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Calculate default probabilty (2007 Practice test 1 Question 28)

Thread starter #1
Hi David,

I'm not sure how to read the transition table in question 28.

The question states:

For a company starting with rating B in year 1, calculate the default (rating D) probability for year 2.
Starting Ending
A 0.98 0.02 0.00 0.00
B 0.12 0.86 0.02 0.00
C 0.00 0.05 0.75 0.20
D 0.00 0.00 0.00 1.00

I've also attached a image of the table.

Can you explain how to read the table and what we are trying to find out? David, I think you may have a presented this kind problem somewhere on the site, but I can't seem to find it.

Thanks in advance.


David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi John,

I have an intro to the transition matrix here (and a related practice question here).

The row starting B gives probabilities (mutually exclusive, cumulatively exhaustive) that you will end up at column rating at end of first year.

So, you have to use the matrix twice: end of first year, then end of second year. You imagine the credit migrating from B at T0 to D at T2. The question is: how many ways can you get to column (D)? There are four ways:

1. start B, migrate to A (end of year 1), migrate to D (end of year 2).
2. start B, remain B (end of year 1), migrate to D (end of year 2)
3. start B, migrate to C (end of year 1), migrate to D (end of year 2)
4. start B, migrate to D (default)

So, then you have to compute the probabilities
1. 0.12 (prob B migrates to A) * 0 (prob A migrates to D) = 0
2. 0.86 (prob B remains) * 0 (prob B migrates to D) = 0
3. 0.02 (prob B migrates to C) * 0.2 (prob C migrates to D) = 0.004
4. 0 (prob B migrates to D) = 0

And add them up because they are mutually exclusive, = 0.004

Note the migration matrix is great practice for Gujarati's terms:

The original cells are marginal/unconditional probabilities; e.g., prob B migrates to C is 2%
Computing the 0.004 above is a joint probability: P(B migrates to C, C migrates to D) is .4%
The same joint probability is the product of two conditional probabilities: P(B migrates to C, C migrates to D) = P(C | start at B)*P(D | start at C)

Thread starter #3
Hi David,

After migrating to the first state, you migrate to D. It looks like matrix multiplication of Row B X Column D. Is that a correct way of interpreting the calculations?