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Answer: 1.33%
## Explanation Using the formula for the joint default probability (Φ ≡ π₁₂) derived from the default correlation: **Formula:** Φ = ρ * π * (1 − π) + π² **Where:** - ρ = default correlation between the two credit assets = 0.45 - π = probability of default of each firm = 0.0285 **Calculation:** Φ = 0.45 * 0.0285 * (1 − 0.0285) + 0.0285² = 0.45 * 0.0285 * 0.9715 + 0.00081225 = 0.45 * 0.02768775 + 0.00081225 = 0.0124594875 + 0.00081225 = 0.0132717375 = 1.33% **Why other options are incorrect:** - **A (1.25%)**: Result obtained by ignoring the addition of π² term in the formula - **C (2.65%)**: Incorrect result obtained by multiplying the correct answer by 2 - **D (2.77%)**: Result obtained by ignoring both default correlation and π² term This calculation demonstrates how default correlation affects joint default probabilities in credit risk modeling.
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The fixed-income trading desk of an investment bank uses several methodologies to estimate joint default probabilities of counterparties. The manager in charge of the desk asks an analyst to calculate the joint default probability of two firms (credits) over a 1-year period using the single-factor model. The portfolio consists of two firms (credits) that have the same beta to the market factor. The probability of default of each firm is 2.85%. If the analyst applies a default correlation between this pair of firms of 0.45, what is the joint default probability of the firms using the single-factor model?
A
1.25%
B
1.33%
C
2.65%
D
2.77%
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