Deriving credit spreads or migration matrices from prob of default

Question

How do I derive credit migration/transition matrices or spreads from default probability? May you please provide references, or do you know what type of articles or authors to find?

Answer 1

You cannot do it.

It is an under-determined problem. That is to say, a whole multitude (subspace of $\mathbb{R}^{N\times N}$) of migration matrices will agree with any given table of default probabilities.

Say you want to find a transition matrix for 2 states (IG, HY) plus default

$$\left(\begin{matrix} p_{11} & p_{12} & p_{1D} \\ p_{21} & p_{22} & p_{2D} \\ 0 & 0 & 1 \end{matrix}\right)$$

and your default probabilities are $(p_{1D},\ p_{2D})$. You have 6 quantities to find in your transition matrix and only two quantities to populate them from (and, by the way, those two quantities are actually entries in the matrix). Topologically, there is no solution.

Now, if you have a term structure of default probabilities, then you can work with a least-squares fit or something to ensure that powers of the transition matrix agree with particular tenors on that term structure.

Answer 2

Actually, there is a practical way to do it.

You can use you PoD estimates to assign a credit rating to your securities and then use a published transition matrix for your purposes.

Or you can estimate transition probabilities by linear interpolation based on the PoD values that you have.

Here is a publication containing transition matrices from Moody's and S&P (go to page 3).