Credit migration includes upgrades and downgrades as well as default. Follow three annual paths to distinguish cumulative default probability from conditional second-year risk.
Credit migration
Credit migration is a change in a borrower's or debt instrument's assessed credit quality over time, often represented as movement between rating categories. It includes upgrades and downgrades; default is a separate destination in a migration model. A bond can lose market value after a downgrade while its issuer continues to make every contractual payment.
This makes migration risk broader than default risk. A credit spread can widen as perceived credit quality deteriorates, reducing the present value of a bond's remaining payments. Forecasting rating destinations and valuing the bond in each destination are separate steps, as the CreditMetrics technical document, §§2.2–2.3 explains.
A downgrade creates another route to default
Consider a fictional borrower observed at annual dates in one of three states: Good, Weak or Default. These are invented model labels, not agency ratings. Stipulate the following annual transition matrix, with today's state in the row and next year's state in the column:
Current state → next annual state
Good
Weak
Default
Good
90%
8%
2%
Weak
10%
85%
5%
Default
0%
0%
100%
Every row sums to 100%. Good → Weak is a downgrade, with an 8% one-year probability. Weak → Good is an upgrade, with a 10% probability. Neither is default.
Assume a time-homogeneous Markov chain: the current state captures all the history used to predict the next state, and the same annual transition probabilities apply each year. Default is an absorbing state, so once default occurs the model keeps that borrower in Default. This is a way to record that default has occurred, not a claim that a real business cannot restructure.
Starting in Good today, the one-year probability of default is 2%. Two-year cumulative default probability adds three mutually exclusive paths:
The first route stays Good before defaulting. The second downgrades to Weak before defaulting at its higher default probability. The last defaults during the first year and remains recorded in Default. Counting that last route does not count a second default.
Three paths from Good to Default by year 2 contribute 1.8%, 0.4% and 2.0%, summing to cumulative default probability 4.2%. The intermediate states are Good, Weak and Default.
Fictional annual model. Multiply along each route and add across routes; all percentages refer to borrowers starting in Good today.
The complete two-year forecast is 81.8% Good, 14% Weak and 4.2% Default. Matrix multiplication gives this distribution as q2=q0P2, where q0=(1,0,0). Simply using 1−(1−0.02)2=3.96% would hold every survivor's annual default risk at the original Good-state level, overlooking migration.
“By year 2” is different from “during year 2, given survival”
The 4.2% figure is cumulative from today's Good starting state. New defaults during the second year account for 4.2%−2%=2.2% of that original population. Among the 98% surviving the first year, the conditional second-year default probability is instead
0.980.022=49011≈2.245%.
It is also different from the 5% next-year default probability for a borrower known to be Weak at year 1. Conditioning only on survival leaves a mixture of Good and Weak borrowers.
Where the model needs judgment
A rating is an assessment of credit quality, not a certain future state. An empirical migration table summarises a particular population and observation period; it does not by itself establish the Markov or time-homogeneity assumptions. Economic conditions, rating history and how withdrawn ratings are handled can change the forecast.
The stipulated probabilities here describe hypothetical real-world outcomes. They are not a calibration to rating-agency data and are not automatically risk-neutral probabilities for pricing. Nor does a migration matrix alone specify loss amounts: valuation in surviving states and exposure and recovery in default still need their own assumptions.
Richard Durrett, Essentials of Stochastic Processes, third edition (Springer, 2016), §§1.1–1.2, printed pp.1–2 and 9–11: the time-homogeneous Markov assumption and multistep transition probabilities.
