Explanation

To calculate the probability of survival in the first year followed by default in the second year, we use the hazard rate model.

Given:

• Hazard rate (λ) = 0.1 per year

Step 1: Calculate probability of survival for 1 year The probability of survival for time t is: P(survival) = e^(-λt)

For t = 1 year: P(survival for 1 year) = e^(-0.1 × 1) = e^(-0.1) ≈ 0.904837

Step 2: Calculate probability of default in the second year The probability of default in year 2 given survival through year 1 is equal to the hazard rate: P(default in year 2 | survival through year 1) = λ = 0.1

Step 3: Calculate joint probability P(survival in year 1 AND default in year 2) = P(survival for 1 year) × P(default in year 2 | survival through year 1) = 0.904837 × 0.1 = 0.0904837 ≈ 9.048%

Step 4: Compare with options

• 8.61% is too low
• 9.00% is close but slightly lower than our calculation
• 9.52% is higher than our calculation
• 19.03% is much too high

Our calculated value of 9.048% is closest to 8.61% among the given options.

Note: The exact calculation using the continuous time model gives: P(survival in year 1 AND default in year 2) = e^(-λ) × (1 - e^(-λ)) = e^(-0.1) × (1 - e^(-0.1)) ≈ 0.904837 × 0.095163 = 0.086106 ≈ 8.61%

This confirms that Option A (8.61%) is the correct answer.