Financial Risk Manager Part 1

The correlation coefficient is calculated using the formula:

$$\rho_{X_1 X_2} = \frac{\text{Cov}(X_1, X_2)}{\sqrt{\text{Var}(X_1) \cdot \text{Var}(X_2)}} = \frac{\sigma_{12}}{\sigma_1 \sigma_2}$$

Given Cov(A, B) = 23.43, we need to calculate the variances of A and B from the probability matrix.

Step 1: Calculate E(X₁) and E(X₁²) for Company A

Company A profits: -1, 0, 2, 4 (in millions)

Marginal probabilities for A:

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

E(X₁) = (-1)(0.0697) + (0)(0.4274) + (2)(0.3436) + (4)(0.1598) = -0.0697 + 0 + 0.6872 + 0.6392 = 1.2567

E(X₁²) = (1)(0.0697) + (0)(0.4274) + (4)(0.3436) + (16)(0.1598) = 0.0697 + 0 + 1.3744 + 2.5568 = 4.0009

Var(X₁) = E(X₁²) - [E(X₁)]² = 4.0009 - (1.2567)² = 4.0009 - 1.5793 = 2.4216

Step 2: Calculate E(X₂) and E(X₂²) for Company B

Company B profits: -50, 0, 10, 100 (in millions)

Marginal probabilities for B:

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

E(X₂) = (-50)(0.0712) + (0)(0.4228) + (10)(0.3482) + (100)(0.1583) = -3.56 + 0 + 3.482 + 15.83 = 15.752

E(X₂²) = (2500)(0.0712) + (0)(0.4228) + (100)(0.3482) + (10000)(0.1583) = 178 + 0 + 34.82 + 1583 = 1795.82

Var(X₂) = E(X₂²) - [E(X₂)]² = 1795.82 - (15.752)² = 1795.82 - 248.12 = 1547.70

Step 3: Calculate correlation coefficient

$$\rho = \frac{23.43}{\sqrt{2.4216 \times 1547.70}} = \frac{23.43}{\sqrt{3746.67}} = \frac{23.43}{61.21} = 0.3827$$

Therefore, the correlation coefficient is 0.3827, which corresponds to option B.