FRM Part I · FRM Exam Part I · Multivariate Random Variables
The covariance between the returns of X and Y is 0.012. Returns are rescaled so that X' = 3X + 2 and Y' = -2Y + 1. What is Cov(X', Y') and what is the correlation of X' and Y' relative to the original correlation ρ?
The covariance becomes -0.072 and the correlation becomes -ρ. Covariance scales by the product of the multipliers, 3 times -2 times 0.012, and shifts do not matter. Correlation is scale-free in magnitude, but the negative multiplier on Y reverses its sign.
- ACov = -0.072; correlation = -ρCorrect
- BCov = -0.072; correlation = ρ
- CCov = 0.072; correlation = -ρ
- DCov = -0.012; correlation = -ρ
Explanation
Cov(aX+b, cY+d) = ac·Cov(X,Y) = 3 × (-2) × 0.012 = -0.072. Constants do not matter. Correlation is unchanged in magnitude by scaling, but one multiplier is negative so the sign flips, giving -ρ. Keeping ρ ignores the negative multiplier.
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