Financial Risk Manager Part 1

Financial Risk Manager Part 1

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Three random variables X, Y, and Z have the 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 X and K given that K=Y+Z?

TTanishq

Explanation:

Explanation

Recall from the properties of the covariance that:

Cov(A,B+C)=Cov(A,B)+Cov(A,C)\text{Cov}(A, B + C) = \text{Cov}(A, B) + \text{Cov}(A, C)

So, we need:

Cov(X,K)=Cov(X,Y+Z)=Cov(X,Y)+Cov(X,Z)=0+0=0\text{Cov}(X, K) = \text{Cov}(X, Y + Z) = \text{Cov}(X, Y) + \text{Cov}(X, Z) = 0 + 0 = 0

This is true because X is independent of both Y and Z, which means:

  • Cov(X, Y) = 0
  • Cov(X, Z) = 0

The correlation between Y and Z (ρ = 0.8) and the variances (σ² = 2) are irrelevant for calculating Cov(X, K) since X is independent of both Y and Z.

Therefore, the covariance between X and K is 0.

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