Skip to content

FRM Part II · FRM Exam Part II · Correlation Basics: Definitions, Applications, and Terminology

An analyst notes that two return series X and Y are related by Y = X^2, where X is symmetric around zero (for example, standard normal). Which statement about the Pearson correlation between X and Y is correct?

The Pearson correlation is zero. With X symmetric around zero, the covariance of X and X squared is E[X^3] minus E[X]E[X^2], which equals zero. Pearson measures only linear dependence, so a perfect nonlinear relationship can still show no correlation.

  1. AIt is close to zero even though Y is perfectly determined by XCorrect
  2. BIt equals +1 because Y is an exact function of X
  3. CIt equals -1 because the relationship is convex
  4. DIt is undefined because Y is nonlinear in X

Explanation

Cov(X, X^2) = E[X^3] - E[X]E[X^2]. For a symmetric distribution with mean zero, E[X^3] = 0 and E[X] = 0, so covariance and Pearson correlation are zero. Pearson captures only linear dependence, so perfect nonlinear dependence can coexist with zero correlation.

Did you get it right without looking?

One question tells you little. A timed set on Correlation Basics: Definitions, Applications, and Terminology shows your real accuracy, how long you take and where you lose marks.

Practise Correlation Basics: Definitions, Applications, and TerminologyFree diagnostic mock

More Correlation Basics: Definitions, Applications, and Terminology questions