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FRM Part II · FRM Exam Part II · Estimating Default Probabilities

Under the Merton model, a firm's current asset value is 120, its zero-coupon debt has face value 100 due in 1 year, asset volatility is 25%, and the expected asset drift is ignored by using the risk-neutral measure with a risk-free rate of 5% (continuous). The distance to default measure d2 = [ln(V/D) + (r - 0.5σ²)T] / (σ√T). Which value of the risk-neutral default probability N(-d2) is closest to the correct one? (N(0.9)=0.816, N(1.0)=0.841, N(0.8)=0.788, N(0.7)=0.758)

The risk-neutral default probability is about 21%, because d2 is roughly 0.80 and N(-0.80) is about 0.21. The calculation uses ln(120/100) plus the drift of 0.01875, divided by volatility of 0.25.

  1. AAbout 18.4%Correct
  2. BAbout 21.2%
  3. CAbout 15.9%
  4. DAbout 24.2%

Explanation

ln(120/100)=0.1823. r - 0.5σ² = 0.05 - 0.03125 = 0.01875. Numerator = 0.20105; divide by 0.25 gives d2 = 0.804, approximately 0.8 to 0.9, so N(d2) is about 0.789 to 0.816. Using interpolation N(0.804) is about 0.789, giving default probability N(-d2) about 21.1%. Recheck: the closest option must therefore be 21.2%.

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