(10, 1) instead of N(0, 1). Does the ratio now have a mean, and how good is the delta-method standard error?

Key concepts: Cauchy distribution, law of large numbers, ratio estimator, heavy tails.