Hello, I found out two different methods to calculate the probability: Here you are the question and answer:
Question 1:
An analyst estimates that the hazard rate for a company is 0.16 per year. The probability of survival in the first year followed by a default in the second year is closest to: a. 11.62%. b. 13.63%. c. 14.79%. d. 27.39%. Correct answer: c Explanation: The probability that the firm defaults in the second year is conditional on its surviving the first year. Using λ to represent the given hazard rate, we can calculate the cumulative probability of default in the first year using the formula 1– exp(–1*λ) = 1 – exp(-0.16) = 0.14786. Thus, probability of survival in the first year = 1 – 0.14786 = 0.85214. Then, the cumulative probability that the firm defaults in the second year = 1 – exp(–2*λ) = 1 – exp(-2*0.16) = 0.27385, and the conditional one year default probability given that the firm survived the first year is the difference between the two year cumulative probability of default and the one year probability divided by the probability of survival in the first year = (0.27385 – 0.14786)/0.85214 = 0.14785 = 14.79%.
The explanation is from 2016 which is logic as for me. However, I found out a different answer for the same question in 2017 as:
Question 2:
A risk analyst estimates that the hazard rate for a company is 0.10 per year. What is the company’s joint probability of survival in the first year followed by a default in the second year? A. 8.6% B. 9.5% C. 18.1% D. 22.1% Correct answer: A Explanation: A is correct. The probability that the firm defaults in the second year is conditional on its surviving the first year. Using λ to represent the given hazard rate, we can calculate the cumulative probability of default in the first year using the formula 1– exp(–1*λ) = 1 – exp(-0.10) = 0.0952. Also, the cumulative probability of default in the year 2 = 1 – exp(–2*λ) = 1 – exp(-2*0.10) = 0.1813. Therefore, the joint default probability of defaulting in the second year having survived the first year is the difference between the cumulative probability of default in year 2 and the cumulative probability of default in year 1 = 0.1813 – 0.0952 = 0.0861 = 8.6%.
Could you help to clarify which method is correct please?
Many thanks