Answer: 2.8%

## Explanation In the single-factor credit risk model (also known as the Merton model or Vasicek model), the conditional default probability is calculated using the formula: \[ P[D|m] = N\left( \frac{N^{-1}(PD) - \beta \cdot m}{\sqrt{1 - \beta^2}} \right) \] Where: - \( PD \) = unconditional default probability = 1% = 0.01 - \( \beta \) = beta coefficient = 0.40 - \( m \) = economic factor = -1.0 - \( N() \) = cumulative standard normal distribution function - \( N^{-1}() \) = inverse cumulative standard normal distribution function **Step-by-step calculation:** 1. Calculate \( N^{-1}(PD) = N^{-1}(0.01) \) - \( N^{-1}(0.01) = -2.3263 \) (from standard normal table) 2. Calculate numerator: \( N^{-1}(PD) - \beta \cdot m \) - \( -2.3263 - (0.40 \times -1.0) \) - \( -2.3263 - (-0.40) \) - \( -2.3263 + 0.40 = -1.9263 \) 3. Calculate denominator: \( \sqrt{1 - \beta^2} \) - \( \sqrt{1 - 0.40^2} = \sqrt{1 - 0.16} = \sqrt{0.84} = 0.9165 \) 4. Calculate the argument: \( \frac{-1.9263}{0.9165} = -2.1016 \) 5. Calculate conditional probability: \( N(-2.1016) \) - From standard normal table: \( N(-2.10) ≈ 0.0179 \) or 1.79% - More precisely: \( N(-2.1016) ≈ 0.0178 \) or 1.78% **Interpretation:** The conditional default probability of approximately 1.78% (rounded to 1.8% in the options) represents how the firm's default risk increases during an economic downturn. The beta of 0.40 indicates moderate sensitivity to economic conditions, and the negative m value (-1.0) represents adverse economic conditions. **Answer: D (2.8%)** Wait, let me recalculate more precisely: - \( N^{-1}(0.01) = -2.3263 \) - Numerator: \( -2.3263 - (0.40 \times -1.0) = -2.3263 + 0.40 = -1.9263 \) - Denominator: \( \sqrt{1 - 0.40^2} = \sqrt{0.84} = 0.9165 \) - Argument: \( -1.9263 / 0.9165 = -2.1016 \) - \( N(-2.1016) = 0.0178 \) or 1.78% Actually, looking at the options again, 1.78% would correspond to option B (1.8%), not D (2.8%). Let me verify with more precise calculation: Using exact values: - \( N^{-1}(0.01) = -2.326347874 \) - \( -2.326347874 - (0.40 \times -1.0) = -1.926347874 \) - \( \sqrt{1 - 0.16} = \sqrt{0.84} = 0.916515139 \) - \( -1.926347874 / 0.916515139 = -2.1016 \) - \( N(-2.1016) = 0.0178 \) or 1.78% This confirms the answer should be **B (1.8%)**. **Correction:** The correct answer is **B (1.8%)**.

Author: LeetQuiz .