Financial Risk Manager Part 1
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Three random variables X, Y, and Z have equal variance of σ² = 2. X is independent of both Y and Z, and that Y and Z are correlated with a correlation coefficient of 0.8. What is the covariance between Z and V given that V = 3X − 2Y.
Explanation:
Explanation
To find the covariance between Z and V where V = 3X - 2Y:
Step 1: Apply covariance properties
Using the linearity property of covariance: [\text{Cov}(Z, V) = \text{Cov}(Z, 3X - 2Y) = 3\text{Cov}(Z, X) - 2\text{Cov}(Z, Y)]
Step 2: Simplify using independence
Since X is independent of both Y and Z: [\text{Cov}(Z, X) = 0] Therefore: [\text{Cov}(Z, V) = 0 - 2\text{Cov}(Z, Y) = -2\text{Cov}(Z, Y)]
Step 3: Calculate Cov(Z, Y)
Given that the correlation coefficient between Y and Z is ρ = 0.8, and all variables have equal variance σ² = 2: [\text{Cov}(Z, Y) = ρ_{ZY} × σ_Z × σ_Y = 0.8 × \sqrt{2} × \sqrt{2} = 0.8 × 2 = 1.6]
Step 4: Final calculation
[\text{Cov}(Z, V) = -2 × 1.6 = -3.2]
Key points:
- When variables are independent, their covariance is zero
- Correlation coefficient formula: ρ = Cov(X,Y)/(σ_Xσ_Y)
- Covariance is linear: Cov(aX + bY, Z) = aCov(X,Z) + bCov(Y,Z)_