Explanation

To calculate the one-month 99.0% CVaR for this three-bond portfolio, we need to:

Step 1: Convert annual default probability to monthly

• Annual cumulative default probability = 4.0% = 0.04
• Monthly default probability = 1 - (1 - 0.04)^(1/12) ≈ 0.0034 or 0.34%

Step 2: Calculate default probabilities for the portfolio

Since the bonds are independent, we can model this as a binomial distribution:

• Probability of 0 defaults: (1 - 0.0034)^3 ≈ 0.9898
• Probability of 1 default: 3 × 0.0034 × (1 - 0.0034)^2 ≈ 0.0102
• Probability of 2 defaults: 3 × (0.0034)^2 × (1 - 0.0034) ≈ 0.0000347
• Probability of 3 defaults: (0.0034)^3 ≈ 0.000000039

Step 3: Determine the 99.0% confidence level

We need to find the loss level that corresponds to the 99th percentile:

• Cumulative probability of 0 defaults: 98.98%
• Cumulative probability of 1 default: 98.98% + 1.02% = 100.00%

Since 99.0% confidence falls between 0 defaults (98.98%) and 1 default (100.00%), the unexpected loss at 99.0% confidence corresponds to 2 defaults.

Step 4: Calculate the CVaR

• With zero recovery rate, each default results in a loss of $1.0 million
• 2 defaults = $2.0 million loss
• The nearest option to $2.0 million is $2.3 million

Therefore, the one-month 99.0% CVaR is approximately $2.3 million.