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

A portfolio manager observes that two assets have a Pearson correlation near zero, but a scatter plot shows that Y is almost exactly X squared, with X symmetric around zero. Which conclusion is most appropriate?

Zero Pearson correlation does not imply independence. Y is a deterministic function of X, yet because X is symmetric the covariance with X squared is zero. Pearson correlation measures only linear dependence and misses nonlinear relationships.

  1. AThe assets are statistically independent because correlation is zero
  2. BZero correlation does not imply independence, since Pearson captures only linear dependenceCorrect
  3. CThe data must contain an error, since Y depends on X so correlation must be positive
  4. DCorrelation is negative because the relationship is convex

Explanation

For symmetric X, Cov(X, X²) = E[X³] − E[X]E[X²] = 0, so Pearson correlation is zero despite perfect functional dependence. Independence implies zero correlation, but not the reverse.

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