Garp 2015 Practice Exam Question #10

[Lina]

An analyst estimates that hazard rate for company is 0.1 per year. Tho probability of survival in the first year followed by a default in the second year is closest to:
a. 8.61%
b. 9.00%
c. 9.52%
d. 19.03%

Correct answer: a

Garp writes: The conditional one year default probability given that the firm survived the first year is the difference betrween the two year cumulative probability of default and the one year probablity: 0.18127-0.09516 = 0.08611

My question: should it not be (two year cumulative PD - one year PD)/ survival probability = (0.18127-0.09516)/ 0.90484 = 0.09517 ??

 

[David Harper CFA FRM]

Hi @Lina

Yes, absolutely! Excellent! Here, 8.611% is the unconditional probability of default; i.e., the probability of default during the second year as seen from time 0.
The answer to the requested "The probability of survival in the first year followed by a default in the second year is closest to:" is given by the conditional probability of default in the second year; i.e., the probability of default in the second year conditional on survival until the end of the first year. This is given by:
unconditional PD in year 2 / Prob survival to end of year 1 = [1-exp(-.1*2)] - [1-exp(-.1*1)] / exp(-.1*1) = 9.5163%

 

[Lina]

Hi, David,
thank you for confirming that my solution is right. But... Which answer should we choose, when we get that kind of question at the exam? GARP wrote, that the correct answer is a. To they mind the same at the exam as well, or is it a mistake in GARP 2015 Practice Exam?

 

[David Harper CFA FRM]

Hi @Lina it's a mistake in the practice exam! never choose the incorrect answer ... the actual exam has a process for impeaching incorrect answers. If this question came up on the actual exam, many people would notice and it would be discarded. Nevermind the actual exam should have a higher quality standard. In truth, the practice exams do contain errors (just like my practice questions contain errors!). Thanks,

 

[David Harper CFA FRM]

Just one more note (based on an observation by @filip313 at https://forum.bionicturtle.com/threads/p2-t6-307-hazard-rate-malz-section-7-2.6932/#post-35562). My revised thinking is that the question is flawed due to the incongruity between the question and answer. Given a 10.0% hazard rate:

• For any given year, the conditional probability of default (i.e., conditional on survival up to the beginning of that year) is 9.52%, per @Lina point above; this will just be true for all years. For example, the conditional PD in year 5 = (39.35% - 32.97%)/(1-32.97%) = 9.52%. Such is the elegance of of the hazard rate: it is already the instantaneous PD.
• The joint probability of survival in the first year and default in the second year = 8.61% = (1-9.52%)*9.52%; this is also called the unconditional probability of a default in the second year. For year (x), the unconditional probability of default equals the probability of survival up to the end of the prior year multiplied by the conditional PD (in this case, 9.52%)

So it's the difference between a joint and conditional probability such that:

• The question "The probability of survival in the first year followed by a default in the second year ..." to me implies a joint/unconditional; hence my agreement with @Lina
• The answer given is correct for the conditional PD.