Answer-first summary for fast verification
Answer: P; R; Q
## Explanation The distance to default (DD) in the Merton model is calculated using the formula: \[ DD = \frac{\ln(V/D) + (r - \frac{1}{2}\sigma_V^2)T}{\sigma_V\sqrt{T}} \] However, for relative ranking purposes with the same time horizon (T=1) and assuming risk-free rate is the same for all companies, we can use the simplified approximation: \[ DD \approx \frac{V - D}{\sigma_V \cdot V} \] Let's calculate for each company: **Company P:** - V = 100, D = 60, σ_V = 10% - DD_P ≈ (100 - 60) / (0.10 × 100) = 40 / 10 = 4.0 **Company Q:** - V = 150, D = 100, σ_V = 7% - DD_Q ≈ (150 - 100) / (0.07 × 150) = 50 / 10.5 = 4.76 **Company R:** - V = 250, D = 160, σ_V = 8% - DD_R ≈ (250 - 160) / (0.08 × 250) = 90 / 20 = 4.5 **Ranking by Distance to Default (lower DD = higher default probability):** - P: 4.0 (highest default probability) - R: 4.5 - Q: 4.76 (lowest default probability) Therefore, the ranking from most likely to least likely to default is: **P; R; Q**
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Q-34. A credit manager in the counterparty risk division of a large bank uses a simplified version of the Merton model to monitor the relative vulnerability of its largest counterparties to changes in their valuation and financial conditions. In order to assess the risk of default of three particular counterparties, the manager calculates the distance to default assuming a 1-year horizon (t = 1). The counterparties: Company P, Company Q, and Company R, belong to the same industry. Selected information on the companies is provided in the table:
| Company | P | Q | R |
|---|---|---|---|
| Market value of asset (EUR million) | 100 | 150 | 250 |
| Face value of debt (EUR million) | 60 | 100 | 160 |
| Annual volatility of asset values | 10.0% | 7.0% | 8.0% |
Using the information above with the assumption that short-term debt is the only liability for each company, and the approximation formula of the distance to default, what is the correct ranking of the counterparties, from most likely to least likely to default?
A
P; R; Q
B
Q; P; R
C
Q; R; P
D
R; Q; P
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