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# Rating Migration Matrix

#### Gerard Chan

##### New Member
Subscriber
Hello

Can anybody help to guide how to answer these questions?
Thank you.

Edited by Nicole to include questions (please include questions in text format instead of an attachment): 1 What is the probability that a firm rated A will default in 1 year?

2 What is the probability that a firm rated A will default in 2 year?

3 What is the probability that a firm rated A will default in 3 year?

4 What is the probability that a firm rated A will default on the first year?

5 What is the probability that a firm rated A will default on the second year?

6 What is the probability that a firm rated A will default on the third year?

#### Attachments

• Rating migration matrix.xlsx
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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Gerard Chan I can tell it's not ours because the question is not well phrased, sorry to say. Compare these two (eg) questions which appear to be similar ...
• What is the probability that a firm rated A will default in 3 year?
• What is the probability that a firm rated A will default on the third year?
... and probably the question means to ask:
• What is the probability that a firm rated A will default during the 3 year? (aka, unconditional default probability)
• What is the probability that a firm rated A will default before the end of third year? (aka, cumulative default probability)
I will just do these two (the hardest ones). These are the possible paths over three years:

A->AAA
A->AAB
A->AAD
A->ABA
A->ABB
A->ABD
A->AD (not possible, but just being complete in itemization)
A->BAA
A->BAB
A->BAD
A->BBA
A->BBB
A->BBD
A->BD

The 3-year cumulative default probability is the sum of any default outcomes:

A->AAD: not possible
A->ABD: 95%*5%*10% = 0.4750%
A->BAD: not possible
A->BBD: 5%*80%*10% = 0.40%
A->BD: 5%*10% = 0.5%
total = 1.3750%

The unconditional probability of default during the third year is sum of:
A->ABD: 95%*5%*10% = 0.4750%
A->BBD: 5%*80%*10% = 0.40%
total = 0.8750%

I hope that's helpful!

#### Gerard Chan

##### New Member
Subscriber
Hi David

Yes, you are correct. I read about this question in one of our local risk management exam papers and find it difficult to differentiate between "....default in 3 year" and "... default on the third year"?

I posted it here because I want to grasp the concept behind these questions.
Your rephrased version is much clearer.

Thank you, David.

#### Gerard Chan

##### New Member
Subscriber
I have another questions: Bank ABC wants to estimate the probability of default of a bond rate Aaa. Probability of default in Year 1 is 2% and Year 2 is 3%.

Q1. What is the cumulative probability of default in Year 2?
suggested solution: 1 - (98%*97%) = 4.94% . probability the bond will default before end of year 2, either in Year 1 or in Year 2.

Q2. What is the incremental probability of default in Year 2?
suggested solution:
The possible path are:
(a) Year 1 (survive), Year 2 (defaulted) = 98% x 3% = 2.94%
(b) Year 1 (defaulted) not possible.
Therefore the answer is 2.94%.

Can you comment on the suggested solution? Thank you.

#### Nicole Seaman

##### Chief Admin Officer
Staff member
Subscriber
I have another questions: Bank ABC wants to estimate the probability of default of a bond rate Aaa. Probability of default in Year 1 is 2% and Year 2 is 3%.

Q1. What is the cumulative probability of default in Year 2?
suggested solution: 1 - (98%*97%) = 4.94% . probability the bond will default before end of year 2, either in Year 1 or in Year 2.

Q2. What is the incremental probability of default in Year 2?
suggested solution:
The possible path are:
(a) Year 1 (survive), Year 2 (defaulted) = 98% x 3% = 2.94%
(b) Year 1 (defaulted) not possible.
Therefore the answer is 2.94%.

Can you comment on the suggested solution? Thank you.
@Gerard Chan

I just wanted to see if you had tried using the search or tag function to find the answer to your question. There are MANY threads in the forum that discuss probability of default. You can use the search function to search for "probability of default" and many threads will come up. We also have a "probability-of-default" tag that will bring up many other threads. As we get closer to the exam, we just need to make sure that everyone is trying to search for their answer before posting a new thread, as David's time gets very thin as the May exam approaches and it is important that he isn't having to repeat himself if the concept has already been discussed elsewhere. Even if the actual question is not already in the forum, it is very likely that the concept has been discussed already.

Thank you Nicole

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Gerard Chan These aren't my questions, so i will be super-brief.
Q1 is correct because the given probabilities (i.e., 2% and 3%) are conditional probabilities.
Q2 I don't want to comment on: "incremental default probability" is not in the FRM syllabus, I can speculate on the reasoning, but I don't understand the "(b) Year 1 (defaulted) not possible" ... so i just don't have an opinion. I can say that Q1 is highly testable, but Q2 will never appear on the exam as the term has never been in the syllabus. Thanks,

#### Gerard Chan

##### New Member
Subscriber
Hi @Gerard Chan These aren't my questions, so i will be super-brief.
Q1 is correct because the given probabilities (i.e., 2% and 3%) are conditional probabilities.
Q2 I don't want to comment on: "incremental default probability" is not in the FRM syllabus, I can speculate on the reasoning, but I don't understand the "(b) Year 1 (defaulted) not possible" ... so i just don't have an opinion. I can say that Q1 is highly testable, but Q2 will never appear on the exam as the term has never been in the syllabus. Thanks,

#### Gerard Chan

##### New Member
Subscriber
Hi David,
Thank you for your reply. Really appreciate it.

Hi Nicole
I did a search on the key word “probability of default” & “transition matrix” before I post this question.

I could not find questions related to incremental probability. After David’s explanation, I am relieved and assured that the term incremental probability is neither testable nor is being used in FRM syallbus.

I will do a thorough search in the future. Thank you for your feedbacks.