Financial Risk Manager Part 1

Explanation

Given:

• Security A: Weight w₁ = 0.10, Standard deviation σ₁ = 0.06
• Security B: Weight w₂ = 0.90, Standard deviation σ₂ = 0.15
• Portfolio standard deviation σₚ = 0.141

We use the portfolio variance formula:

σₚ² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁,R₂)

Where Cov(R₁,R₂) = ρσ₁σ₂ (covariance = correlation × product of standard deviations)

Step 1: Calculate portfolio variance σₚ² = (0.141)² = 0.019881

Step 2: Calculate the first two terms w₁²σ₁² = (0.10)² × (0.06)² = 0.01 × 0.0036 = 0.000036 w₂²σ₂² = (0.90)² × (0.15)² = 0.81 × 0.0225 = 0.018225

Sum of first two terms = 0.000036 + 0.018225 = 0.018261

Step 3: Solve for covariance 0.019881 = 0.018261 + 2 × 0.10 × 0.90 × Cov(R₁,R₂) 0.019881 = 0.018261 + 0.18 × Cov(R₁,R₂) 0.00162 = 0.18 × Cov(R₁,R₂) Cov(R₁,R₂) = 0.00162 / 0.18 = 0.0090

Alternative approach using correlation: We can also find that the correlation ρ = 1.0, then: Cov(R₁,R₂) = ρσ₁σ₂ = 1 × 0.06 × 0.15 = 0.0090

Key insight: The portfolio standard deviation of 14.1% equals the weighted average of individual standard deviations (0.1×6 + 0.9×15 = 14.1), which only occurs when correlation is perfect (ρ = 1). This makes the covariance calculation straightforward.

Units note: The covariance of 0.0090 represents 0.0090 in decimal terms, which corresponds to option D.