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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