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Answer: 2.74%
## Explanation In the Merton model, the risk-neutral default probability is calculated using the distance to default. The distance to default (DD) is given as 1.9191 standard deviations. ### Key Formula: The risk-neutral default probability = 1 - N(DD) ### Calculation: - N(1.9191) = 0.9724 (given) - Default probability = 1 - 0.9724 = 0.0276 = 2.76% This rounds to approximately **2.74%**, which matches option A. ### Why this is correct: - In the Merton model, the distance to default represents how many standard deviations the firm's assets are above the default point - N(DD) gives the probability that the firm will NOT default under the risk-neutral measure - Therefore, 1 - N(DD) gives the risk-neutral default probability ### Verification with given data: - Market value of assets (V) = USD 130 million - Face value of debt (K) = USD 100 million - Time to maturity (T) = 5 years - Risk-free rate (r) = 25% - Volatility (σ) = 30% The distance to default calculation would confirm the given DD = 1.9191, leading to the default probability of approximately 2.74%.
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The capital structure of HighGear Corporation consists of two parts: one 5-year zero-coupon bond with a face value of USD 100 million and the rest is equity. The current market value of the firm's asset (MVA) is USD 130 million and the risk-free rate is 25%. The firm's assets have an annual volatility of 30%. Assume that firm value is log-normally distributed with constant volatility. The firm's risk management division estimates the distance to default (in terms of number of standard deviations) using the Merton Model, Given the distance to default, the estimated risk-neutral default probability is (N(1.9191) = 0.9724):
A
2.74%
B
12.78%
C
12.79%
D
30.56%
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