Financial Risk Manager Part 1

Explanation

To calculate the covariance between company A and B, we need to use the formula:

Cov(X₁, X₂) = E(X₁X₂) - E(X₁)E(X₂)

We are given that E(X₁X₂) = 43.23, so we need to calculate E(X₁) and E(X₂) from the joint probability distribution table.

Step 1: Calculate E(X₁) - Expected value of Company A profits

First, we need to find the marginal distribution of X₁ by summing probabilities across each column (for each value of X₁).

Values of X₁: -1, 0, 2, 4 (in millions)

Marginal probabilities for X₁:

• For X₁ = -1: 0.0197 + 0.0390 + 0.011 + 0 = 0.0697
• For X₁ = 0: 0.0395 + 0.230 + 0.127 + 0.0309 = 0.4274
• For X₁ = 2: 0.010 + 0.124 + 0.144 + 0.0656 = 0.3436
• For X₁ = 4: 0.002 + 0.0298 + 0.0662 + 0.0618 = 0.1598

Check sum: 0.0697 + 0.4274 + 0.3436 + 0.1598 = 1.0005 (≈ 1, minor rounding differences)

E(X₁) = Σ [x₁ × P(X₁ = x₁)] = (-1 × 0.0697) + (0 × 0.4274) + (2 × 0.3436) + (4 × 0.1598) = -0.0697 + 0 + 0.6872 + 0.6392 = 1.2567

Step 2: Calculate E(X₂) - Expected value of Company B profits

Now find the marginal distribution of X₂ by summing probabilities across each row (for each value of X₂).

Values of X₂: -50, 0, 10, 100 (in millions)

Marginal probabilities for X₂:

• For X₂ = -50: 0.0197 + 0.0395 + 0.010 + 0.002 = 0.0712
• For X₂ = 0: 0.0390 + 0.230 + 0.124 + 0.0298 = 0.4228
• For X₂ = 10: 0.011 + 0.127 + 0.144 + 0.0662 = 0.3482
• For X₂ = 100: 0 + 0.0309 + 0.0656 + 0.0618 = 0.1583

Check sum: 0.0712 + 0.4228 + 0.3482 + 0.1583 = 1.0005 (≈ 1)

E(X₂) = Σ [x₂ × P(X₂ = x₂)] = (-50 × 0.0712) + (0 × 0.4228) + (10 × 0.3482) + (100 × 0.1583) = -3.56 + 0 + 3.482 + 15.83 = 15.752

Step 3: Calculate Covariance

Cov(X₁, X₂) = E(X₁X₂) - E(X₁)E(X₂) = 43.23 - (1.2567 × 15.752) = 43.23 - 19.796 = 23.434

Rounding to two decimal places: 23.43

Therefore, the correct answer is B. 23.43.

Note: The question states that E(X₁X₂) = 43.23 is given, which saves us from having to calculate it directly from the joint distribution table. This value would be computed as ΣΣ x₁x₂ × P(X₁=x₁, X₂=x₂) across all combinations in the table.