Explanation

Step 1: Calculate Initial Portfolio Variance

For an equally weighted two-asset portfolio:

$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2\rho_{12}$

Given:

• w₁ = w₂ = 0.5
• σ₁ = σ₂ = 0.02
• ρ₁₂ = initial correlation

$\sigma_p^2 = 0.5^2(0.02)^2 + 0.5^2(0.02)^2 + 2\times0.5\times0.5\times0.02\times0.02\times\rho_{12}$ $\sigma_p^2 = 0.0001 + 0.0001 + 0.0002\times\rho_{12}$ $\sigma_p^2 = 0.0002\times(1 + \rho_{12})$

Step 2: Find Initial Correlation

Given 10-day 99% VaR = USD 2.33 million

1-day VaR formula: $\text{VaR}_{1-day} = z_{0.99} \times \sigma_p \times V_p$

10-day VaR: $\text{VaR}_{10-day} = \sqrt{10} \times z_{0.99} \times \sigma_p \times V_p$

Where:

• z₀.₉₉ = 2.326 (for 99% confidence)
• V_p = USD 20,000,000

$2,330,000 = \sqrt{10} \times 2.326 \times 20,000,000 \times \sigma_p$

First calculate: $\sqrt{10} \times 2.326 \times 20,000,000 = 2,080,123$

So: $2,330,000 = 2,080,123 \times \sigma_p$ $\sigma_p = \frac{2,330,000}{2,080,123} = 1.1201$

But this is incorrect - σ_p should be a volatility (0.01414), not 1.1201. Let's recalculate:

$\sigma_p = \sqrt{0.0002 \times (1 + \rho_{12})} = 0.01414 \times \sqrt{1 + \rho_{12}}$

$2,330,000 = \sqrt{10} \times 2.326 \times 20,000,000 \times 0.01414 \times \sqrt{1 + \rho_{12}}$ $2,330,000 = 2,080,123 \times \sqrt{1 + \rho_{12}}$ $\sqrt{1 + \rho_{12}} = \frac{2,330,000}{2,080,123} = 1.1201$ $1 + \rho_{12} = (1.1201)^2 = 1.2547$ $\rho_{12} = 0.2547$

Step 3: Calculate New VaR with Doubled Correlation

New correlation: ρ₁₂ = 2 × 0.2547 = 0.5094

New portfolio variance: $\sigma_p^2 = 0.0002 \times (1 + 0.5094) = 0.0002 \times 1.5094 = 0.00030188$ $\sigma_p = \sqrt{0.00030188} = 0.01737$

New 10-day VaR: $\text{VaR}_{10-day} = \sqrt{10} \times 2.326 \times 20,000,000 \times 0.01737$ $\text{VaR}_{10-day} = 2,080,123 \times 0.01737 = 2,549,000$

This is approximately USD 2.55 million, which matches option B.

Key Insight: When correlation increases, portfolio diversification decreases, leading to higher portfolio volatility and higher VaR.