Risk Rating Article 2004

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Les Cahiers du CREF ISSN: 1707-410X

The Ordered Qualitative Model forCredit Rating Transitions

Dingan FengChristian GourirouxJoann Jasiak

CREF 04-05

April 2004

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The Ordered Qualitative Model for Credit RatingTransitions

Dingan Feng

Post Doctoral Student

CREF and York University

Department of Economics

4700 Keele Street

Toronto, Ontario M3J 1P3

E-Mail:[email protected]

Christian Gouriroux

Professor

CREF, CREST, CEPREMAP

and University of Toronto

15, boulevard Gabriel Pri

92245 Malakoff

E-Mail: [email protected]

Joann Jasiak

Professor

CREF and York University

Department of Economics

4700 Keele Street

Toronto, Ontario M3J 1P3

E-Mail:[email protected]

April 2004

Les Cahiers du CREF

CREF 04-05
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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Les Cahiers du CREF CREF 04-05

The Ordered Qualitative Model for Credit Rating Transitions

Abstract

The dynamic analysis of corporate credit ratings is needed for predicting therisk included in a credit portfolio at different horizons. In this paper, we presentthe estimation of an ordered probit model with factors for the migrationprobabilities, with its application to aggregate data regularly reported byStandard & Poor's.

Keywords: Credit Rating, Migration

JEL : C23, C35, G11


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1 Introduction

Credit ratings for firms and bond issuing are regularly reported by specialized rating agencies such

as Moodys, Standard &Poors and Fitch. A credit rating provides a measure of risk quality, and

is a basic tool for risk management. This paper is concerned by the dynamic analysis of ratings

for a homogenous population of firms (bonds). The analysis is based on an ordered qualitative

model explaining how the transition probabilities between credit rating categories depend on some

underlying unobservable factors. In Section 2, we present the deterministic and stochastic ordered

probit specifications for the transition probabilities. In particular we explain why it is necessary to

introduce stochastic transition to define and study joint migration, which arises when several firms

are jointly down- or up-graded. Statistical inference is discussed in Section 3, for both determin-

istic and stochastic specifications. We explain how the panel data on ratings can be aggregated

per year without any loss of information. This possibility of aggregation is used to define a two

step estimation approach for the nonlinear latent factor model of transition. In the first step an

approximated latent factor model is estimated from the aggregate observed transition frequencies.

Then this first step estimator is used as a starting value before applying a (more efficient) simulated

maximum likelihood approach.

The methodology is applied to the aggregate transition frequencies regularly reported by Stan-

dard & Poors, in Section 4. We first explain how to correct for missing data (the so-called not

rated companies). Then the deterministic ordered qualitative model is estimated independently for

each year. This allows to obtain time dependent risk summaries, such as average risk, and risk

volatility per rating class. Different tests are performed on this basic specification to indicate

which summaries can be assumed time independent. Then the approach is extended to stochastic

models featuring either serial independence between transition, or serial dependence by means of

a small number of factors. The estimated stochastic migration model is used in Section 5 to predict

future ratings. Section 6 concludes.

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2 Specification of migration dynamics

The aim of migration (or transition) models is to analyze the credit rating histories of several firms.

The basic models arise as specifications of the transition matrix, which consists of the probabilities

of migration from one rating category to another rating category for a given firm in a given period.

According to the selected specification, the transition matrix can be time independent (time homo-

geneity assumption), or can vary in time in either a deterministic (heterogeneity assumption), or a

stochastic way (stochastic transition model). In this section, we present the approach based on the

ordered qualitative dependent variable models.

The credit rating categories are denoted by

Thus the individual histo-

ries (

correspond to qualitative processes with state space

.

2.1 Deterministic models

2.1.1 Dependence assumption

The basic simple specification is obtained under the following assumptions,

Assumption A1: The individual histories (

are independent,

identically distributed;

Assumption A2: Any individual history satisfies the Markov condition, that is the most recent

individual rating summarizes efficiently the whole individual history.

Under Assumptions A1, A2, the joint dynamics of rating histories of all firms is characterized

by the sequence of transition matrices

. The matrix

is a

matrix. Its

elements provide the transition probabilities from a rating

to another rating

between dates

and

,

(1)

The transition probabilities are the same for different individuals, which is the homogeneity as-

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sumption accross the population of firms . They are non-negative and sum up to one by column.

When the Markov process is time heterogenous [resp. time homogenous], the transition probabili-

ties depend on [resp. do not depend on ].

2.1.2 The ordered qualitative model

It can be useful to constrain the transition matrices in order to diminish the number of parameters

to be estimated [curse of dimensionality], and to robustify the results. A structural model for

transition matrices is based on the fact that the individual qualitative ratings are usually determined

from an underlying continuous score. More precisely, it is common to assign a continuous grade

(or score) , which is an increasing function of estimated default probability, to each firm at every

date. Let us denote

the value of the score. The qualitative rating is obtained by discretizing the

score values. More precisely, let us introduce a partition

of admissible values

of the score. Then the observed rating is defined by,

if and only if

(2)

where by convention,

and

. The relation (2) expresses the link between the

observable endogenous variable

and the score

, which is generally not publicly diffused

(when it exists), and has to be considered as a latent variable.

Then the model is completed by specifying the (conditional) distribution of the quantitative

score. In the ordered polytomous model, the underlying scores are such that,

Assumption A1 : The individual score histories are independent, identically distributed;

Assumption A2 : The conditional distribution of

given the lagged score values depends

on the past through the most recent qualitative rating only. This distribution is such that:

if

(3)

see Gagliardini, and Gourieroux(2003)a for a discussion of the homogeneity assumption.

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where

are i.i.d. variables with identical cumulative density function (cdf) . Thus the

conditional mean and variance depend on the current rating.

The expressions of transition probabilities follow directly from Assumptions A1, A2. We get,

or:

(4)

For example,

is the probability of default of a firm rated

;

similarly

is the probability of migration to the highest category 1, or AAA.

The parameters

are identifiable up to some linear

affine transformation on and a linear transformation on . Under identification restriction,

different transition probabilities now depend on a smaller number of parameters equal

to .

The ordered probit model is obtained when the error variable

is standard normal, and the

cdf is replaced by the cdf of the standard normal. This type of ordered probit model is frequently

encountered in both the applied and theoretical literature [see e.g. Gupton, Finger, Bhatia (1997),

Nickell, Perraudin, Varotto (2000), Bangia, Diebold, Kronimus, Schlagen, Schuermann(2002),

Albanese et alii. (2003)a].

Remark 1: In the application to firm rating, one of the states corresponds to default and is by

definition an absorbing state. If this state has index , then we have:

if

otherwise

or implicitly

. In this framework the (non degenerate) ordered qualitative model applies

to the remaining columns of the transition matrix.

It is usually called the asset value model by reference to Merton(1974). However the latent variable

does not

necessarily admit an interpretation as a difference between liabilities and asset values as in Mertons model.

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2.2 Stochastic transition models

The deterministic models can be extended to allow for stochastic transition matrices. In this re-

spect, they extend the stochastic intensity model, largely used for modeling default [see e.g. Lando

(1998), Duffie and Singleton (1999)]. The advantage of specifying stochastic transition matrices

is twofold. First the time heterogeneous deterministic Markov chain can be used for prediction

purpose, only if the dynamics of transition matrices is clearly defined . Thus it is necessary to

introduce such a stochastic dynamics, which will involve a small number of underlying factors for

tractability and robustness purposes. Second the migration correlations, which measure the joint

up- or down-grades of firms, can be defined in the stochastic framework only [see e.g. Gagliardini

and Gourieroux (2003)a,b] . In Section 2.2.1, we describe a specification with i.i.d. transition

matrices. The factor ordered qualitative model is discussed in Section 2.2.2.

2.2.1 I.I.D. transition matrices

A simple specification is obtained when the dated transition matrices (

) are as-

sumed independent, identically distributed (i.i.d.). Let us explain how the assumption of stochastic

transition modifies the probabilistic properties of the rating histories.

Under the deterministic model considered in Section 2.1:

i) The individual histories are independent (Assumption A.1),

ii) Any individual history satisfies a Markov process, with given time dependent parametric

transitions.

In particular, the prediction of future states at all horizons is easy to perform. Let us define the

vector of state indicators:

(5)

where :

Contrary to a usual belief, The method of estimation (of joint credit quality co-movements) has the advantage

that it does not make assumption on the underlying process (Gupton et alii.(1997)), the migration correlations and

their estimation can only be considered within a precise specification of default.

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if firm is in state at date

otherwise

The knowledge of rating histories

is equivalent to the knowledge of indicator histories.

Moreover the prediction of

performed at date is simply:

E

(6)

Let us now assume that the transition matrices are iid and unobserved. Then:

i) The individual histories become dependent.

Indeed let us consider the covariance between two firm ratings,

, when their pre-

vious ratings are known. By the covariance decomposition equation, we get:

cov

E cov

cov E

E

cov

Therefore,

cov

cov

These covariances are generally different from zero. For instance, let us assume current identical

ratings , and consider joint up-grades by one bucket: . We get:

cov

var

which is strictly positive due to the stochastic assumption on the transition matrix. This com-

putation shows that stochastic transition matrices are introduced to define non-zero cross-sectional

correlations.

ii) Any individual history is a homogeneous Markov process.

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Indeed, from the prediction formula (6) and the iterated expectation theorem, it follows that:

E

E

E

E

E

E

by the i.i.d. assumption on transition matrices.

Thus the rating history of a given individual satisfies a Markov property with a time indepen-

dent transition matrix, equal to the expected stochastic transition:

E

(7)

iii) Joint analysis of two firms histories.

The results given above can be extended to a joint analysis of rating histories of two firms

and

, say. Typically the joint transition probabilities are given by:

E

E

The matrix is a matrix of all joint transition probabilities. In general this matrix

is different from the matrix with elements

as a consequence of migration correlation.

2.2.2 Factor ordered qualitative model

The stochastic transition model with iid transition matrices is simple to apply to credit analysis,

since, as mentioned above, it provides homogenous Markov rating histories. Equivalently, in fi-

nancial term, it provides a flat term structure of migration correlations and thus flat term structures

of spreads of interest rates.

For a more flexible term structure specification, it is necessary to introduce serial dependence

between the transition matrices. This can be done in the ordered qualitative model by writing the

time dependent parameters

as functions of (unobservable) dynamic factors (

).

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i) For instance, let us assume that

, independent of ,

, independent of , and

consider a linear factorial representation for the latent means:

(8)

where the factors satisfy a Gaussian Vector Autoregressive (VAR) model:

(9)

and the error terms are iid standard normal vectors

.

By introducing the factor representation (4)-(8)-(9), the parameters of time dependent

latent means are replaced by parameters. Note that some sensitivity

coefficients can be close to zero in practice. This can arise when one factor,

, say, is driving

the extreme risks. Thus the coefficient corresponding to this factor and to the high ratings will be

small. It can be easily checked that in the factor ordered qualitative model the rating histories are

no longer Markov processes. Each current rating is influenced by all past ratings, including also

the ratings of other firms which provide information on the (unobservable) past factor values.

In practice, the factors can correspond to some observable variables or be considered as un-

observables. The first approach has been followed for instance by Bangia et alii (2002), who

consider a single factor model and select the indicator of recession-expansion regularly reported

by the NBER as the factor. A similar approach has also been implemented by Nickell, Perraudin

and Varotto (2000) with several individual explanatory variables and a time dependent variable

related to business cycle, which is based on the GDP growth rate by country. They distinguish if

the country growth rate is in upper, middle or lower third of growth rate recorded in the sample

period. The approach with observable factors is simple to implement from the statistical point of

view, since we get a standard ordered probit regression model. However it can lead to misspecifi-

Since the factors are defined up to a linear invertible transformation, it is always possible to fix the covariance

matrix of the error term at identity.

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cation if the factor variable is not well selected. For instance, the importance of the U.S. business

cycle can be questioned on a set of firms, which includes almost 30% of foreign firms, and can

gather industrial sectors with different cycles. Moreover as mentioned earlier a model with ob-

servable factor can not handle efficiently the migration correlation feature and is difficult to use

for prediction purpose. Indeed it is usually more difficult to predict the future business cycles than

directly the migrations.

In a first step, it is preferable to search for factors intrinsic to the credit problem, to try to

reconstitute the factors (filtering step) and then possibly to interpret them ex-post as a function of

observed macro-variables.

3 Statistical inference

The parameters of interest can be estimated by exact or approximated maximum likelihood (ML)

for all models introduced above.

3.1 Deterministic transition matrices

When the transition matrices are deterministic and parameterized by , the -likelihood function

is:

(10)

where

denotes the number of firms which migrates from to between and .

In particular, the expression of the -likelihood function shows that the set of counts (

)

provide a sufficient statistic for parameters . This is a consequence of the cross-sectional homo-

geneity hypothesis.

3.1.1 Unconstrained model

Let us assume that the different transition matrices

are unconstrained, except the positivity and

unit mass restrictions. Then the parameter is

,

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and satisfies the constraints:

The unconstrained ML estimator of the transition matrices is given by :

(11)

where

denotes the number of firms in grade

at the beginning of period

. The uncon-

strained estimator corresponds to the transition frequencies for date t.

3.1.2 Constrained model

It is useful to introduce the transition frequencies in the expression of the

-likelihood function.

We get :

Both the transition frequencies

and the sample structure per grade (

) are

available aggregate information, and then the likelihood estimator can be used (if

is identifiable).

The fact that the structure per grade is now required is due to the constraints between transition

matrices of different dates by means of the common parameters

. Indeed the population of firms,

that are their size and structure per grade, change with time. The counts

are used to weight in

an appropriate way the information of the different dates . The objective criterion involves a mea-

sure

for the discrepancy between sample and theoretical transition probabilities,

called information criterion. It should seem natural to replace this measure by a chi-square mea-

sure

say. However the chi-square criterion has to be used with care; indeed,

whereas the number of firms is large, the major part of sample frequencies observed in practice are

see De Servigny, Renault (2002) for a discussion of weighting in transition models.

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equal to zero. In this situation in which the chi-square approximation (which is based on central

limit theorem) is not very accurate.

3.2 Stochastic transition matrices

The maximum likelihood approach is easily extended to stochastic transition models. Let us con-

sider for the discussion a factor model, where the transition probabilities depend on a (multivariate)

factor value

:

say (12)

and the factor satisfies a Gaussian VAR model:

(13)

where the errors

are i.i.d. standard normal. The parameters to be estimated are (i) the

parameters defining the transition probabilities, (ii) the parameter which characterizes the

factor dynamics. We assume that these parameters are identifiable. They are called micro-and

macro-parameters, respectively, in the general approach of error-in-factor model developped in

Gourieroux-Monfort (2004).

3.2.1 Simulated maximum likelihood

If both rating and factor histories were observed, the likelihood function would be:

(14)

When the factors are not observed, the distribution has to be integrated with respect to factor

values

. We get a likelihood function based on the rating histories only:

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where denotes the joint distribution of the factor values

. Thus the log-likelihood:

involves a multivariate integral with a dimension equal to , which can be very large. This

integral can not be computed easily and is replaced in practice by an approximation computed by

simulation to get the so-called simulated likelihood [see e.g. Gourieroux and Monfort (1995) for a

survey].

The simulated maximum likelihood (SML) estimator is defined as:

A

A

A A

where the simulated factor values are computed recursively by:

(15)

with the initial condition

, and the errors

independently drawn in the standard

normal. The initial condition is fixed at a past date to ensure that some stationary behaviour of the

factor is reached for the period of interest starting at .

3.2.2 Approximated linear factor model

The SML approach can be rather time consuming and it is useful to also present an estimation

method providing consistent results, even if they are partly subefficient. This first step estimation

will be used for the preliminary analysis of the stochastic migration model, in particular for deter-

mining the number of factors and constraining their dynamics. In a second step, it will be used as

initial value of the numerical algorithm used to maximize the simulated likelihood function. Let us

consider the factor ordered qualitative model where the Gaussian autoregressive factors are driving

the latent means [see equations (4)-(8)-(9)]. If the number of firms per rating class, that are

, are

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sufficiently large, the estimators

of the latent means, computed per year, will be close to the

true latent means. Thus we can write:

(16)

where the errors (

) are Gaussian . Thus we get approximately a Gaussian linear factor

model, which can be analyzed in the usual way by means of the Kalman filter. It provides estimates

of parameters A and var , but also approximated factor value (smoothing). It is proved

in Gourieroux, Monfort (2004) that these approximations are rather accurate. Typically if both

the time dimension and the cross-sectional dimension tend to infinity the estimators and the

smoothed values are consistent. Moreover if

tend to zero, the approximated estimator of

the macro-parameter are

consistent and efficient. If tend to zero the estimator of the

microparameter is

- consistent and the error on the smoothing value is of order .

4 Application

The deterministic and stochastic ordered qualitative transition models will be estimated from the

aggregate data regularly reported by Standard & Poors [ See Brady,Bos(2002), Brady, Vazza, Bos

(2003)]. In the first section, we describe the data set and explain how to correct the bias for missing

data on not rated companies. In Section 4.2, the deterministic models are estimated independently

for each year. This allows to derive the estimated cutting points

, as well

as the estimated mean

and variances

per rating category

and to observe how they vary in time. Different tests are performed to check for the time stability

of the parameters. In Section 4.3, we focus on the model with time independent cutting points:

independent of , and on the time series properties of the conditional mean and variance

They can be assumed Gaussian by the Central Limit Theorem applied to the estimator

.

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as a preliminary step before estimating an i.i.d. stochastic transition model in Section 4.4 and a

stochastic ordered qualitative transition model in Section 4.5.

4.1 Data set

The data set used in this paper has been obtained from Rating Performance 2002 provided by

Standard & Poors, which is free down-loadable at www.standardandpoors.com. The data con-

sists of yearly transition matrices from year 1981 to year 2002, reported in [Table 15: Static Pool

One-Year Transition Matrices in Standard & Poors (2003)], [see also Brady, Vazza and Bos

(2003)]. According to S&P rating system, there are

rating grades. They are AAA,AA, A,

BBB, BB, B, CCC and D from lowest risk to highest risk up to default state D [for

the definition of each rating and the correlation with other rating systems, refer to Foulcher et alii.

(2003)]. Since the published ratings focus on individual bond issuers, S&P convert their bond

rating to issuer ratings by considering the implied long-term senior unsecured rating [see Bangia

et alii (2002)] . Here for convenience, we use one-digital number 1,2, up to 8 for the rating

grades, and for example, 1 is for the highest rating category AAA, 8 for default D. The

number of states is . Therefore we have the following scheme:

Table 1: Scheme 1

R.C

AAA AA A BBB BB B CCC D

C.P

Note: R.C. and C.P. stand for Rating category and cutting point respectively.

The yearly transition matrix displays all rating movements during one year period and account

for missing data . A typical example of transition matrix is given in Table 2 and corresponds to

year 1997.

Similarly Nickell, Perraudin and Varotto (2000) considered long term corporate and sovereign bond ratings on the

Moodys data base.

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Table 2: Transition matrix of year 1997

Issuers

1 199 94.47 4.02 0.00 0.00 0.00 0.00 0.00 0.00 1.51

2 586 0.85 91.30 2.90 0.85 0.00 0.34 0.00 0.00 3.75

3 1161 0.00 1.64 89.15 3.70 0.17 0.43 0.00 0.00 4.91

4 846 0.00 0.35 3.66 86.29 2.72 0.71 0.12 0.35 5.79

5 557 0.00 0.00 0.18 8.62 76.12 4.67 0.00 0.18 10.23

6 479 0.00 0.00 0.63 0.42 7.10 74.53 2.51 3.34 11.48

7 28 0.00 0.00 0.00 0.00 0.00 14.29 53.57 10.71 21.43

This table has columns and rows. The rows represent the rating category at the beginning

of year 1997. For example, the first row refers to the rating category 1, or AAA, the last row

refers to the rating level 7, or CCC. The 8 or D is excluded because once the firm defaults,

it remains in default forever. The columns contain different information. The first column, named

Issuers, provides the numbers of long-term rated issuers on 12:01 a.m. January 1, 1997 per

rating, that is the structure per rating

at the beginning of the period . The columns 2 to 9

correspond to rating levels 1, to 8 at the end of year 1997. The last column, 9 corresponds

to the alternative N.R., which means not rated. It refers to issuers which are not rated at the

end of year, but were rated at the beginning of the year. As pointed out by Brady, Vazza and Bos,

(2003), Ratings are withdrawn when an entitys entire debt is paid off or when the program or

programs rated are terminated and the relevant debt extinguished. They may also occur as a result

of mergers and acquisitions. Others are withdrawn because of a lack of cooperation. From the

statistical point of view, the rating cannot be assigned due to a lack of information concerning the

balance sheet of firms. Thus they correspond to missing data. The proportions of missing data

in the available data bases from S&P and Moodys are rather high (between 10% and 20%). Let

us now discuss more precisely the data in columns 1 to 9. They provide the observed transition

frequencies for year 1997, including the N.R. alternative. For example, the third row shows that

See Brady Vazza and Bos, (2003) for the precise definition of the so-called static pool.

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out of 1161 firms rated A at the beginning of the year 1997: no one were rated as AAA at

the end of the year 1997; 1.64% were upgraded to AA; 89.15% stayed in the same category; the

proportion downrated to BBB, BB,B were 3.7%, 0.17% and 0.12%, respectively. The last

number, 4.91%, stands for the proportion of firms which were not rated.

In order to describe the rating migration, a complete rating migration structure is required. This

is not the case with the matrix given in Table 2, since the transition probabilities for the companies

not rated at the beginning of the year are not provided. Then two approaches can be followed:

i) we can include the alternative N.R. in the state space; or

ii) just consider the rating alternatives 1 to 8, that are AAA to D.

However the first approach requires including additional row for companies, which are not

rated at the beginning of 1997. Generally this information is not provided by the rating companies,

likely for confidentiality reasons. Indeed it could be used to find out the evolution of the population

of firms, which ask to be rated by the rating agency, and also its structure with respect to risk

quality. Because the first possibility requires additional information about firms, which is not

easily available, we will follow the second option, which excludes the N.R. alternative from

the transition matrix, as in [Foulcher et. al. (2003)]. For this purpose, the incomplete transition

matrix given by S&P is normalized, by assigning proportionally the N.R. firms among the other

categories (see below) . We get the so-called N.R.-adjusted transition matrix of year 1997 given

in Table 3 (as above, the row 8 or D is not reported).

Let us consider the third row of this matrix for example. The transition frequency from A

to AA is 1.72%, which is computed as . The N.R.-adjusted transition

matrices are used in the analysis below as measures of the unconstrained transition frequencies

.

The matrix defined in Table 3 is typical of an observed rating transition matrix. The frequen-

cies are close to one on the diagonal, which shows that the changes of categories are not very

This assignment of NR companies assumes no selection bias.

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Table 3: N.R. adjusted transition matrix of year 1997

1 95.92 4.08 0.00 0.00 0.00 0.00 0.00 0.00

2 0.88 94.87 3.01 0.88 0.00 0.35 0.00 0.00

3 0.00 1.72 93.75 3.89 0.18 0.45 0.00 0.00

4 0.00 0.37 3.89 91.60 2.89 0.75 0.13 0.37

5 0.00 0.00 0.20 9.60 84.79 5.20 0.00 0.20

6 0.00 0.00 0.71 0.47 8.02 84.19 2.84 3.77

7 0.00 0.00 0.00 0.00 0.00 18.19 68.18 13.63

frequent. The transition frequencies on the two diagonals below and above the main diagonal are

also significant, while the other ones are generally equal to zero. Indeed the changes of categories

(down- or up-grades) are at most by one bucket. In one year it takes some time to get a significant

change for more than one bucket. It happens mainly to firms, close to a failure which has not

been predicted by the rating agency; then the agency will quickly perform several down-grades to

correct for its prediction error. This effect can be viewed at the bottom of Table 3. We also observe

more heterogeneity in low ratings than in high ratings.

4.2 Estimation of the deterministic ordered qualitative model

In the first step, we consider the estimation of the deterministic ordered qualitative model from

the NR adjusted transition matrices computed from S&P data set. The estimation is performed

separately for each year, which provides approximation of the cutting points

, of the latent

means

and latent standard errors

,

Different estimation methods have been considered:

1) The maximum likelihood approach, for which the objective function corresponds to the

information criterion:

for model identification and 1 corresponds to the highest rating AAA.

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2) The chi-square type ( ) criterion, where only sufficiently large observed transition proba-

bilities are taken into account. The criterion is:

where

denotes the indicator function with value , if

, value , otherwise. The

threshold has been fixed to and .

The three estimation procedures provide similar results, except for the beginning of the period

. The estimated values of the different parameters for the maximum likelihood ap-

proach are displayed in Table 16, Table 17 and Table 18 in the appendix and the goodness of fit

measure,

is computed per year and reported in Table 4.

Table 4: Goodness of fit by ML

Year

0.005 0.017 0.023 0.021 0.037 0.038 0.017 0.022

Year 1989 1990 1991 1992 1993 1994 1995

0.024 0.027 0.028 0.032 0.050 0.006 0.008

Year 1996 1997 1998 1999 2000 2001 2002

0.016 0.007 0.099 0.013 0.016 0.025 0.032

The -goodness of fit statistics seems stable in time, but its value is difficult to interpret,

since the measure depends on the accuracy of transition probabilities close to zero and transition

probabilities close to .

The chi-square measure is not weighted by the inverse of the frequencies to avoid the problem of null observed

frequencies (see Table 3).

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Let us now discuss more carefully the estimated parameters. By definition the cutting points

are in increasing order. Moreover, for a firm in category (AAA) the probability to be downrated

to or less is equal to

. Since these probabilities are small, we expect

positive cutting points, which are observed in Table 16. Similarly we expect increasing positive

values for the mean per category, since the risk increases with and the mean has been fixed to

zero for the highest category [See Table 17]. Concerning the variances, the highest ones are for

category B, AAA (by convention,

), AA and BB.

The dynamics of the estimated parameters can be visualized by reporting their values as func-

tion of time. These time series are given in Figures 1.a, 1.b, 1.c. It has to be interpreted with

caution, since it can concern ratios of different parameters due to identification constraints.

For years , the estimates are more erratic. It seems that this is not caused by a

change in the risk on corporates, but rather by different data collecting techniques. Indeed the

quality of data has improved in recent years. In particular, the data base for the initial years 1981-

1987 is currently under revision. This explains for instance the big differences between transition

matrices reported by S&P in 2002 and 2003 for these years 1981-1987. Moreover the structure

of the population of firms by geography (North America, Western Europe, Asia) and by industry

(Manufacturing, Utilities, Financial Institute,...) has been more stable after 1990, as shown in

[Bangia et alii (2002) Figure 5, or in Nickell et alii (2000) for Moodys data set], Since we are

interested in time varying factor model, a use of the data base including the first years, during

which the proportions of foreign firms and of non manufacturing firms increase will provide a first

factor measuring the change of structure of the data base instead of measuring the risk fundamental.

For both reasons, in the sequel, we keep only the reliable data from 1990 to 2002.

4.3 Test of the ordered probit model

In Section 4.2 the deterministic ordered probit model has been estimated per year without in-

tertemporal constraints on the parameters. However it is important to check if some parameters

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can be assumed constant in the period 1990-2002 before introducing stochastic factors. Three con-

strained estimation procedures of the model for the whole sampling period 1990-2002 have been

performed with i) constant cutting points, ii) constant cutting points and constant latent variances

and iii) constant cutting points and constant latent means. We provide in Table 19 in the appendix

the -goodness of fit measures for the different years and constraints. These -measures are

small, that is the constrained model provides good fit, when either the cutting points, or the latent

variances are constrained to be constant. In the sequel, we consider the model with time invari-

ant cutting points, that is we assume that S&P does not modify the definition of the rating levels

in the different years. Under this restriction, the estimated latent means and stan-

dard deviations (squared root of variance) are given in Figure 2 and Figure 3 in the appendix for

the period :

As expected, the estimated latent means are in the right order: the higher the rating, the smaller

are the mean and the default probability (see Figure 2). However, contrary to what can be expected,

a similar ordering does not appear for the latent variance. This can be explained in two ways. First,

this is a consequence of the data themselves. When we consider the probit transformation on the

probabilities to stay in the same state

, and compute their historical variance,

we get the values: 0.21 for AAA, 0.15 for AA, 0.12 for A, 0.08 for BBB, 0.10 for BB, 0.09

for B and 0.25 for CCC. Thus the fact that the heterogeneity decreases when the rating category

improves is not observed. If the heterogeneity is the largest for CCC, we also observe rather large

values for AAA and AA. Second, the latent variances are more difficult to estimate than the latent

means. They require more observations and are very sensitive to the numerical computation of

the cumulative density function (cdf) of the standard normal in the tails for example. This lack of

robustness will be diminished in the sequel by constraining the latent variances ( and the cutting

points) to be time independent. Finally the latent means seem less stable than the latent variances,

especially for the risky classes. This is an incentive for introducing the factors through the latent

means in a first step. Finally the standardized cutting points

, where

is the

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cdf of the standard normal are provided in Table 5. They give the limiting migration probabilities

to categories worse than , for a firm rated 1, or AAA, and are compatible with the observed

probabilities (see the first row of Table 3).

Table 5: Standardized cutting points

1 2 3 4 5 6 7

0.051 2.12e-10 3.36e-11 7.30e-14 1.91e-17 1.35e-35 5.94e-38

4.4 Stochastic model with iid transition matrices

Let us now consider a stochastic specification of the dated transition matrices. Under the iid as-

sumption, we have seen in Section 2.2.1 that any individual rating history defines a Markov process

with transition matrix:

E

This matrix can be estimated by averaging the observed frequencies over time:

(17)

The estimated value of

is given below:

Table 6: Estimated aggregate transition matrix

1 94.88 5.12 0.00 0.00 0.00 0.00 0.00 0.00

2 0.00 91.64 6.14 2.21 0.00 0.00 0.00 0.00

3 0.00 0.00 94.64 5.36 0.00 0.00 0.00 0.00

4 0.00 0.39 4.66 87.98 6.98 0.00 0.00 0.00

5 0.00 0.00 0.01 6.42 82.94 10.64 0.00 0.00

6 0.00 0.12 0.11 1.03 5.06 82.66 4.62 6.39

7 0.00 0.00 0.00 0.00 0.00 10.05 61.12 28.83

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An idea of the dynamics of the associated chain is provided by the spectral decomposition of

the matrix. The estimated eigenvalues and eigenvectors are given in Table 7.

Table 7: Eigenvalues and eigenvectors of the aggregate estimated transition matrix

Values 1.000 0.988 0.949 0.920-0.005i 0.920+0.005i 0.844 0.749 0.590

V 1.000 0.942 1.000 -6.890-4.714i -6.890+4.714i -0.058 -0.004 0.000

E 1.000 0.717 0.000 3.430+3.381i 3.430-3.381i 0.119 0.015 0.001

C 1.000 0.652 0.000 0.580-0.111i 0.580+0.111i -0.445 0.124 0.003

T 1.000 0.503 0.000 -0.299-0.001i -0.299+0.001i 0.849 -0.455 -0.020

O 1.000 0.304 0.000 -0.749-0.092i -0.749+0.092i -0.142 0.771 0.080R 1.000 0.149 0.000 -0.461-0.041i -0.461+0.041i -0.531 -0.309 -0.169

S 1.000 0.040 0.000 -0.150-0.016i -0.150+0.016i -0.229 -0.226 0.784

1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Note:

is imaginary sign.

Since the expected transition matrix has mainly nonzero elements on the three main diagonals

and elements close to one on the main diagonal, it is rather close to an identity matrix, which

explains why the different eigenvalues have a large modulus , close to one.

Finally the migration risk created by the assumption of stochastic transition matrix can be

measured by means of covariances between the transition probabilities cov

. There is

a large number of such covariances, that is

. We provide in Table 8, the estimated standard

errors:

var

, which correspond to migration of the same type for two

firms and can be directly compared to the expectations given in Table 6.

4.5 Factor models

Let us now consider the possibility of introducing serially dependent stochastic transition matrices,

by means of latent factors which determine the latent means.

Recall that the eigenvalues of any transition matrix have a modulus smaller or equal to .

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Table 8: Standard error of stochastic transition probabilities

1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

2 0.000 0.116 0.047 0.016 0.000 0.000 0.000 0.000

3 0.000 0.000 0.050 0.050 0.000 0.000 0.000 0.000

4 0.000 0.000 0.014 0.033 0.068 0.000 0.000 0.000

5 0.000 0.000 0.000 0.069 0.067 0.253 0.000 0.000

6 0.000 0.000 0.000 0.005 0.057 0.033 0.023 0.084

7 0.000 0.000 0.000 0.000 0.000 0.517 0.246 1.179

4.5.1 Static factor analysis

In the first step, we perform a static factor analysis based on the latent means estimated per year.

For this purpose, we consider the series of estimated means in Table 20 [in the appendix], where

column number

, or AAA, corresponds to the identifying constraints. Each column represents

a time series of latent means. We provide in Table 21[in the appendix] the historical variance-

covariance matrix of these time series, and perform its spectral decomposition [see Table 22 in

the appendix]. It is observed that the largest eigenvalue is significantly larger than the other ones,

which indicates a one factor model.

4.5.2 Approximate linear analysis of the factor model

Let us now consider the one factor model:

(18)

where

,

are independent standard Gaussian variables, and

are de-

terministic coefficients. The

parameter provides a measure of the expected risk averaged on

time, whereas the time effect is captured by the sensitivity coefficient

with respect to the factor

. When the factor increases with aggregate risk, the larger

, the larger is the sensitivity with

respect to the aggregate risk.

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Since the latent means can be well approximated by the cross- sectional estimates

, we first

consider the approximate model:

where

are , and the measurement errors are approximately

Gaussian. Note that a factor is defined up to a multiplicative factor and that the variance of the

innovation

can been fixed to one for identification . The advantage of this specification is that

it is a special case of linear factor models, for which standard softwares are available .

Table 9 displays the first and second order autocorrelations of the estimated latent means for

each rating class. Indeed, before applying the usual Kalman filter methodology, it is important to

test for stationarity; we observe immediately that there is no unit root in the dynamics of the latent

means, and then in the factor dynamics.

Table 9: First and second order correlation of means estimates

First 0.265 0.498 0.286 0.507 0.590 0.416

Second 0.270 0.186 0.084 0.155 0.156 -0.064

The M.L. estimation of the approximated latent one-factor model is reported in Table 10,where

the error terms

have been assumed uncorrelated, with different variances

.

In a one factor model the factor is identified up to a scale effect. In the present estimation

the identification restriction provides a factor which is in a positive relationship with default risk.

Larger the factor value, larger the expected score, that is the expected probability of default. As

expected the sensitivity coefficients have all the same sign and tend to be much larger for the risky

The last identification restriction consists in choosing a factor which increases with aggregate risk, that is fixing

the sign in order to get positive sensitivity coefficients. As already mentioned, the error terms

are due to the estimation error. They are approximately Gaussian by

Central Limit Theorem. They are also conditionally heteroscedastic. The heteroscedasticity is not completely taken

into account in this first step, providing inefficient, but consistent estimators.

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Table 10: Estimation result of the approximate model

5.657 6.524 7.012 7.929 10.700 12.818

(0.082) (0.005) (0.041) (0.053) (0.384) (0.085)

0.003 0.019 0.03 0.231 0.049 5.829

(2.0e-4) (2.11e-3) (2.29e-3) (1.62e-2) (3.72e-3) 0.121

1.5e-3 0.028 0.029 0.151 0.039 0.361

(4.3e-9) (1.2e-4) (6.0e-4) (0.021) (1.8-4) (0.013)

0.02

(0.011)

Note: The standard errors are in parentheses.

class (remind that

for the AAA).

Of course, the analysis of heterogeneity is more difficult, since we have to distinguish between

the whole heterogeneity and the residual one. More precisely the quantitative score is given by:

Therefore its variance is equal to:

var

var

It involves two components corresponding to the factor effect and the innovation, respectively.

The ordering of whole heterogeneity discussed in Section 4.3 is expected on the sum of the compo-

nents, but a large value of the variance of the score can be obtained in different ways: for instance,

for bad risk CCC, the effect of the factor ( ) is very large, with a residual variance of

. Conversely for high rating AAA, there is by convention no factor effect (

), and a

rather large residual variance

.

The filtered factor values are given in Table 11 and reported in Figure 4. The evolution of

the factor is similar to the evolution of the total default rate as reported for instance in Brady,

Bos(2002), Chart 5, or Exhibit 4 in Hamilton, Cantor and Ou (2002).

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Table 11: filtered latent variable

0.967 0.653 -0.602 -1.192 -0.625 -0.730 -1.607

-0.852 -0.03 0.493 0.774 1.095 1.607

In fact the factor is essentially capturing the parallel evolution of downgrade probabilities, as

seen in Figure 5,

It is natural to compare this factor to some indicator of growth or business cycle. As noted in

Bangia et alii (2002), such a comparison is not easy to perform since the (U.S.) economy is mainly

in expansion during the period. More precisely the months of recession are from the third of 1981

to the fourth of 1982, the third of 1990 to the first of 1991, and the whole year of 2001 from the

NBER report [at NBER website]. Moreover the evolution is not in the same direction, which may

be due to either a lag, or an advance of the credit cycle with respect to the general cycle. We also

observe that the underlying factors do not feature jumps, but has some smoother variation. This

observed feature means that a dynamic of the factor by means of a three state Markov chain, to

distinguish cycle through, cycle normal, cycle peak [as in Nickell et alii (2000), Bangia et

alii (2002)] is likely misspecified.

4.5.3 The number of factors

Before implementing a more complicated estimation method, we have to check if it would be

necessary to introduce more factors. Different diagnostic tools are considered below.

First, the estimated means and their predictions deduced from the approximated latent factor

models are displayed in Figure 6, for different rating categories. The goodness of fit is rather good

for such a single factor model and the limited number of available temporal observations. The

factor model tends to smooth the time series of means and lacks the small risk observed in 1993, for

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the bad rating categories (BB,B,CC) or the high risk observed in 2003 for the high rating categories.

Clearly the introduction of a two-factor model will provide a first factor measuring the global risk

and a second factor to opposite the investment grades (AA,A,BBB) from the speculative categories

(BB,B,CCC). Second, we have computed the difference between the average square estimates

(

and the estimated (diagonal) variance-covariance matrix and computed the

spectral decomposition of the difference (see Table 24 in the appendix). Even if the first eigenvalue

is significantly larger than one, its value is rather small.

4.5.4 Simulated maximum likelihood of the factor probit model

Finally the one-factor probit model has been estimated by a simulated maximum likelihood ap-

proach, which is (asymptotically) more efficient from a theoretical point of view than the approxi-

mated ML approach, essentially for the sensitivity (micro) parameters. The estimation results are

given in Table 12, where the number of replications has been fixed to 2000. Whereas the estimated

and

coefficients are rather similar to those obtained with the approximated Kalman filter, the

estimation of

and the cutting points can be different, which reveal the lack of robustness for

heterogeneity estimation:

Table 12: SML estimation result of one factor model

5.661 6.776 7.326 8.2540 10.950 12.71

0.003 0.437 0.017 0.2467 0.057 0.823

0.431 0.022 0.280 0.242 1.257 1.229

0.01

To facilitate the interpretation of heterogeneity in terms of the rating category, the total latent

variance of the quantitative score is given in Table 13, with the proportion of the variance explained

by the factor.

The total variances per grade are ranked approximately with the same order than the historical

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Table 13: Heterogeneity

Rating AAA AA A BBB BB B CCC

Total variance 1.00 0.19 0.19 0.08 0.18 1.58 23.32

Part of explained variance(in %) 0.00 .01 99.19 0.35 55.26 0.20 30.98

variance of the probabilities to stay in the same category. Some improved smoothing latent factor

values can be derived by using the large cross-sectional dimension, which allows to avoid the

complicated nonlinear filtering. If

denote the coefficient estimates of Table 12, we have

approximately:

Then a smoothed value of

taking into account the improved estimates will be the OLS esti-

mator in the regression above:

(19)

These values have to be demeaned and standardized to satisfy the identification restriction. The

standardization is performed by dividing these factor values by the standard deviation of

E

E

. The improved smoothed values are given in Table 14.

Table 14: Smoothed factor values

0.775 0.715 -0.431 -1.879 -0.440 -0.151 -1.331

-1.052 -0.101 0.829 0.619 1.262 1.185

and reported in Figure 7 in the Appendix.

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5 Prediction of future ratings

An advantage of the unobservable factor ordered probit model is to be convenient for prediction

purpose, and to allow for an analysis of migration correlation. Let us first recall the prediction

formulas and apply them on the S&P data set.

5.1 Prediction for a given firm

Let us first consider a given firm. If the future values of the factor

were known,

the transition at horizon

would be:

and the distribution of its rating at date when its rating at date is would correspond

to the

row of the matrix

.

When the future factor values are not observed, the matrix above becomes stochastic, and has

to be integrated with respect to

conditional on

. The integrated matrix is:

E

This matrix has no explicit expression, but the prediction of the rating at

can be per-

formed by simulation along the following steps:

Step 1: Simulate a future path of the noise

, and deduce the associated future

factor values

by applying the autoregressive formula;

Step 2: Compute the matrix

and its row number

:

,

say.

Step 3: Replicate the simulation

times and compute

, which is

an approximation of the row of the matrix

.

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5.2 Prediction for two given firms

Let us now consider two firms and , say, where ratings are known at date . Their joint transition

are summarized by a matrix , which gives the probability that firms and currently in

grate

and

, respectively, are in grade

and

at , say. If the future value of the factor is

known, this joint transition matrix is:

where

denotes the tensor product, which associates with the matrices

, the matrix

with block decomposition

. If the sequence of future values is known, the joint migration

matrix at horizon becomes:

When only the current factor value is known, this matrix has to be integrated to get:

E

E

This matrix can not be written in general as a tensor product, which means that migration

correlations have been created by the common effect of the unobservable factor. As above this

matrix has no explicit expression, but can be well approximated by simulation.

5.3 Prediction of future ratings

Let us now consider the prediction of future ratings for the S&P data base. For this purpose the

factor value will be fixed to its filtered value computed for 2002, that is

, and the

parameters to their estimates given in Table 12.

We perform replications of a simulated factor path to get the matrices

,

at horizon 1 year, 2 year, 5 year and 10 year. The matrices for are given in Table

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25 in the appendix. We also compute the difference

for

the following joint migrations corresponding to joint up grades , to joint

stability and to joint downgrades . (see Table 15, Table

26, Table 27 and Table 28 in the appendix) to get more information about the term structure of the

migration correlations.

Table 15: The key cells in

for 1 year horizon

co-movement AAA AA A BBB BB B CCC

AAA

AA 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

up A 0.0000 0.0298 0.0010 0.0231 0.0002 0.0274

1 BBB 0.0000 0.0010 0.0002 -0.0003 0.0000 0.0034

bucket BB 0.0000 0.0237 0.0012 0.0234 1e-04 0.0367

B 0.0000 0.0002 0.0000 -0.0004 0.0000 0.0006

CCC 0.0000 0.0274 0.0034 0.0307 0.0006 0.1184

AAA -0.0001 -0.0001 0.0000 0.0000 0.0705 0.0000 0.0000

AA -0.0001 0.0000 0.0001 0.0000 0.0608 0.0000 0.0000

unchanged A 0.0000 0.0001 0.1300 -0.0008 0.1627 0.0016 0.0733

BBB 0.0000 0.0000 -0.0008 0.0001 -0.0616 0.0001 -0.0005BB 0.0705 0.0680 0.1627 0.0616 0.2037 0.0599 0.1244

B -1e-04 -1e-04 0.0017 0.0000 0.0020 0.0000 0.0011

CCC 0.0000 0.0000 0.0733 -0.0005 0.1244 0.0011 0.1407

AAA 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

AA 0.0000 0.0000 0.0002 0.0000 0.0003 0.0000 0.0004

down A 0.0000 0.0003 0.1099 0.0011 0.0868 0.0018 0.0886

1 BBB 0.0000 0.0000 0.0011 0.0000 -0.0001 0.0000 0.0018

bucket BB -0.0019 -0.003 0.0868 -0.0001 0.0823 -0.0034 0.1085

B 0.0000 0.0000 0.0018 0.0000 -0.0034 0.0001 0.0029

CCC 0.0000 4e-04 0.0886 0.0018 0.1085 0.0029 0.1789

Rather large values can be observed even for rather high rating sector A, but these values tend

to diminish when the horizon increases. This result is not surprising since the current rating is less

informative when the horizon increases.

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6 Concluding remarks

The aim of this paper was to implement a factor probit model for rating transitions as suggested

in the theoretical and applied literature, to see if such a specification is suitable for regular compu-

tation of CreditVaR. The message is fourfold: i) a one-factor model seems to produce reasonable

prediction of the expected risk per rating category,

ii) but the estimated heterogeneity is not robust, which can have severe consequences for the

reliability of CreditVaR computed from these estimations. This is partly a consequence of the data

bases, which are currently available and do not include a large number of years, with data of good

quality. But this is also a consequence of the transitions themselves, and of the few number of

down- or up-grades of more of one bucket in a given year.

(iii) It seems preferable to introduce an unobservable factor instead of assuming it related to

some macromeasure of the business cycle. In some sense the available data set on credit migration

is sufficiently rich at the aggregate level to reveal the credit cycles.

iv) Finally the estimation of such a nonlinear dynamic model involves rather sophisticated

estimation techniques such as simulated based estimation methods or nonlinear filtering, which

are not commonly used by the research-development groups of the banks and not available in

standard software.

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Table 16: Estimates of cutting points by ML

Year

1981 1.3661 4.0453 6.7862 7.9719 11.1906 14.4934 21.7740

1982 1.2328 5.4060 5.7112 6.3915 7.4043 12.5313 13.1165

1983 0.8836 4.9459 5.3035 6.3230 7.3473 14.3599 14.5940

1984 0.4762 4.5752 5.0654 6.0260 7.0337 13.5765 13.8405

1985 1.4730 5.6924 5.9981 6.6785 7.6913 12.8185 13.4035

1986 1.3981 6.1111 6.8695 8.2936 9.3165 15.6508 16.2461

1987 1.6781 3.0130 3.4471 4.9318 5.3918 10.2598 10.6971

1988 1.7516 6.3559 6.5843 7.4906 8.5181 12.4404 12.8253

1989 1.6096 6.2486 6.4738 7.3797 8.4072 12.3099 12.6922

1990 1.7523 6.3558 6.5841 7.4904 8.5179 12.4398 12.82471991 1.7528 6.3558 6.5841 7.4904 8.5178 12.4403 12.8253

1992 1.3120 6.3407 6.8013 7.4614 8.4754 12.4825 13.0590

1993 1.7714 6.4200 6.6501 7.5566 8.5842 12.5176 12.8985

1994 1.3740 5.6667 6.1096 6.8535 7.8650 12.6246 13.1608

1995 1.7346 6.4124 6.7819 7.6652 8.7394 12.6521 12.9990

1996 1.4842 5.9733 6.3003 7.0845 8.0982 12.3535 12.9919

1997 1.7430 6.3529 6.5814 7.4876 8.5151 12.4344 12.8191

1998 1.7367 6.3444 6.5732 7.4797 8.5071 12.4302 12.8151

2000 1.7455 6.2718 6.4996 7.4063 8.4337 12.3694 12.7540

2001 1.7523 6.3540 6.5822 7.4885 8.5160 12.4418 12.82682002 1.7521 6.3554 6.5837 7.4900 8.5175 12.4392 12.8243

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Table 17: Estimates of latent means by ML

Year

1981 2.9240 5.5025 7.3725 10.4454 11.2920 14.5371

1982 4.8464 5.6891 6.1221 6.9520 10.2301 12.93371983 4.3121 5.2506 5.8132 6.8925 11.0118 14.4256

1984 3.9321 4.9777 5.5373 6.4559 10.1781 13.8242

1985 5.2212 5.7261 6.4060 7.2337 10.5059 12.9768

1986 4.2245 6.1843 7.6656 8.7729 13.5479 16.1361

1987 2.4434 3.3273 4.2742 5.1666 7.8905 10.4983

1988 5.8423 6.5490 7.0491 8.0426 10.5814 12.6925

1989 5.5406 6.4430 6.9350 7.7662 10.3837 12.6760

1990 5.7538 6.5526 7.0769 8.0415 10.9218 12.7546

1991 5.6897 6.5506 7.0729 8.0420 11.0193 12.7536

1992 5.7174 6.5560 7.1403 7.8629 10.5612 12.85171993 5.6975 6.6095 7.1416 8.0309 10.0311 12.6133

1994 5.0663 5.8367 6.4616 7.3098 10.2538 12.9480

1995 5.7881 6.7248 7.1926 8.1587 10.6233 12.8957

1996 5.6740 6.2695 6.6451 7.5289 10.0106 12.6047

1997 5.6163 6.5420 7.0434 7.9396 10.3595 12.6089

1998 5.6073 6.5370 7.0784 8.0356 10.5839 12.6892

1999 5.6149 6.4652 6.9496 7.9675 10.8201 12.6900

2000 5.7456 6.5506 7.0677 8.0415 10.8981 12.7421

2001 5.6990 6.5483 7.0740 8.0409 11.2624 12.7868

2002 5.9045 6.5550 7.1595 8.0407 11.0484 12.7882

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Table 18: Estimates of latent standard deviations by ML

Year

1981 0.7560 0.8638 0.3619 1.5557 0.0624 0.0341

1982 0.4357 0.0172 0.2267 0.3773 1.6596 0.28891983 0.4133 0.0318 0.3698 0.4718 2.1354 0.0598

1984 0.4167 0.0483 0.4454 0.3881 1.9095 0.0169

1985 0.4360 0.0172 0.2272 0.3774 1.6596 0.2888

1986 1.4233 0.0468 0.5999 0.3542 2.2799 0.1247

1987 0.3647 0.0017 0.5493 0.1703 1.5875 0.1695

1988 0.4667 0.0224 0.2960 0.3593 1.3576 0.2073

1989 0.4655 0.0204 0.2957 0.3701 1.3576 0.2132

1990 0.4668 0.0224 0.2960 0.3593 1.3577 0.2072

1991 0.4668 0.0224 0.2960 0.3593 1.3577 0.2072

1992 0.4551 0.0178 0.2457 0.3470 1.6591 0.29051993 0.4650 0.0225 0.2887 0.3657 1.3589 0.2068

1994 0.4259 0.0205 0.2093 0.3103 1.5746 0.2509

1995 0.4298 0.0313 0.2647 0.3636 1.3586 0.1959

1996 0.1866 0.0148 0.2276 0.3624 1.4301 0.2482

1997 0.4670 0.0224 0.2963 0.3593 1.3582 0.2071

1998 0.4665 0.0225 0.2963 0.3591 1.3585 0.2075

1999 0.4665 0.0218 0.2974 0.3596 1.3607 0.2046

2000 0.4668 0.0224 0.2960 0.3593 1.3576 0.2072

2001 0.4669 0.0224 0.2955 0.3595 1.3576 0.2074

2002 0.4668 0.0224 0.2960 0.3594 1.3577 0.2073

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Table 19:

-goodness of fit by ML under three constraintsYear Case 1 Case 2 Case 3

1990 0.030 0.032 0.028

1991 0.027 0.031 0.023

1992 0.046 0.037 0.154

1993 0.067 0.067 0.117

1994 0.013 0.014 0.033

1995 0.011 0.006 0.012

1996 0.052 0.034 0.121

1997 0.004 0.005 0.087

1998 0.113 0.122 0.0121999 0.013 0.009 0.012

2000 0.014 0.012 0.011

2001 0.032 0.030 0.034

2002 0.031 0.030 0.050

Note: Case1: Only cutting points are unchanged; Case 2: Cutting points and variances are unchanged, Case 3: Cutting

points and means are unchanged.

Table 20: Estimated latent means when cutting points and variances are constant

Year 1 2 3 4 5 6 7

c.p. 1.6497 6.2765 6.5597 7.4233 8.4603 12.5324 12.9579

1.000 0.433 0.022 0.28 0.343 1.410 0.218

m 1990 0.000 5.7248 6.5291 7.0309 7.9798 10.9591 12.8786

e 1991 0.000 5.6636 6.5271 7.0242 7.9362 11.0634 12.8781

a 1992 0.000 5.6705 6.5206 7.0042 7.8307 10.5696 12.8042

n 1993 0.000 5.6101 6.5203 7.0259 7.9093 9.9879 12.6414

s 1994 0.000 5.6519 6.5215 6.9756 7.8951 10.4830 12.7901

1995 0.000 5.6429 6.5202 6.9593 7.8950 10.4332 12.8302

1996 0.000 5.5652 6.5144 6.9489 7.8718 10.2800 12.7038m 1997 0.000 5.5719 6.5215 7.0002 7.8775 10.3869 12.7262

e 1998 0.000 5.5827 6.5245 7.0415 7.9467 10.6204 12.8053

a 1999 0.000 5.5991 6.5254 6.9858 7.9901 10.9386 12.8863

n 2000 0.000 5.7165 6.5275 7.0214 7.9708 10.9364 12.8643

s 2001 0.000 5.6751 6.5266 7.0283 7.9909 11.3175 12.9118

2002 0.000 5.8703 6.5315 7.1108 7.9834 11.0926 12.9162

Note: c.p. stands for cutting point.

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Table 21: Variance and covariance matrix of mean estimates

0.00676 2.826e-04 0.00247 0.00198 0.01856 0.00444

0.00028 2.143e-05 0.00015 0.00020 0.00146 0.00031

0.00246 1.525e-04 0.00172 0.00121 0.00792 0.00147

0.00198 1.992e-04 0.00121 0.00279 0.01515 0.00310

0.01856 1.464e-03 0.00792 0.01515 0.14754 0.03078

0.00444 3.147e-04 0.00147 0.00310 0.03078 0.00720

Table 22: Eigenvalues and eigenvectors of variance and covariance matrix

1 2 3 4 5 6 7

Values 1.584e-01 4.942e-03 1.487e-03 8.490e-04 3.082e-04 1.541e-06

V 0.126 0.917 -0.191 -0.063 -0.321 0.002

E 0.009 0.023 0.035 -0.009 0.043 -0.998

C 0.053 0.358 0.478 0.427 0.675 0.051

T 0.100 0.033 0.782 -0.574 -0.218 0.024O 0.964 -0.158 -0.009 0.185 -0.103 -0.001

R 0.203 0.070 -0.350 -0.671 0.617 0.024

Table 23: Variance matrix of prediction errors

1 2 3 4 5 6 71 0.004 0.000 0.001 0.001 0.004 0.001

2 0.000 0.000 0.000 0.000 0.001 0.000

3 0.001 0.000 0.001 0.001 0.002 0.000

4 0.001 0.000 0.001 0.002 0.009 0.002

5 0.004 0.001 0.002 0.009 0.073 0.016

6 0.001 0.000 0.000 0.002 0.016 0.004

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Table 24: Eigenvalues and eigenvectors of the difference of matrices

1 2 3 4 5 6

Values 0.074 0.004 0.000 -0.001 -0.021 -0.127

V 0.057 0.950 0.307 0.005 -0.010 -0.008

E 0.010 0.022 -0.053 -0.998 -0.003 -0.001

C 0.029 0.304 -0.950 0.058 -0.023 -0.001

T 0.093 0.011 -0.018 -0.001 0.995 -0.010

O 0.990 -0.065 0.012 0.008 -0.092 -0.077

R 0.078 0.003 0.003 0.000 0.003 0.997

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Table 25: The prediction of transition matrix,

Horizon AAA AA A BBB BB B CCC D

AAA 0.9505 0.0495 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

AA 0.0000 0.9141 0.0847 0.0012 0.0000 0.0000 0.0000 0.0000

1 A 0.0000 0.0334 0.8192 0.1472 0.0002 0.0000 0 .0000 0.0000

year BBB 0.0000 0.0001 0.1191 0.8432 0.0375 0.0000 0.0000 0.0000

BB 0.0000 0.0000 0.0000 0.0487 0.7141 0.2372 0.0000 0.0000

B 0.0000 0.0001 0.0007 0.0053 0.0177 0.7803 0.1441 0.0519CCC 0.0000 0.0000 0.0000 0.0000 0.0000 0.2014 0.4125 0.3860

AAA 0.9034 0.0923 0.0042 0.0001 0.0000 0.0000 0.0000 0.0000

AA 0.0000 0.8381 0.1477 0.0141 0.0000 0.0000 0.0000 0.0000

2 A 0.0000 0.0550 0.7060 0.2336 0.0054 0.0000 0 .0000 0.0000

year BBB 0.0000 0.0038 0.1993 0.7295 0.0585 0.0088 0.0000 0.0000

BB 0.0000 0.0000 0.0060 0.0753 0.5281 0.3463 0.0326 0.0117

B 0.0000 0.0002 0.0017 0.0096 0.0268 0.6419 0.1719 0.1479

CCC 0.0000 0.0000 0.0001 0.0011 0.0036 0.2418 0.2057 0.5476

AAA 0.7758 0.1884 0.0307 0.0050 0.0001 0.0000 0.0000 0.0000

AA 0.0000 0.6598 0.2563 0.0802 0.0033 0.0005 0.0000 0.0000

5 A 0.0000 0.1037 0.4968 0.3627 0.0278 0.0080 0 .0007 0.0004

year BBB 0.0000 0.0283 0.3069 0.5355 0.0764 0.0422 0.0059 0.0048

BB 0.0000 0.0015 0.0319 0.1085 0.2205 0.4029 0.0936 0.1411

B 0.0000 0.0006 0.0060 0.0185 0.0325 0.4040 0.1279 0.4105

CCC 0.0000 0.0001 0.0012 0.0048 0.0105 0.1805 0.0634 0.7395

AAA 0.6019 0.2736 0.0886 0.0332 0.0019 0.0006 0.0001 0.0001

AA 0.0000 0.4636 0.3207 0.1906 0.0159 0.0071 0.0010 0.001110 A 0.0000 0.1298 0.3861 0.3885 0.0475 0.0330 0.0059 0.0093

year BBB 0.0000 0.0652 0.3266 0.4114 0.0673 0.0741 0.0162 0.0393

BB 0.0000 0.0082 0.0589 0.1040 0.0721 0.2760 0.0789 0.4018

B 0.0000 0.0019 0.0124 0.0241 0.0232 0.2023 0.0629 0.6732

CCC 0.0000 0.0005 0.0036 0.0080 0.0093 0.0906 0.0285 0.8596

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for 2-year horizon


AAA AA 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

up A 0.0000 0.0445 -0.0009 0.0292 0.0001 0.0260

1 BBB 0.0000 -0.0009 0.0023 -0.0099 0.0005 0.0037

bucket BB 0.0000 0.0292 -0.0099 0.0187 -0.0011 0.0230

B 0.0000 0.0001 0.0005 -0.0011 0.0001 0.0009

CCC 0.0000 0.0260 0.0037 0.0230 0.0009 0.1008

AAA 0.0001 0.0000 0.0000 -0.0012 0.0930 -0.0006 0.0001

AA 0.0000 0.0001 -0.0014 -0.0013 0.0856 -0.0003 -0.0003

A 0.0000 -0.0014 0.1678 -0.0129 0.1915 0.0006 0.0513

unchanged BBB -0.0012 -0.0013 -0.0129 -0.0002 0.0655 -0.0020 -0.0039

BB 0.0930 0.0856 0.1915 0.0655 0.2104 0.0678 0.0861

B -0.0006 -0.0003 0.0006 -0.0020 0.0678 0.0018 -0.0021

CCC 0.0001 -0.0003 0.0513 -0.0039 0.0861 -0.0021 0.0672

AAA 0.0000 0.0000 0.0000 0.0003 -0.0037 0.0000 -0.0001

AA 0.0000 0.0009 -0.0070 0.0007 -0.0107 0.0008 -0.0024

down A 0.0000 -0.0070 0.1425 -0.0005 0.0907 -0.0053 0.0821

1 BBB 0.0003 0.0007 -0.0005 0.0004 -0.0021 0.0009 0.0017

bucket BB -0.0037 -0.0107 0.0907 -0.0021 0.0672 -0.0131 0.0817

B 0.0000 0.0008 -0.0053 0.0009 -0.0131 0.0028 -0.0050

CCC -0.0001 -0.0024 0.0821 0.0017 0.0817 -0.0050 0.1685

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for 5-year horizon


AAA AA 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

up A 0.0000 0.0643 -0.0080 0.0287 0.0002 0.0174

1 BBB 0.0000 -0.0080 0.0191 -0.0172 0.0034 0.0030

bucket BB 0.0000 0.0287 -0.0172 0.0124 -0.0006 0.0124

B 0.0000 0.0002 0.0034 -0.0006 0.0005 0.0018

CCC 0.0000 0.0174 0.0030 0.0124 0.0018 0.0454

AAA 0.0000 0.0000 0.0000 -0.0045 0.0623 -0.0011 0.0000

AA 0.0000 0.0036 -0.0070 -0.0045 0.0508 0.0013 -0.0002

A 0.0000 -0.0070 0.1413 -0.0385 0.1110 0.0015 0.0113

unchanged BBB -0.0045 -0.0045 -0.0385 0.0094 0.0249 -0.0052 -0.0035

BB 0.0623 0.0508 0.1110 0.0249 0.0813 0.0363 0.0177

B -0.0011 0.0013 0.0015 -0.0052 0.0363 0.0065 0.0006

CCC 0.0000 -0.0002 0.0113 -0.0035 0.0177 0.0006 0.0079

AAA 0.0000 -0.0002 -0.0004 0.0021 0.0005 -0.0002 -0.0001

AA -0.0002 0.0113 -0.0258 0.0053 -0.0090 0.0033 -0.0039

down A -0.0004 -0.0258 0.1216 -0.0007 0.0531 -0.0087 0.0385

1 BBB 0.0021 0.0053 -0.0007 0.0024 0.0023 0.0028 0.0076

bucket BB 0.0005 -0.0090 0.0531 0.0023 0.0404 -0.0062 0.0360

B -0.0002 0.0033 -0.0087 0.0028 -0.0062 0.0046 -0.0048

CCC -0.0001 -0.0039 0.0385 0.0076 0.0360 -0.0048 0.0893

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for 10-year horizon


AAA AA 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

up A 0.0000 0.0531 -0.0097 0.0145 0.0007 0.0070

1 BBB 0.0000 -0.0097 0.0354 -0.0166 0.0041 0.0002

bucket BB 0.0000 0.0145 -0.0166 0.0072 0.0000 0.0052

B 0.0000 0.0007 0.0041 0.0000 0.0005 0.0009

CCC 0.0000 0.0070 0.0002 0.0052 0.0009 0.0122

AAA 0.0000 0.0000 -0.0007 -0.0055 0.0175 0.0005 0.0000

AAA 0.0000 0.0125 -0.0100 -0.0083 0.0123 0.0030 0.0000

AA -0.0007 -0.0100 0.0763 -0.0422 0.0293 0.0013 0.0020

A -0.0055 -0.0083 -0.0422 0.0232 0.0033 -0.0030 -0.0014

unchanged BBB 0.0175 0.0123 0.0293 0.0033 0.0114 0.0076 0.0024

BB 0.0005 0.0030 0.0013 -0.0030 0.0076 0.0047 0.0006

B 0.0000 0.0000 0.0020 -0.0014 0.0024 0.0006 0.0017

CCC 0.0004 -0.0013 -0.0025 0.0036 0.0086 -0.0001 -0.0002

AA -0.0013 0.0308 -0.0351 0.0083 0.0068 0.0026 -0.0018

down A -0.0025 -0.0351 0.0677 0.0021 0.0220 -0.0039 0.0056

1 BBB 0.0036 0.0083 0.0021 0.0026 0.0051 0.0016 0.0119

bucket BB 0.0086 0.0068 0.0220 0.0051 0.0260 0.0011 0.0289

B -0.0001 0.0026 -0.0039 0.0016 0.0011 0.0014 -0.0017

CCC -0.0002 -0.0018 0.0056 0.0119 0.0289 -0.0017 0.0282

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Figure 1a Estimates of the mean

m1

85 90 95 00

3

4

5

6

m2

85 90 95 00

4

5

6

m3

85 90 95 00

5

6

7

m4

85 90 95 00

5

6

7

8

9

10

m5

85 90 95 00

8

9

10

11

12

13

m6

85 90 95 00

11

12

13

14

15

16

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Figure 1c :Estimated of cutting points

a1

85 90 95 00

0.6

0.8

1.0

1.2

1.4

1.6

1.8

a2

85 90 95 00

3

4

5

6

Estimates of cutting points

a3

85 90 95 00

4

5

6

7

a4

85 90 95 00

5

6

7

8

a5

85 90 95 00

6

7

8

9

10

11

a6

85 90 95 00

11

12

13

14

15

a7

85 90 95 00

12

14

16

18

20

2

2

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Figure 2 Estimated means, constant cutting points

89 91 93 95 97 99 01 035

6

7

8

9

10

11

12

13

Year

estimates

m2

m3

m4

m5

m6

m7

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Figure 3 Estimated standard deviations, constant cutting points

89 91 93 95 97 99 01 030

0.5

1

1.5

Year

estimates

2

3

4

5

6

7

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Figure 4 Smoothed latent variable

Year

1990 1992 1994 1996 1998 2000 2002

-1.

5

-1.

0

-0.

5

0.

0

0.

5

1.

0

1.

5

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Figure 5 Evolution of downgrade probabilities

time

Probability

1990 1992 1994 1996 1998 2000 2002

0.0

4

0.0

6

0.0

8

0.1

0

0.1

2

0.1

4

0.1

6

0.1

8

AA

time

Probability

1990 1992 1994 1996 1998 2000 2002

0.0

2

0.0

4

0.0

6

0.0

8

0.1

0

A

time

Probability

1990 1992 1994 1996 1998 2000 2002

0.0

2

0.0

4

0.0

6

0.0

8

0.1

0

BBB

time

Probability

1990 1992 1994 1996 1998 2000 2002

0.0

5

0.1

0

0.1

5

BB

time

Probability

1990 1992 1994 1996 1998 2000 2002

0.0

5

0.1

0

0.1

5

0.2

0

B

time

Probability

1990 1992 1994 1996 1998 2000 2002

0.1

0.2

0.3

0.4

CCC

Figure 6 Means and predicted means

89 94 99 045.55

5.6

5.65

5.7

5.75

5.8

5.85

5.9AA

Year

Means

Predictions

89 94 99 046.51

6.515

6.52

6.525

6.53

6.535

6.54A

Year

Means

Predictions

89 94 99 046.9

6.95

7

7.05

7.1

7.15BBB

Year

Means

Predictions

89 94 99 047.8

7.85

7.9

7.95

8BB

Year

Means

Predictions

89 94 99 049.5

99

10.5

11

11.5B

Year

Means

Predictions

89 94 99 0412.6

12.65

12.7

12.75

12.8

12.85

12.9

12.95CCC

Year

Means

Predictions

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Figure 7 Smoothed Factor Values

Year

Smoothed

1990 1992 1994 1996 1998 2000 2002-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Documents

Risk Rating Article 2004