FRM Part I · FRM Exam Part I · Measuring Return, Volatility, and Correlation
X is a standard normal random variable and Y = X^2. Which is the value of the Pearson correlation between X and Y, and the best interpretation?
The correlation is zero. Covariance equals E[X^3] minus E[X]E[X^2], and both terms vanish for a symmetric zero-mean normal variable. Despite Y being exactly determined by X, Pearson correlation only measures linear dependence and therefore reports no relationship.
- AZero, even though Y is completely determined by XCorrect
- B1.0, because Y is a perfect function of X
- C0.5, because Y is a convex function of X
- D-1.0, because squaring removes the sign of X
Explanation
Cov(X, X^2) = E[X^3] - E[X]E[X^2]. For a symmetric standard normal, E[X^3] = 0 and E[X] = 0, so covariance is 0 and correlation is 0. Yet Y is a deterministic function of X, showing correlation misses nonlinear dependence.
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