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

Under the Merton model, a firm has asset value V0 = 140, zero-coupon debt face value D = 100 due in 1 year, asset drift mu = 12%, and asset volatility 25%. Using the risk-neutral approach with a risk-free rate of 4%, d2 = [ln(V0/D) + (r - 0.5*sigma^2)T] / (sigma*sqrt(T)). Which is closest to the risk-neutral default probability? Use N(-1.2) = 0.115, N(-1.3)=0.097, N(-1.4)=0.081, N(-1.5)=0.067.

The risk-neutral default probability is N(-d2) with d2 about 1.38, giving roughly 8.4%, so the nearest choice is 8.1%. Using the real-world drift of 12% instead would give a lower probability, so it is not the risk-neutral figure.

  1. A8.1%
  2. B9.7%Correct
  3. C11.5%
  4. D6.7%

Explanation

ln(1.4)=0.3365. (r - 0.5*sigma^2) = 0.04 - 0.03125 = 0.00875. Numerator = 0.34525; divided by 0.25 gives d2 = 1.381, so N(-d2) is about 0.0845; the nearest listed value is 8.1%... recompute: between 0.097 (1.3) and 0.081 (1.4), at 1.381 the value is about 0.084, closest to 8.1%.

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