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

A risk analyst states that two returns series X and Y have a Pearson correlation of zero. Which conclusion is most appropriate?

A zero Pearson correlation means there is no linear relationship, but nonlinear dependence can still exist. Independence implies zero correlation, not the other way around, except for jointly normal variables. Y equal to X squared is a classic example of zero correlation with full dependence.

  1. AX and Y are statistically independent
  2. BThere is no linear relationship between X and Y, but a nonlinear dependence may still existCorrect
  3. CX and Y must have identical marginal distributions
  4. DX and Y move in opposite directions half of the time and never jointly in the tails

Explanation

Pearson correlation measures only linear dependence. Zero correlation does not imply independence, for example Y = X^2 with X symmetric about zero has zero covariance with X yet is fully dependent. The independence option wrongly reverses the implication, which holds only in special cases such as joint normality.

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