Explanation

To calculate the unilateral Credit Valuation Adjustment (CVA), we need to compute the expected loss due to counterparty default over the life of the contract. The formula for CVA is:

$\text{CVA} = \sum_{i=1}^{n} \text{DF}_i \times \text{PD}_i \times \text{LGD}_i \times \text{EE}_i$

Where:

• DF = Discount factor
• PD = Probability of default
• LGD = Loss given default = 1 - Recovery rate
• EE = Expected exposure (adjusted for collateral)

Step 1: Calculate Net Expected Exposure

Since there is AUD 11 million collateral, the net exposure is:

• Net EE = Max(Expected Exposure - Collateral, 0) = Max(14 - 11, 0) = AUD 3 million for each year

Step 2: Calculate Hazard Rates from CDS Spreads

The hazard rate (λ) can be approximated from CDS spreads using: $\lambda \approx \frac{\text{CDS spread}}{1 - \text{Recovery rate}}$

• Year 1: λ₁ = 200 bps / (1 - 0.80) = 200 / 0.20 = 1,000 bps = 0.10
• Year 2: λ₂ = 300 bps / (1 - 0.70) = 300 / 0.30 = 1,000 bps = 0.10
• Year 3: λ₃ = 400 bps / (1 - 0.60) = 400 / 0.40 = 1,000 bps = 0.10

Step 3: Calculate Survival Probabilities and Default Probabilities

With constant hazard rate λ = 0.10:

• Survival probability to time t: S(t) = e^{-λt}

• Marginal default probability in year t: PD(t) = S(t-1) - S(t)

• S(0) = 1.0000

• S(1) = e^{-0.10×1} = 0.9048

• S(2) = e^{-0.10×2} = 0.8187

• S(3) = e^{-0.10×3} = 0.7408

Marginal PDs:

• PD(1) = S(0) - S(1) = 1.0000 - 0.9048 = 0.0952
• PD(2) = S(1) - S(2) = 0.9048 - 0.8187 = 0.0861
• PD(3) = S(2) - S(3) = 0.8187 - 0.7408 = 0.0779

Step 4: Calculate Discount Factors

With risk-free rate r = 3%:

• DF(1) = 1/(1+0.03) = 0.9709
• DF(2) = 1/(1+0.03)² = 0.9426
• DF(3) = 1/(1+0.03)³ = 0.9151

Step 5: Calculate CVA

CVA = Σ [DF(t) × PD(t) × LGD(t) × Net EE(t)]

• Year 1: 0.9709 × 0.0952 × (1-0.80) × 3 = 0.0555
• Year 2: 0.9426 × 0.0861 × (1-0.70) × 3 = 0.0731
• Year 3: 0.9151 × 0.0779 × (1-0.60) × 3 = 0.0855

Total CVA = 0.0555 + 0.0731 + 0.0855 = AUD 0.2141 million

Therefore, the correct estimate of the unilateral CVA is AUD 0.214 million.