Why Default Probabilities Do Not Determine Credit Migration Matrices

Summary

The document asks whether default probabilities alone can determine credit spreads or a rating transition matrix. Its central answer is that a single set of default probabilities is insufficient to identify a full migration matrix: many combinations of transitions among surviving rating categories can produce the same probabilities of reaching default. A small example contrasts the number of unknown matrix entries with the limited default-probability information available.

With default probabilities across multiple horizons, a matrix can instead be fitted so its powers align with the observed term structure, for example through least squares. The answers also suggest practical alternatives: assign securities to ratings using probability-of-default estimates and use a published rating transition matrix, or interpolate transition probabilities from default estimates. These alternatives add assumptions or external information; they do not make the original inverse problem uniquely determined. The discussion offers no derivation of credit spreads and no validation of the proposed interpolation method, so those outputs require additional modeling choices and evidence.

Key ideas

• Default probabilities at a given horizon do not uniquely identify transitions among non-default rating states.
• A term structure of default probabilities can constrain matrix fitting across multiple horizons.
• Published rating transition matrices can supply information absent from default probabilities alone.
• Interpolation is a practical proposal, but it adds assumptions and is not validated in the discussion.

Tags

# Deriving credit spreads or migration matrices from prob of default


# Deriving credit spreads or migration matrices from prob of default












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 by Brian B (score 5)

https://quant.stackexchange.com/a/15725

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 by jaamor (score 4)

https://quant.stackexchange.com/a/15728

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).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.