Explanation

For a Poisson process with an average of 6 defaults per year, we need to find the probability that the next default occurs within one month.

Step-by-step calculation:

1. Convert annual rate to monthly rate:

   • Annual default rate λ = 6 defaults/year
   • Monthly default rate = 6/12 = 0.5 defaults/month

2. Time between defaults in Poisson process:

   • In a Poisson process, the time between events follows an exponential distribution
   • The probability that the time to next default is less than or equal to t is: $$P(T \leq t) = 1 - e^{-\lambda t}$$

3. Apply to our case:

   • λ = 0.5 defaults/month
   • t = 1 month
   • $$P(T \leq 1) = 1 - e^{-0.5 \times 1} = 1 - e^{-0.5}$$

4. Calculation:

   $$1 - e^{-0.5} = 1 - 0.6065 = 0.3935 = 39.35\%$$

Verification:

• The exponential distribution parameter is λ = 0.5
• The probability density function is f(t) = 0.5e^{-0.5t}
• Integrating from 0 to 1 gives the cumulative probability

Therefore, the correct answer is C. 39.35%.