详细解题步骤

问题如下：

Question A credit risk analyst at a wholesale bank is estimating annual default probabilities of a 5-year loan that has just been extended to a corporate borrower. The analyst determines from rating agency data that the 5-year cumulative default probability of bonds from this borrower with identical terms and seniority is 6.2%, and uses this information to calculate the 5-year survival rate for the borrower. If the borrower’s average hazard rate for the first 4 years of the loan is 1.1%, what is the unconditional default probability of the borrower during year 5 of the loan?

选项：

A.

1.71%

B.

1.80%

C.

1.90%

D.

1.98%

解释：

Explanation:

C is correct. The unconditional default probability between the end of year 4 and the end of year 5 is calculated as

exp(-ℎ̅ 4 ∗ 4) − exp(-ℎ̅ 5 ∗ 5)

where ℎ̅4 and ℎ̅5 are the average hazard rates between today and end-of-year 4 and end-of-year 5, respectively. The term

exp(-ℎ̅5 ∗ 5)

in the equation above represents the probability of survival (or survival rate) to the end of year 5. This is equal to one minus the cumulative default probability to the end of year 5, given as 6.2%. Therefore, the 5-year survival rate is

1 − 0.062 = 0.938

and the unconditional default probability during the fifth year of the loan is

exp(−0.011 ∗ 4) − 0.938 = 0.956954 − 0.93800 = 0.01895 or 1.895%

A is incorrect. This incorrectly calculates the survival rate as exp(−0.062) = 0.939883, and uses this to calculate the unconditional default probability = 0.956954 – 0.939883 = 0.017071 = 1.71%.

B is incorrect. This incorrectly calculates the unconditional default probability as 0.062 − 4 ∗ 0.011.

D is incorrect. This is the conditional default probability during the fifth year, or 0.01895/exp(0.011 ∗ 4).

Section: Valuation and Risk Models

Learning Objective: Define and use the hazard rate to calculate the unconditional default probability of a credit asset

Reference: Global Association of Risk Professionals. Valuation and Risk Models. New York, NY: Pearson, 2022. Chapter 4. External and Internal Credit Ratings