Rating-Based Term Structure Models
Often, we don't just care about "Default vs No Default". We care about Downgrades. A portfolio manager holding AAA bonds loses money if they get downgraded to BBB (even without default).
Transition Matrix (Migration Matrix)
We model credit quality as a "State".
- States: AAA, AA, A, BBB, ... Default.
- We use a Markov Chain to define the probability of moving from State A to State B in one year.
Example Matrix (Simplified)
| From \ To | AAA | AA | BBB | Default |
|---|---|---|---|---|
| AAA | 90% | 9% | 1% | 0% |
| BBB | 1% | 4% | 90% | 5% |
- A BBB firm has a 5% chance of defaulting.
- An AAA firm has a 0% chance of defaulting directly, but a 9% chance of becoming AA.
Credit Term Structure
Using this matrix, we can calculate the Cumulative Probability of Default for any term (1 year, 5 years, 10 years).
- For 1 year: Use the matrix once.
- For 2 years: Multiply the matrix by itself (Matrix^2).
Pricing Bonds
Since different ratings trade at different Yield Spreads, we can price a bond by taking the expected value of its future price across all possible rating states.
Note
Jarrow-Lando-Turnbull (JLT): This is a famous model that combines Rating Transitions with Intensity Models to build a full term structure of credit spreads.
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