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# Transition matrix cumulative pd

#### lowhueyyi

##### New Member
Hi @David Harper CFA FRM , suppose that matrix A is a transition matrix (I.e one with Markov property). Suppose we have calculated two and three year transition matrix by taking the square and cube of matrix A. How do we get the cumulative PD from the example in this link:

https://www.bionicturtle.com/images/forum/e1.41b.png

I have read through certain papers where they add up the confusing term “marginal pd” which I wish to avoid using to get cumulative pd. But from what I understand, transition matrix is a conditional pd. Why does adding up conditional pd give us a cumulative pd? Adding conditional pd might results in probability exceeding 1.

Thanks!

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Hi @David Harper CFA FRM , suppose that matrix A is a transition matrix (I.e one with Markov property). Suppose we have calculated two and three year transition matrix by taking the square and cube of matrix A. How do we get the cumulative PD from the example in this link:

https://www.bionicturtle.com/images/forum/e1.41b.png

I have read through certain papers where they add up the confusing term “marginal pd” which I wish to avoid using to get cumulative pd. But from what I understand, transition matrix is a conditional pd. Why does adding up conditional pd give us a cumulative pd? Adding conditional pd might results in probability exceeding 1.

Thanks!
Hello @lowhueyyi

Are you referencing a specific practice question? If so, I would like to move your post so we can keep similar questions in the same threads to retain organization in the forum. I just did a quick search of the forum and there are a lot of threads discussing transition matrix so you may be able to find the answer to your question without waiting if you use the search function here: https://www.bionicturtle.com/forum/search/?type=post. Also using the tag, "transition-matrix" brings up a number of threads discussing this.

Thank you,
Nicole

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @lowhueyyi The typical transition/migration matrix, such as the copy you've linked above, does indeed contain single-period conditional probabilities; most often, one-year conditional probabilities. If we cube the single-period matrix, as above, then T^3 is a three-year matrix. Further, the illustrated Prob(A defaults) = 1.375% can be called a 3-year cumulative default probability; this is demonstrated by the fact that 1.375% represents the three possible paths that reach default within three years. Now, cubing the matrix requires us to assume this is a Markov chain; aka, Markovian see https://en.wikipedia.org/wiki/Markov_chain ... which boils down to independence: each year (e..g, in the 2nd and 3rd year), the conditional probabilities are "memoryless" or have no memory of the prior path.

Alternative, we can do a typical thing to retrieve the 3-year cumulative PD for, say bond B, which has a one-year PD of 10.0%. When PD = 10%, we often say the 3-year cumulative PD is given by 1 - (1 - 10%)^3 = 27.1%. Notice this is different than 24.5% shown under the Markovian-assumed T^3. That's because our 27.1% assumes i.i.d. instead of just independence: it assumes also "identical," that each year the bond has an identical (conditional) probability. I haven't checked the theory, so don't quote me, but i think it's true that i.i.d. implies Markovian but Markovian does not imply i.i.d. But you can see that, in my opinion, we have at least two valid ways to retrieve a 3-year cumulative PD. One is shown by T^3 and assumes Markovian chain; the other is more rigid (less realistic because it ignores interim migrations) and assumes i.i.d.

Please note: with respect to your question "Why does adding up conditional pd give us a cumulative pd?" ... the summation to 1.375% shown in the red cell is adding the probability of each possible path (in this case, only 3 paths) that ends in a default before the end of the 3rd year. If we added the probabilities of all possible paths (regardless of outcome), we should get exactly 100.0%. So here we are just adding the subset (of all possible paths) that includes the three outcomes that result in default. As shown, it matches the 1.375% that is produced by cubing the matrix. I hope that's helpful,

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