Rating migration · RiskAtoms Academy

By the end you will be able to

1Read a one-year rating transition matrix: staying, upgrades, downgrades and the absorbing default state.
2Compute multi-year cumulative, marginal and conditional default probabilities by matrix powers.
3Explain why the term structure of PD rises for strong grades and falls for weak ones.
4Name the limits of the Markov assumption and what banks do about them.

Why it matters

Thornvale Engineering, one of RiskAtoms Bank's borrowers, is rated BBB today. Its one-year PD on the bank's matrix is 0.20%. Its loan runs for five years. If the PD stayed at 0.20% a year, the five-year PD would be 1.00%. The matrix says 2.86%, almost three times as much, because a BBB borrower can slip to BB or B first, and from there the yearly PD is far higher. Ratings move, and a loan's risk moves with them long before anyone defaults.

Core explanation

Ratings move, one step at a time

Illustrative dataRiskAtoms Bank plc's credit book

One made-up lending book on one made-up date (31 December, year 0), in pounds: a ten-grade master scale, a one-year rating transition matrix, twelve invented corporate borrowers (£587m of exposure, expected loss £2.81m), retail portfolios and twenty past workout cases. The firms are invented and the numbers are round illustrative values, not anyone's data. The same book runs through every lesson in Module 3.3.

This lesson uses the book's one-year transition matrix on seven agency-style grades plus default. It is shaped like published long-run averages but is nobody's data.

A rating is a snapshot. Over a year, a borrower may keep its rating, be upgraded or downgraded (migration), or default. A transition matrix collects those chances for every starting rating. Row BBB says where today's BBB borrowers are a year from now:

One-year transition matrix (illustrative). Rows: rating at the start of the year; columns: at the end.
From ↓ to →	AAA	AA	A	BBB	BB	B	CCC	D
AAA	90.00%	9.00%	0.70%	0.20%	0.05%	0.03%	0.02%	0
AA	0.60%	90.50%	8.00%	0.60%	0.15%	0.10%	0.03%	0.02%
A	0.05%	2.00%	91.50%	5.50%	0.60%	0.20%	0.10%	0.05%
BBB	0.02%	0.20%	4.00%	89.50%	4.50%	1.00%	0.58%	0.20%
BB	0.01%	0.05%	0.40%	6.00%	83.00%	8.00%	1.54%	1.00%
B	0	0.04%	0.20%	0.40%	6.00%	83.00%	6.36%	4.00%
CCC	0	0	0.20%	0.40%	1.20%	11.00%	60.00%	27.20%
D	0	0	0	0	0	0	0	100.00%

Read the BBB row: 89.5% stay BBB, 4.22% are upgraded, 6.08% are downgraded without defaulting and 0.20% default. Three patterns run through every row. The diagonal is large (most borrowers keep their rating). Big jumps are rarer than small ones. And the weaker the rating, the more it moves: only 60% of CCC borrowers are still CCC a year later. The last row is special: a defaulted borrower stays in default with probability 1. Default is an absorbing state.

The matrix uses agency-style letters, so its BBB one-year PD (0.20%) is not exactly the bank's grade 4 PD of 0.28% on the master scale (L3.3.04: Internal ratings and master scales). Banks build the same kind of matrix on their own grades from their own history.

Where a matrix comes from

The simplest estimate is the cohort method: take every firm rated BBB on 1 January, count where each is on 31 December and divide by the number at the start. Pool many years to get enough firms. Firms whose rating is withdrawn during the year are usually dropped or spread over the other columns. Agencies publish such matrices every year from decades of history; the duration method uses the exact dates of every rating change instead of year-end snapshots, and so picks up a firm that was downgraded twice in one year.

Two years: multiply the matrix by itself

The matrix answers one-year questions. For longer horizons, chain the years together (M-C13: Markov chains and transition matrices). To default within two years, a borrower either defaults in year 1, or moves to some rating in year 1 and defaults from there in year 2. Adding up those routes is exactly a matrix multiplication (M-E02: Matrix multiplication and quadratic forms). A small case shows every number. Collapse the book's scale into investment grade (IG: BBB or better), speculative grade (SG: BB to CCC) and default, using the BBB row for IG and the BB row for SG:

P = \begin{array}{c} \text{IG} \\ \text{SG} \\ \text{D} \end{array}\!\begin{pmatrix} 0.9372 & 0.0608 & 0.0020 \\ 0.0646 & 0.9254 & 0.0100 \\ 0.0000 & 0.0000 & 1.0000 \end{pmatrix}

P^2 = P \times P = \begin{pmatrix} 0.8823 & 0.1132 & 0.0045 \\ 0.1203 & 0.8603 & 0.0194 \\ 0.0000 & 0.0000 & 1.0000 \end{pmatrix}

The top-right entry is the two-year PD of an IG borrower. It is the sum over the three places the borrower can be after one year:

\begin{aligned} (P^2)_{\text{IG},\text{D}} &= P_{\text{IG},\text{IG}}P_{\text{IG},\text{D}} + P_{\text{IG},\text{SG}}P_{\text{SG},\text{D}} + P_{\text{IG},\text{D}}P_{\text{D},\text{D}} \\ &= 0.9372 \times 0.0020 + 0.0608 \times 0.0100 + 0.0020 \times 1 \\ &= 0.0018744 + 0.0006080 + 0.0020000 = 0.448\% \end{aligned}

Five years is $P^5$ : the IG five-year PD in the small chain is 1.42%. The full 8 × 8 matrix gives a BBB borrower 0.624% over two years, more than the small chain's 0.448%. Lumping BB, B and CCC into one state hides how far a downgraded borrower has fallen, and that changes its next move. Coarse grades are one way the Markov assumption can fail.

The term structure of default

Powers of the full matrix give each rating's cumulative PD year by year. The extra default in year $t$ is the marginal PD; divided by the share still alive at the start of the year it is the conditional PD (or forward PD).

Cumulative and conditional PDs from powers of the book's matrix
Rating	1 year	3 years	5 years	10 years	Conditional, year 2	Conditional, year 10
AAA	0.00%	0.03%	0.10%	0.52%	0.01%	0.12%
AA	0.02%	0.11%	0.29%	1.25%	0.04%	0.26%
A	0.05%	0.30%	0.77%	2.94%	0.10%	0.56%
BBB	0.20%	1.23%	2.86%	8.57%	0.42%	1.37%
BB	1.00%	4.59%	9.37%	22.32%	1.60%	3.15%
B	4.00%	14.52%	24.78%	43.95%	5.32%	5.30%
CCC	27.20%	54.62%	66.45%	77.75%	23.04%	6.06%

BBBBBBCCC

The lines converge. A BBB borrower's conditional PD climbs from 0.20% to 1.37% by year 10, because some BBBs slip. A CCC borrower's falls from 27.20% to 6.06%: the weakest default early, and survivors have often been upgraded. Published default studies show the same shape. And because default is absorbing, the matrix eventually sends everyone there: a BBB borrower's cumulative PD is 54% after 50 years and 96% after 200. No real loan lasts that long, but it is a reminder that the matrix is a model, not a law.

The Markov assumption and its limits

Matrix powers assume the Markov property: tomorrow's move depends only on today's rating, and the same matrix applies every year to every borrower. Real ratings bend both rules. Sort each finding by the part of the assumption it breaks:

Sort itWhich part of the assumption does it break?

A firm downgraded last year is more likely to be downgraded again than a firm long at the same rating.
Downgrades outnumber upgrades by far more in recessions than in booms.
Banks' BBB borrowers migrate differently from industrial BBB borrowers.
A firm that has kept the same rating for many years is less likely to move than one rated recently.
The bank's point-in-time grades move more in a downturn than agency ratings do.

Banks live with this. They estimate matrices by sector or region where data allow, condition them on the economic cycle (a downturn matrix as well as an average one), use the duration method, and add expert overlays. A Bank of England working paper (Nickell, Perraudin and Varotto) found that transition probabilities vary with the business cycle, the issuer's industry and its domicile, and that ignoring this can misstate credit risk.

What migration is used for

Lifetime PDs for loan-loss provisions (L3.3.18: Provisioning basics); the value of a loan or bond that is marked to market, which falls on a downgrade even without default; pricing a long loan; rating triggers in contracts; and portfolio models that simulate migrations, such as CreditMetrics (M6.5: Portfolio credit models).

The formal version

Two-step transitions. Let $R_t$ be the rating at the end of year $t$ . By the law of total probability over the rating after one year, and the Markov property:

\begin{aligned} \Pr(R_2 = j \mid R_0 = i) &= \sum_k \Pr(R_1 = k \mid R_0 = i)\,\Pr(R_2 = j \mid R_1 = k,\ R_0 = i) \\ &= \sum_k P_{ik}\,P_{kj} \qquad \text{(Markov: } R_0 \text{ drops out)} \\ &= (P^2)_{ij} \end{aligned}

Repeating the argument, $\Pr(R_n = j \mid R_0 = i) = (P^n)_{ij}$ and $P^{m+n} = P^m P^n$ (the Chapman-Kolmogorov equation). A row vector of probabilities $\pi_0$ moves to $\pi_n = \pi_0 P^n$ .

Cumulative, marginal and conditional PD. Because $P_{\mathrm{DD}} = 1$ , being in D at year $n$ is the same as having defaulted at some time up to $n$ :

\begin{aligned} \mathrm{PD}^{\text{cum}}_i(n) &= (P^n)_{i,\mathrm{D}}, \qquad S_i(n) = 1 - \mathrm{PD}^{\text{cum}}_i(n) \\ \mathrm{PD}^{\text{marg}}_i(t) &= \mathrm{PD}^{\text{cum}}_i(t) - \mathrm{PD}^{\text{cum}}_i(t-1) \\ h_i(t) &= \frac{\mathrm{PD}^{\text{marg}}_i(t)}{S_i(t-1)}, \qquad S_i(n) = \prod_{t=1}^{n} \bigl(1 - h_i(t)\bigr) \end{aligned}

Only with one performing state is $h$ constant and $\mathrm{PD}^{\text{cum}}(n) = 1 - (1-\mathrm{PD})^n$ . With several grades the conditional PD drifts towards a common level, as the chart above shows.

Try it

Pick a rating and a horizon. The bars show the row of $P^n$ : where the borrower can be after $n$ years. The downturn matrix multiplies every downgrade and default probability by the factor and divides upgrades by it; the diagonal takes up the difference. Compare a BBB and a CCC borrower at ten years.

Try itWhere will the borrower be in n years?

Rating today

Horizon5 years

Downturn factor on downgrades and defaults

Book matrixDownturn (× 1.5)

5-year PD, book matrix2.86%

5-year PD, downturn5.81%

If the 1-year PD (0.20%) never changed1.00%

Still BBB61.0%

Upgraded15.3%

Downgraded, not defaulted20.8%

Cumulative PD, book matrixCumulative PD, downturnConditional PD in each year, book matrix

Worked example

Thornvale Engineering over three years

Thornvale Engineering Ltd is internal grade 4 (BBB+ to BBB on the master scale), so the bank uses the BBB row. Its £60m facility has five years to run. Start with three.

From one year to three

Step 1 of 5

1
Today
$\pi_0 = (0, 0, 0, 1, 0, 0, 0, 0)$ over (AAA, AA, A, BBB, BB, B, CCC, D).

Regulatory lens · looking ahead

Capital rules use a one-year PD per grade (M4.2: Credit risk capital), so migration enters capital only through the grade a borrower holds. Accounting is different: IFRS 9 needs lifetime PDs for loans whose credit risk has increased significantly since the loan was made. The test compares the risk of default over the loan's life now with that when it was made, and banks often measure it through a downgrade in the loan's rating or a rise in its lifetime PD. Many banks build those lifetime PDs from transition matrices conditioned on economic scenarios (L3.3.18: Provisioning basics and M4.7: Accounting for risk). Portfolio models that simulate migrations, and continuous-time and cycle-dependent matrices, are in M6.4: Credit risk modelling and M6.5: Portfolio credit models.

Spark Atoms

Spark AtomChains from poetry

Andrey Markov tested his chains on literature. In 1913 he wrote out the first 20,000 letters of Pushkin's Eugene Onegin and counted how often a vowel followed a vowel or a consonant. The same arithmetic now turns one-year rating matrices into lifetime default probabilities.

Spark AtomA zero that does not stay zero

In the book's matrix an AAA borrower cannot default within a year: the AAA → D entry is 0. Yet its five-year PD is 0.10% and its ten-year PD 0.52%, because an AAA can be downgraded first. Matrix powers give strong grades a positive multi-year PD even when no default has ever been seen from that grade.

Key takeaways

A transition matrix gives the chance of moving from each rating to every other within a year; rows add up to 1 and default is absorbing.
n-year probabilities are the n-th power of the matrix: the default column of Pⁿ is the cumulative PD.
Migration makes multi-year PDs higher than a constant PD suggests: BBB five-year 2.86% against 1.00%.
Conditional PDs rise over time for strong grades and fall for weak ones.
The Markov assumption (no memory, one matrix for every year and borrower) is convenient but broken by momentum, the cycle and differences between sectors.

Reaction

Check what stuck

Glossary

Rating migration: A change in a borrower's rating over time, up (upgrade) or down (downgrade), short of default or into it.
Transition matrix: A table of probabilities: row i, column j is the chance that a borrower rated i at the start of a period is rated j at the end. Each row adds up to 1.
Absorbing state: A state the chain never leaves. Default is absorbing: a defaulted borrower is not re-rated back to health.
Markov property: The next move depends only on the rating today, not on how the borrower got there or how long it has been there.
Cumulative PD: The probability of defaulting at any time within the first n years.
Marginal PD: The probability of defaulting in year t itself, seen from today: cumulative PD at t minus cumulative PD at t − 1.
Conditional (forward) PD: The probability of defaulting in year t given survival to the start of year t: the marginal PD divided by the survival probability.
Cohort method: Estimating a matrix by counting: of the firms rated i at the start of a year, the share rated j at the end.
Rating momentum: The finding that a recently downgraded borrower is more likely than others with the same rating to be downgraded again.

Deep Dives & Origins

Deep Dives

D. Lando, Credit Risk Modeling: Theory and Applications, Princeton University Press, 2004, the chapter on rating-based term structure models. Advanced.
A. Saunders and L. Allen, Credit Risk Measurement In and Out of the Financial Crisis, 3rd ed., Wiley, 2010, the chapter on the VaR approach and CreditMetrics, which is built on rating migration. Intermediate.

Origins

P. Nickell, W. Perraudin and S. Varotto, “Stability of ratings transitions”, Bank of England Working Paper 133, 2001.
D. Lando and T. M. Skødeberg, “Analyzing rating transitions and rating drift with continuous observations”, Journal of Banking & Finance 26(2-3), 2002.
O. Schwartz, “Andrey Markov & Claude Shannon counted letters to build the first language-generation models”, IEEE Spectrum, 2019.

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