Financial Risk Manager Part 1

Explanation

To find Cov[E, F], we use the covariance properties:

Cov[E, F] = Cov[X + Y, X - Y]

Using the bilinearity property of covariance: = Cov[X, X] - Cov[X, Y] + Cov[Y, X] - Cov[Y, Y]

Since:

• Cov[X, X] = V[X] (variance of X)
• Cov[Y, Y] = V[Y] (variance of Y)
• Cov[X, Y] = Cov[Y, X] (covariance is symmetric)

We get: = V[X] - Cov[X, Y] + Cov[X, Y] - V[Y] = V[X] - V[Y]

Given that V[X] = 4V[Y], we substitute: = 4V[Y] - V[Y] = 3V[Y]

Key properties used:

• Cov[X, X] = V[X]
• Cov[X, Y] = Cov[Y, X]
• Cov[aX + bY, cZ + dW] = acCov[X, Z] + adCov[X, W] + bcCov[Y, Z] + bdCov[Y, W]