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The ECB's approach towards credit risk modelling: issues and parameter choices

The ECB's approach towards credit risk modelling: issues and parameter choices

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123

Credit risk modelling for public institutions’ portfolios

There are a number of competing approaches to model credit risk, each
with their own strengths and weaknesses. The purpose of this chapter is not
to give an overview of various approaches – the interested reader is referred
to a growing list of textbooks1 – but to describe and motivate the ECB’s
approach, with a focus on issues and parameters that are particularly relevant
for central banks and other public investors.
The selection of a credit risk model at the ECB was driven by theoretical as
well as practical considerations. It was foreseen that the model would primarily be used for ex post risk assessments, and that it might be integrated in
strategic asset allocation decisions (see Chapter 2) and be employed for limit
setting (Chapter 4), but it was considered unlikely that the model would be
used for trading. This setting has a number of implications, two of which are
worth mentioning here. The first is that speed of calculations does not have a
very high priority. A simulation-based approach can therefore be used, which
can be made very flexible and intuitive, also for non-insiders, even if the
technical details may be complex. The second implication is that there is no
need for a very precise pricing model; for our purpose, a crude approximation of (relative) prices based on ratings and generic credit curves is sufficient.
Given that, in addition, all issuers and counterparties of the ECB are rated by
at least one of the major rating agencies, it is natural to use a ratings- and
simulation-based approach for credit risk modelling.
At the time the decision was made to model credit risk more formally –
around 2005 – an off-the-shelf system already existed that seemed to fulfill
most of the ECB’s requirements and needs, CreditManagerÒ from the
RiskMetrics Group, based on the well-known CreditMetricsÔ methodology
(Gupton et al. 1997). It was, however, decided to develop an in-house
system. Aside from the more general considerations regarding the choice
between build and buy, discussed in Chapter 4, a particular argument in
favour of an in-house development was the learning experience in terms of
modelling and improved understanding of credit markets. At the time, these
were fairly underdeveloped areas of expertise, which deserved more attention. Moreover, commercial systems seemed primarily targeted at ‘pure’
asset managers and investors, and were considered not necessarily optimal
for central bank portfolios.
At the same time, it was recognized that an in-house model does not
undergo rigorous testing by the market. It was therefore decided to
1

This list includes Bluhm et al. (2003), Cossin and Pirotte (2007), Duffie and Singleton (2003), Lando (2004),
Saunders and Allen (2002). A particularly good introduction for practitioners is Ramaswamy (2004a).

124

Van der Hoorn, H.

‘benchmark’ the model against similar models used by several Eurosystem
NCBs. This benchmark study has culminated in an ECB Occasional Paper
by a Task Force of the Markets Operations Committee of the European
System of Central Banks (2007). Some of its main findings are also discussed in this chapter.
There are some apparent limitations to the use of external ratings, most of
which have been well known for many years (see for example BCBS 2000a).
Despite these limitations, ratings are still believed to add value and are
considered an efficient instrument for resource-constrained investors, such as
central banks, although they should not and are not taken as a substitute for
own risk analysis. When treated with care, an external rating can be used as an
initial assessment of credit risk. In order to reduce the risk of using an overly
optimistic rating, a second-best rating rule is in place (see also Chapter 4).
In what follows, a risk horizon of one year is assumed, although longer
or shorter horizons can also be considered. Note that rating agencies claim
that their ratings reflect a ‘through-the-cycle’ opinion of credit risk, which
obviously impacts the migration probabilities. Actual probabilities over a
one-year point-in-time horizon fluctuate around the through-the-cycle
averages and depend on economic and financial market conditions. The
ECB’s credit risk model distinguishes eight rating levels: AAA, AA, A, BBB,
BB, B, CCC-C and D (¼ default, all using the S&P/Fitch classification). As
explained in detail in Chapter 4, one of the eligibility criteria for issuers and
counterparties is that they have at least one rating (of a certain level) by one
of the major rating agencies. Consequently, the initial rating of each obligor
in the portfolios is known and can be mapped onto one of the eight rating
levels. Probabilities of default and up- or downgrades for all obligors are
readily obtained from historical default and migration studies and summarized in so-called ‘migration (or transition) matrices’, provided the
maturities of the positions exceed the horizon of the migration probabilities
(if not, adjustments will have to be made that are discussed in the next
section). These matrices are published and updated at least annually by all
the rating agencies. It is important to realize that the ECB model does not
provide PD estimates; instead, PDs are important input parameters.2 Given
also the magnitude of credit spreads and recovery rates, it is relatively easy
to estimate the expected value and, hence, expected credit loss over a given
horizon for every obligor in portfolio. The horizon is set at one year. The
2

The most common alternative approaches are estimating these probabilities from bond prices and spreads, using a
reduced form model, and from the volatility of stock prices, using a structured model in the spirit of Merton (1974).

125

Credit risk modelling for public institutions’ portfolios

derivation of summary statistics other than expected loss involves more
steps and is discussed in the next sections. A distinction is made between
those results that can be derived analytically and those for which simulation
is needed.
3.2 Analytical results
The core of the ECB’s credit risk model is a large-scale simulation engine,
but some results – expected loss and unexpected loss – are also derived
analytically. Whenever available, analytical results are preferred over simulation results, which are essentially random and therefore subject to finitesample noise. Moreover, analytical results play an important role in the
validation of simulation results, since expected and unexpected losses are
also estimated from the simulation output. In order to formalize the analytical derivation of expected loss, already touched upon in the previous
section, it is useful to introduce a number of concepts that will facilitate
notation, in particular for the derivation of unexpected loss.
Define the forward value FV as the value of a position that is found by
moving one year forward in time, while keeping the rating of the obligor
unchanged. It is computed by discounting all cash flows at the relevant
discount rate. Formally, for a position in obligor i and portfolio P:
X
À Á
FVi ¼
CFij df icri tij
ð3:1Þ
j

FVP ¼

n
X

FVi

ð3:2Þ

i¼1

Here, CFij represents the jth cash flow (in EUR) by obligor i, tij is the time
(in years) of the cash flow and df cr(t) is the one-year forward discount
factor for a cash flow at time t from an obligor with a credit rating equal to
cr (icri is the initial credit rating of obligor i). This discount rate is derived
from the relevant spot (zero coupon) rates y at maturities 1 and t years.
Assuming, in addition, that any cash flows received during the year are not
reinvested, so that the value of any of these cash flows at time t ¼ 1 is simply
equal to the cash flow itself, the expression for the forward discount factors,
using continuous compounding, is as follows:
&
exp½y cr ð1Þ À t Á y cr ðt ފ; t > 1
cr
ð3:3Þ
df ðt Þ ¼
1;
t 1

126

Van der Hoorn, H.

The conditional forward value CFV is the forward value of a position,
conditional upon a rating change:
À Á
(P
fcr
CF
df
tij ; fcr 6¼ D
ij
fcr
j
ð3:4Þ
CFVi ¼
fcr ¼ D
Nomi rri ;
In this expression, fcr is the forward credit rating (D is default), Nomi is the
nominal investment (in EUR) in obligor i and rri is the recovery rate (in per
cent) of obligor i.
Finally, the expected forward value EFV is the forward value of the position,
taking into account all expected rating changes. It is therefore a weighted
average of conditional forward values, with weights equal to the probabilities
of migration p:
X
fcr
EFVi ¼
pðfcr jicri ÞCFVi
ð3:5Þ
fcr

With these concepts, we can simply write the expected (forward) loss EL as
the differences between the forward and expected portfolio value:
ELP ¼ FVP À

n
X

ð3:6Þ

EFVi

i¼1

Note that expected loss is defined as the difference between two portfolio
values, both at time t ¼ 1. Hence, the current market value of the portfolio
is not used; if it were, expected loss would be ‘biased’ by the time return
(carry and possibly roll-down) of the portfolio. Defining expected loss as in
equation (3.6) ensures a ‘pure’ credit risk concept. It is useful to decompose
expected loss into the contribution of migration (up- and downgrades) and
default. Substituting (3.1), (3.2), (3.4) and (3.5) in (3.6) and rearranging, it
is easy to verify that
8
>
>
>
n >
X
X
Â
À Á
À ÁÃ
ELP ¼
pðfcr jicri Þ
CFij df icri tij À df fcr tij
>fcr6¼D
j
i¼1 >
>
>
:|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
contribution of migration

þ pðD jicri Þ

X

À Á
CFij df icri tij À Nomi rri

9
>
>
!>
>
=

>
>
>
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} >
;
j

contribution of default

ð3:7Þ

127

Credit risk modelling for public institutions’ portfolios

The first element of the right-hand side of equation (3.7) represents the
contribution of migration per obligor. It is equal to a probability-weighted
average of the change in forward value of each cash flow. For high-quality
portfolios, a reasonably good first-order approximation of this expression is
usually found by multiplying the modified duration of the bond one year
forward by the change in the forward credit spread. The second element is
the contribution of default.
Unexpected loss UL, defined as the standard deviation of losses in excess
of the expected loss, is derived in a similar way, although the calculations
are more involved and need assumptions on the co-movement of ratings.
Building on the concepts already defined, a convenient way to compute
unexpected loss analytically involves the computation of standard deviations of all two-obligor subportfolios (of which there are n [n À 1] / 2), as
well as individual obligor standard deviations. First note that, by analogy of
expected loss, the variance (unexpected loss squared) of each individual
position is given by
ULi2 ¼

X


2
fcr
pðfcr jicri Þ CFVi
À EFVi2

ð3:8Þ

fcr

In this formula, it is assumed that there is uncertainty only in the ratings
one year forward, and that conditional forward values of each position are
known. It could be argued that there is also uncertainty in these values, in
particular the recovery value, in which case the standard deviation needs to
be added to the conditional forward values.
A similar calculation can be made for each two-obligor portfolio, but
the probabilities of migration to each of the 8 · 8 possible rating combinations depend on the joint probability distributions of ratings. Rather than
modelling this directly, it is common and convenient to assume that rating
changes are driven by an underlying asset return x and to model joint asset
returns as standard bivariate normal with a given correlation q, known as
asset correlation. The intuition of this approach should become clear in the
next section on simulation. The joint probability of migrating to ratings fcri
and fcrj, given initial ratings icri and icrj, and correlation pij equals

À
Á
p fcri ; fcrj icri ; icrj ; pij ¼


Zfcri jicri



fcri jicri



Zfcrj jicrj



j

fcrj icrj

 

1
qffiffiffiffiffiffiffiffiffiffiffiffiffi exp À12 xi2 þ xj2 À 2qij xi xj dxj dxi
2p 1 À q2ij

ð3:9Þ

128

Van der Hoorn, H.

where the b represent the boundaries for rating migrations from a standard
normal distribution (also explained in the next section). The probabilities
allow the variance computation for each two-obligor portfolio:
2
ULiþj
¼


XX À

Á
fcr 2
fcr
p fcri ; fcrj icri ; icrj ; pij CFVi i þ CFVj j
fcri

À

fcrj

À EFVi þ EFVj

Á2

ð3:10Þ

With the results from equations (3.8)–(3.10), it is easy to compute the
unexpected loss of the portfolio:3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u nÀ1 n
n
X
uX X
2
t
ð3:11Þ
ULP ¼
ULiþj À ðn À 2Þ
ULi2
i¼1 j¼iþ1

i¼1

3.3 Simulation approach
Expected and unexpected loss capture only the first two moments of the
credit loss distribution. In fact, formally, expected loss is not even a risk
measure, as risk is by definition restricted to unexpected events. Unexpected
loss measures the standard deviation of losses and is also of relatively
limited use, as a credit loss distribution is skewed and fat tailed. In order to
derive more meaningful (tail) risk measures such as VaR and ES (expected
shortfall), a simulation approach is needed.
To understand the simulation approach at the portfolio level, consider
first a single bond with a known initial rating, e.g. A. Over a one-year
period, there is a high probability that the rating remains unchanged. There
is also a (smaller) probability that the bond is upgraded to AA, or even to
AAA, and, conversely, that it is downgraded or even that the issuer defaults.
Assume that rating changes are driven by an underlying asset value of the
issuer, the same as was used for the analytical derivation of unexpected loss
in the previous section. The bond is upgraded if the asset value increases
3

This result is derived from a standard result in statistics. If X1, . . . , Xn are all normal random variables with variances
r2i and covariances rij, then Y ¼ RXi is also normal and has variance equal to r2Y ¼

n
P
i¼1

r2i þ 2

nP
À1

n
P

rij .

i¼1 j¼iþ1

Rearranging the formula for a two-asset portfolio Xi þ Xj yields an expression for each covariance pair:


rij ¼ 12 r2iþj À r2i À r2j which, when substituted back into the formula for r2Y , gives the desired result.

129

Credit risk modelling for public institutions’ portfolios

Table 3.1 Migration probabilities and standard normal boundaries for bond with initial rating A
Rating after one
year (fcr)
Migration
probability
Cumulative
probability
Lower migration

boundary ( fcrjA )
Upper migration

boundary ( fcrjA )

D

CCC-C

B

BB

BBB

A

AA

0.06%

0.03%

0.17%

0.41%

5.63%

91.49%

2.17%

0.06%

0.09%

0.26%

0.67%

6.30%

97.79%

99.96%

AAA
0.04%
100.00%

À1

À3.24

À3.12

À2.79

À2.47

À1.53

2.01

3.35

À3.24

À3.12

À2.79

À2.47

À1.53

2.01

3.35

þ1

Source: Standard & Poor’s (2008a, Table 6 – adjusted for withdrawn ratings).

beyond a certain threshold and downgraded in case of a large decrease in
asset value. The thresholds are set such that the ratings derived from the
(pseudo-) random asset returns converge to the migration probabilities,
given a certain density for the asset returns. For the latter, it is common to
assume standard normally distributed asset returns, and this is also the
approach of the ECB model, although other densities may be used as well.4
Note that normal asset returns do not imply that migrations and therefore
bond prices are normally distributed. The process is illustrated using actual
migration probabilities from a recent S&P study in Table 3.1.
The table shows, for instance, that the simulated rating remains
unchanged, for simulated asset returns between thresholds bAÀjA ¼ À1:53
À
and bAþjA ¼ 2:01. If the simulated asset return is between bAA
jA ¼ 2:01 and
þ
À
bAA
jA ¼ 3:35, the bond is upgraded to AA, and if it exceeds bAAAjA ¼ 3:35,
þ
À
the bond is upgraded to AAA. Note that bfcr
jicr ¼ bfcrþ1jicr for any com-

bination of initial credit rating icr and forward credit rating fcr (where ‘þ1’
refers to the next highest rating). The same levels are also used in the
analytical derivation of unexpected loss in equation (3.9). The mechanism
applies to downgrades. It is easy to verify that the simulated frequency of
ratings should converge to the migration probabilities. Combining simulated ratings with yield spreads and recovery rates yields asymptotically the

4

The normal distribution of asset returns is merely used for convenience, because the only determinant of codependence is the correlation. It is quite common to use the normal distribution, but in theory alternative probability
distributions for asset returns can also be used. These do, however, increase the complexity of the model.

Van der Hoorn, H.

Downgrade to BBB

Probability density

130

Rating unchanged (A)

Upgrade to AA

Default

b +D|A b +CCC|Ab +B|A b +BB|A b +BBB|A

b –AA|A

b –AAA|A

Asset return over horizon

Figure 3.1

Asset value and migration (probabilities not according to scale).

same expected loss as in the analytic approach. The simulation approach is
illustrated graphically in Figure 3.1.
At the portfolio level, random asset returns are sampled independently
and subsequently transformed into correlated returns. This is achieved via a
Cholesky decomposition of the correlation matrix into an upper and a
lower triangular matrix and by subsequently pre-multiplying the vector of
independent asset returns (of length n) by the lower triangular matrix (of
dimension n · n).5 The result is a vector of correlated asset returns, each of
which is converted into a rating using the approach outlined above.
Assuming deterministic spread curves, the ratings are subsequently used to
reprice each position in the portfolio. The model can also be used for multistep simulations, whereby the year is broken down in several subperiods, or
5

A correlation matrix R is decomposed into a lower triangular L and an upper triangular matrix L0 in such a way that
R ¼ LL0 . A vector of independent random returns x is transformed into a vector of correlated returns xc ¼ Lx.
It is easy to see that xc has zero mean, because x has zero mean, and a correlation matrix equal to
À
Á
E xc ðxc Þ0 ¼ E ðLxx0 L0 Þ0 ¼ LE ðxx0 ÞL ¼ LIL0 ¼ LL0 ¼ R, as desired. Since correlation matrices are symmetric and
positive-definite, the Cholesky decomposition exists. Note, however, that the decomposition is not unique. It is, for
1
0
l11 0 Á Á Á 0
.
.
B
. . .. C
l22
C
Bl
example, easily verified that if L ¼ B 21
C is a valid lower triangular matrix, then so is
@ ... . . . . . . 0 A
ln1 Á Á Á ln2 lnn
1
0
0
l11 0 Á Á Á
.. C
..
B
. C
.
l22
Bl
LÃ ¼ B 21
C. Any of these may be used to transform uncorrelated returns into correlated returns.
.
.
.
@ ..
.. ..
0 A
ln1 Á Á Á ln2 Àlnn

131

Credit risk modelling for public institutions’ portfolios

where the horizon consists of several one-year periods. In those cases, the
vector of returns becomes a matrix (of dimension n · # periods), but
otherwise the approach is essentially the same as for a one-step simulation.
As shown in Chapter 2, it is also possible to use stochastic spreads, thus
integrating market (spread) and credit risk, but this chapter considers
deterministic spreads only.
In order to generate reliable estimates of (tail) risk measures, a large
number of iterations are needed, but the number can be reduced by
applying importance sampling techniques. Importance sampling is based on
the idea that one is really only concerned with the tail of the distribution,
and should therefore sample more observations from the tail than from the
rest of the distribution. With importance sampling, the original distribution
from which observations are drawn is transformed into a distribution which
increases the likelihood that ‘important’ observations are drawn. These
observations are then weighted by the likelihood ratio to ensure that estimates are unbiased. The transformation is done by shifting the mean of the
distribution. Technical details of importance sampling are discussed in
Chapter 10.
The simulation approach is summarized in the following steps:
Step 0 Create a matrix (of dimension # names · # ratings), consisting of
the conditional forward values of the investment in each obligor
under each possible rating realization, as given by equation (3.4).
Step 1 Generate n independent (pseudo-) random returns from a standard
normal distribution, but sampling with a higher probability from
the tail of the distribution. Store the results in a vector x.
Step 2 Transform the vector of independent returns into a vector of
correlated returns xc via xc ¼ Lx, where LL0 ¼ R ¼
1
0
1 q21
ÁÁÁ
q1n
..
.. C
B
B q21 1
.
. C
C is the (symmetric) correlation matrix.
B
C
B ..
..
..
@ .
.
.
qnÀ1;n A
qn1 Á Á Á qn;nÀ1
1
Step 3 Transform the vector of correlated returns into a vector of ratings
n h
i h
io
via fcri ¼ arg max 1 xic ! bcrÀjicri · 1 xic < bcrþjicri , where 1[·] is an
cr

indicator function, equal to unity whenever the statement in
brackets is true, and zero otherwise.
Step 4 Select, in each row of the matrix created in step 0, the entry
(conditional forward value) corresponding to the rating simulated

132

Van der Hoorn, H.

ð1Þ

in step 3. Compute the simulated (forward) portfolio value SFVP
as the sum of these values, where the (1) indicates that this is the
first simulation result.
Step 5 Repeat steps 1–4 many times and store the simulated portfolio
ðiÞ
values SFVP .
Step 6 Sort the vector of simulated portfolio values in ascending order and
compute summary statistics (sim is the number of iterations):
1
 SFVP ¼ sim

sim
P
i¼1

ðiÞ

SFVP ;

 ELðsÞ ¼ FVP À SFVP ;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sim 
À
Á2
P
ðiÞ 2
1
 ULðsÞ ¼ sim
SFVP
À SFVP ;
i¼1

ðaÞ

 VaR ¼ SFVP À SFVP ; where a ¼ sim · (1 À confidence level)
rounded to the nearest integer,
1
 ES ¼ SFVP À aÀ1

aP
À1
i¼1

ðiÞ

SFVP :

Each of these can also be expressed as a percentage of FVP, the
market value in one year time if ratings remain unchanged.
Step 7 Finally, the results may be used to fit a curve through the tail of the
distribution to avoid extreme ‘jumps’ in time series of VaR or ES.
This, however, is an art as much as science for which there is no
‘one size fits all’ approach.6
The creation of a matrix with all conditional forward values in step 0
makes the simulation very fast and efficient. Without such a matrix, it would
be necessary to reprice every position within the loop (steps 1-4) and computing time would increase significantly. If programmed efficiently, even a
simulation with 1,000,000 iterations need not take more than one minute on
a modern computer. For many portfolios, this number of iterations is more
than enough and fairly accurate estimates of credit risk are obtained, even at
the highest confidence levels. In practice, therefore, importance sampling or
other variance reduction techniques are not always needed.
6

A possible strategy, depending on the composition of the portfolio, is to make use of a well-known result by Vasicek
(1991), who found that the cumulative loss distribution of an infinitely granular portfolio in default mode (no
pffiffiffiffiffiffi À1

ðx ÞÀN À1 ðpd Þ
ffiffiq
, where q is the (positive) asset correlation and N (x)
recovery) is in the limit equal to F ðx Þ ¼ N 1ÀqN p
denotes the cumulative standard normal distribution (N–1 being its inverse) evaluated at x, representing the loss as a
proportion of the portfolio market value, i.e. the negative of the portfolio return.

133

Credit risk modelling for public institutions’ portfolios

VaR and ES for credit risk are typically computed at higher confidence
levels than for market risk. This is a common approach, despite increasing
parameter uncertainty, also for commercial issuers aiming at very low
probabilities of default to ensure a high credit rating. For instance, a 99.9
per cent confidence level of no default corresponds only to approximately
an A rating. The Basel II formulas for the Internal Ratings Based (IRB)
approach compute capital requirements for credit risk at the 99.9 per cent
confidence level as well, whereas a 99 per cent confidence level is applied to
determine the capital requirements for market risk (BCBS 2006b). Arguably,
a central bank – with reputation as its main asset – should aim for high
confidence levels, also in comparison with commercial institutions.
The discussion of the analytical and simulation approach has so far
largely ignored the choice of parameters and data sources. There are,
however, a number of additional complexities related to data and parameters, in particular for central banks and other conservative investors. The
remainder of this section is therefore devoted to a discussion of the main
parameters of the model, i.e. the probabilities of migration (including
default), asset correlations and recovery rates. This discussion is not
restricted to the ECB, but includes a comparison with other Eurosystem
central banks, more details of which can be found in the paper by the Task
Force of the Market Operations Committee of the European System of
Central Banks (2007).
3.4 Probabilities of default/migration
Probabilities of default and migration can be obtained from one of the
major rating agencies, which publish updated migration matrices frequently. These probabilities typically have a one-year horizon, i.e. equal to
the standard risk horizon of the model. As the migration matrices of the
rating agencies are normally fairly similar for any given market segment, the
selection of a particular rating agency does not seem terribly important,
although clearly, in order to generate meaningful time series, one should
try to use the same source as much as possible. Also in practice there is not
a clear preference among Eurosystem NCBs for any of the three major
agencies, Standard & Poor’s, Moody’s and Fitch.
The methodologies used by the rating agencies for estimating default and
migration probabilities rely on counting the number of migrations for a
given rating within a calendar year. This number is divided by the total
number of obligors with the initial rating and corrected for ratings that have