GARP Part 2 Questions 76 and 33 (garp16-p2-76) (garp16-p2-33)

[equanimity]

Question 76 on the 2016 FRM Part II practice exam:

A major regional bank has determined that a counterparty has a constant default probability of 5.5% per year. What is the probability of this counterparty defaulting in the fourth year?

The probability of default in year 4 = (1–0.055)(1–0.055)(1–0.055)(0.055) = 4.64%.

Question 33 on the 2016 FRM part II practice exam is:

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:

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%.

MY QUESTION:

These two questions seem to be asking the same thing, namely:

"What is the probability of this counterparty defaulting in the fourth year?"
"The probability of survival in the first year followed by a default in the second year..."

But, why are the formulas different for each solution? Thanks!

 

[Nicole Seaman]

Question 76 on the 2016 FRM Part II practice exam:

A major regional bank has determined that a counterparty has a constant default probability of 5.5% per year. What is the probability of this counterparty defaulting in the fourth year?

The probability of default in year 4 = (1–0.055)(1–0.055)(1–0.055)(0.055) = 4.64%.

Question 33 on the 2016 FRM part II practice exam is:

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:

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%.

MY QUESTION:

These two questions seem to be asking the same thing, namely:

"What is the probability of this counterparty defaulting in the fourth year?"
"The probability of survival in the first year followed by a default in the second year..."

But, why are the formulas different for each solution? Thanks!

Hello @equanimity

I'm sure that David or one of our members can answer your question, but I want to point out that Question 33 has been discussed in detail in the paid section of our forum in these threads:

• https://forum.bionicturtle.com/threads/garp-2016-practice-question-33-garp16-p2-33-garp16-p2-6.9988/
  
• https://forum.bionicturtle.com/thre...-true-or-what-garp16-p2-33-garp15-p2-10.9592/

I just wanted to point this out so our members know that there are other threads where David has discussed this specific question, as there are many GARP practice questions being discussed in the forum.

Thank you,

Nicole

 

[equanimity]

Thanks, Nicole. Now that I've read through all of the thread you provided and other associated threads, my question is now:

When do you use each of the following? Are there certain trigger words we should be looking for?

[1– exp(-λ * time_2)] - [1– exp(-λ * time_1)]

vs.

( [1– exp(-λ * time_2)] - [1– exp(-λ * time_1)] ) / ( 1 - [1– exp(-λ * time_1)] )

Thanks!

 

[Matthew Graves]

The first equation is giving you the probability of default in year 2. The second is giving you the probability of default in year 2 conditional on there being no default in year 1.

Imagine you are given a two year period and told there is a certain probability of default during this time. The first equation is effectively saying what is the probability that the default occurred in Year 2. The second is saying what's the probability of the default occurring Year 2 given that it didn't occur in Year 1.

I don't think I've ever seen a question where the answer was from equation 1. It's always conditional on surviving up to the point of default as I recall.

 

[Arnaudc]

There are both in practice exam.
But Matthew made a point. The conditional probability asks you to find the probability of default in the future given default has not occured yet.

When the question refers to default inyear 2,3,4,.. it usually asks you to find the probability of default considering all the different probabilities on this "path" (default or not, while conditional explicitely assume no default occured). This assumption is also factored into the conditional probability in the denominator.

But I must admit I am in the same situation as you as GARP is somewhat unclear in the question as to what do they expect...

 

[equanimity]

This makes perfect sense now. Thank you to Matthew Graves and Arnaudc.

 

[David Harper CFA FRM]

Just to add to the helpful comments already here:

• Question 76 is asking for the unconditional PD in year 4, which can be also answered by subtracting th 3-year cumulative PD from the 4-year cumulative PD: (1-5.5%)^4 - (1-5.5%)^3 = 4.64%. I like this calculation because to me it is somewhat intuitive: as the unconditional probability is the probability "from the perspective of today," if the outcome at the end of four years is a default, either it happened in the fourth year or it happened earlier. We have requested the term "unconditional" for this because that's Hull's term, but it can also be thought of as a joint probability (ie, the joint probability of survive 3 years plus default on the 4th year).
• Question 33 (as noted in @Nicole Seaman 's links) is asking for a conditional PD and, clearly due to the prevalence of the term, this question should help the user by specifying that it wants "conditional"
• The other assumption that (in my opinion, at least) requires specification in these questions is the issue of continuous versus discrete. Notice that question 76 assumes the 5.5% is discrete per annum, while question 33 assumes continuous. The term "hazard rate" is a conditional PD but it also connotes (if not denote) continuous conditional PD. But Q 76 does not say hazard rate, so notice how we are not there solving Q 76 with 1-exp(5.5%*4) - [1-exp(5.5%*3)] = 4.5375%; it has also been my expressed opinion that continuous/discrete frequency should be explicitly specified.