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%).