The Ordered Qualitative Model for Credit Rating Transitions

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ISSN: 1707-410X

The Ordered Qualitative Model for Credit Rating Transitions Dingan Feng Christian Gouriéroux Joann Jasiak CREF 04-05 April 2004

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The Ordered Qualitative Model for Credit Rating Transitions

Dingan Feng Post Doctoral Student CREF and York University Department of Economics 4700 Keele Street Toronto, Ontario M3J 1P3 E-Mail:[email protected]

Christian Gouriéroux Professor CREF, CREST, CEPREMAP and University of Toronto 15, boulevard Gabriel Péri 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

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 the risk included in a credit portfolio at different horizons. In this paper, we present the estimation of an ordered probit model with factors for the migration probabilities, with its application to aggregate data regularly reported by Standard & Poor's.

Keywords: Credit Rating, Migration JEL : C23, C35, G11

1

Introduction

Credit ratings for firms and bond issuing are regularly reported by specialized rating agencies such as Moody’s, Standard &Poor’s 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 deterministic 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 Standard & Poor’s, 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.

1

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 homogeneity 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 ries (

Thus the individual histo-

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-

2

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 probabilities 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,

where by convention,

if and only if

and

(2)

. 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:

see

if

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

3

(3)

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,

½

or:

(4)

is the probability of default of a firm rated ¼¼; similarly is the probability of migration to the highest category “1”, or AAA. are identifiable up to some linear The parameters For example,

½

½

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:

or implicitly

if otherwise

. 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 Merton’s model.

4

2.2 Stochastic transition models The deterministic models can be extended to allow for stochastic transition matrices. In this respect, 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 Gouriéroux (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:

¼

(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.

5

firm is in state at date ifotherwise 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 previous ratings are known. By the covariance decomposition equation, we get:

cov Ecov covE 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 computation shows that stochastic transition matrices are introduced to define non-zero cross-sectional correlations. ii) Any individual history is a homogeneous Markov process.

6

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

E EE 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 independent 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 financial 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 ( ).

7

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 unobservables. 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.

8

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 observable 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 homogeneity 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

9

,

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 unconstrained 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-

for the discrepancy between sample and theoretical transition probabilities,

sure

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

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

¾

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.

10

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 consider 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:

"

11

" #" #"

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. Gouriéroux 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:

" " & with the initial condition "

, and the errors

(15)

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 determining 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 12

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 microparameter is

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

- 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 & Poor’s [ 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 the estimated mean

and variances

, as well

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:

They

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

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

13

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 & Poor’s, which is free down-loadable at “www.standardandpoors.com”. The data consists of yearly transition matrices from year 1981 to year 2002, reported in [Table 15: Static Pool One-Year Transition Matrices” in Standard & Poor’s (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 C.P

AA

A

BBB

BB

B

CCC D

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

Moody’s data base.

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Table 2: Transition matrix of year 1997

1 2 3 4 5 6 7

Issuers 199 94.47 4.02 0.00 0.00 0.00 0.00 0.00 0.00 1.51 586 0.85 91.30 2.90 0.85 0.00 0.34 0.00 0.00 3.75 1161 0.00 1.64 89.15 3.70 0.17 0.43 0.00 0.00 4.91 846 0.00 0.35 3.66 86.29 2.72 0.71 0.12 0.35 5.79 557 0.00 0.00 0.18 8.62 76.12 4.67 0.00 0.18 10.23 479 0.00 0.00 0.63 0.42 7.10 74.53 2.51 3.34 11.48 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 entity’s 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 Moody’s 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.

15

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 frequencies 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.

16

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.

17

2) The chi-square type (* ) criterion, where only sufficiently large observed transition probabilities are taken into account. The criterion is:

½

where ½ 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).

18

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 function 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 19811987 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 Moody’s 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 intertemporal constraints on the parameters. However it is important to check if some parameters 19

can be assumed constant in the period 1990-2002 before introducing stochastic factors. Three constrained 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 invariant 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 standard 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 20

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 assumption, 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:

The estimated value of is given below:

(17)

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

21

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 V E C T O R S

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.988 0.942 0.717 0.652 0.503 0.304 0.149 0.040 0.000

0.949 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.920-0.005i -6.890-4.714i 3.430+3.381i 0.580-0.111i -0.299-0.001i -0.749-0.092i -0.461-0.041i -0.150-0.016i 0.000

0.920+0.005i -6.890+4.714i 3.430-3.381i 0.580+0.111i -0.299+0.001i -0.749+0.092i -0.461+0.041i -0.150+0.016i 0.000

0.844 -0.058 0.119 -0.445 0.849 -0.142 -0.531 -0.229 0.000

0.749 0.590 -0.004 0.000 0.015 0.001 0.124 0.003 -0.455 -0.020 0.771 0.080 -0.309 -0.169 -0.226 0.784 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 errors:

var

. We provide in Table 8, the estimated standard

¾ ¾

, 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 .

22

Table 8: Standard error of stochastic transition probabilities 1 2 3 4 5 6 7

0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.116 0.000 0.000 0.000 0.000 0.000

0.000 0.047 0.050 0.014 0.000 0.000 0.000

0.000 0.016 0.050 0.033 0.069 0.005 0.000

0.000 0.000 0.000 0.068 0.067 0.057 0.000

0.000 0.000 0.000 0.000 0.253 0.033 0.517

0.000 0.000 0.000 0.000 0.000 0.023 0.246

0.000 0.000 0.000 0.000 0.000 0.084 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 ¼¼ ¼¼ , 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 variancecovariance 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:

where "

." ,

"

(18)

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. 23

Since the latent means can be well approximated by the cross- sectional estimates , we first consider the approximate model:

where "

."

" '

are //0 , 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.

24

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:

1 " 1

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

) 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 . for bad risk “CCC”, the effect of the factor (

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). 25

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

26

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 approach, which is (asymptotically) more efficient from a theoretical point of view than the approximated 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 27

Table 13: Heterogeneity Rating AAA AA A Total variance 1.00 0.19 0.19 Part of explained variance(in %) 0.00 .01 99.19

BBB BB B CCC 0.08 0.18 1.58 23.32 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 estimator 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

-1.052 -0.101

-0.431 -1.879 -0.440 -0.151 -1.331

0.829

0.619

1.262

1.185

and reported in Figure 7 in the Appendix. 28

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:

& " " " "

& when its rating at date is would correspond " .

and the distribution of its rating at date 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 an approximation of the row of the matrix & " .

29

& "

" , which is

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 2 , the matrix 2 with block decomposition 2 . 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 30

25 in the appendix. We also compute the difference & " & " & " for the following joint migrations corresponding to joint up grades 3

3 , to joint . (see Table 15, Table

stability 3 3 and to joint downgrades 3 3

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

up 1 bucket

unchanged

down 1 bucket

AAA AA AAA — — AA — 0.0000 A — 0.0000 BBB — 0.0000 BB — 0.0000 B — 0.0000 CCC — 0.0000 AAA -0.0001 -0.0001 AA -0.0001 0.0000 A 0.0000 0.0001 BBB 0.0000 0.0000 BB 0.0705 0.0680 B -1e-04 -1e-04 CCC 0.0000 0.0000 AAA 0.0000 0.0000 AA 0.0000 0.0000 A 0.0000 0.0003 BBB 0.0000 0.0000 BB -0.0019 -0.003 B 0.0000 0.0000 CCC 0.0000 4e-04

A BBB — — 0.0000 0.0000 0.0298 0.0010 0.0010 0.0002 0.0237 0.0012 0.0002 0.0000 0.0274 0.0034 0.0000 0.0000 0.0001 0.0000 0.1300 -0.0008 -0.0008 0.0001 0.1627 0.0616 0.0017 0.0000 0.0733 -0.0005 0.0000 0.0000 0.0002 0.0000 0.1099 0.0011 0.0011 0.0000 0.0868 -0.0001 0.0018 0.0000 0.0886 0.0018

BB B CCC — — — 0.0000 0.0000 0.0000 0.0231 0.0002 0.0274 -0.0003 0.0000 0.0034 0.0234 1e-04 0.0367 -0.0004 0.0000 0.0006 0.0307 0.0006 0.1184 0.0705 0.0000 0.0000 0.0608 0.0000 0.0000 0.1627 0.0016 0.0733 -0.0616 0.0001 -0.0005 0.2037 0.0599 0.1244 0.0020 0.0000 0.0011 0.1244 0.0011 0.1407 0.0000 0.0000 0.0000 0.0003 0.0000 0.0004 0.0868 0.0018 0.0886 -0.0001 0.0000 0.0018 0.0823 -0.0034 0.1085 -0.0034 0.0001 0.0029 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.

31

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 computation 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.

32

REFERENCES Albanese, C., Campolieti, G., Chen, O., and A., Zavidonov (2003)a: “Credit Barrier Models”, forthcoming in Risk, 16. Albanese, C., and O., Chen (2003)b: “Implied Migration Rates from Credit Barrier Models”, Working paper. Avellaneda, M., and J., Zhu (2001): “Distance to Default”, Risk, 14, 125-129. Bangia,A., Diebold, F., Kronimus, A., Schlagen, C., and T., Schuermann (2002): “Ratings Migration and Business Cycle, with Application to Credit Portfolio Stress Testing”, Journal of Banking and Finance, 26,445-474. Basle Committee on Banking Supervision (2002): “Quantitative Impact Study 3, Technical Guidance”, Bank of International Settlement, Basle. Brady, B., and R., Bos (2002): “Record Default in 2001. The Result of Poor Credit Quality and a Weak Economy”, Standard & Poor’s, February. Brady, B., Vazza, D., and R. J. Bos (2003), “Ratings Performance 2002: Default, Transition, Recovery, and Spreads ”. down-loadable on the website of Standard & Poor’s. Carty, L. (1997): “Moody’s Rating Migration and Credit Quality Correlation, 1920-1996”, Moody’s Investor Service, July. Cheung, S., (1996): “Provincial Credit Ratings in Canada: An Ordered Probit Analysis”. Working paper 96-6, Bank of Canada, Ottawa. Das, S., and G., Geng (2002): “Modeling the Processes of Correlated Default”, Working paper, Santa-Clara University. De Servigny, A., and O., Renault (2002): “Default Correlation: Empirical Evidence”, Standard & Poor’s, September. Duffie, D., and K., Singleton (1999): “Modeling the Term Structure of Defaultable Bonds”, Review of Financial Studies, 12, 687-720. Erturk, E. (2000): “Default Correlation Among Investment Grade Borrowers’,, Journal of 33

Fixed Income, 9,55-60. Foulcher, S., Gouriéroux, C., and A., Tiomo (2003): “Term Structure of Defaults and Ratings”, forthcoming Assurance et Gestion du Risque. Frey, R., and A., McNeil (2001): “Modeling Dependent Defaults”, Working paper, Zurich. Gagliardini, P., and C., Gouriéroux (2003) a: “Stochastic Migration Models”, CREST DP . Gagliardini, P., and C., Gouriéroux (2003) b: “Migration Correlation: Definition and Consistent Estimation”, forthcoming Journal of Banking and Finance. Gollinger, T., and J., Morgan (1993): “Calculation of an Efficient Frontier for a Commercial Loan Portfolio”, Journal of Portfolio Management, 39-46. Gouriéroux, C., and A., Monfort (1995): “Simulated Based Estimation Methods”, Oxford University Press. Gouriéroux, C., and A., Monfort (2003): “Equidependence In Qualitative and Duration Models with Application to Credit Risk”, CREST DP. Gouriroux, C., and A., Monfort (2004): ”Error-in-Factor Model”, CREST DP. Gupton, G., Finger,C., and M., Bhatia (1997): “Creditmetrics: Technical Document”, Technical Report, Hamilton, D., Cantor, R., and S., Ou (2002): “Default & Recovery Rates of Corporate Bond Issuers:A Statistical Review of Moody’s Ratings Performance 1970-2001.” Working paper, downloadable at Moody’s website. Jarrow, R., Lando, D., and S., Turnbull (1997): “A Markov Model for the Term Structure of Credit Risk Spreads”, Review of Financial Studies, 10(2), 481-523. Kijima, M., and K., Komoribayashi (1998): “A Markov Chain Model for Valuing Credit Risk Derivatives”, Journal of Derivatives 6(1), 97-108. Kijima, M., and E., Suzuki (2002): “A Multivariate Markov Model for Simulating Correlated Defaults”, Journal of Risk. Klaassen, L., and A., Lucas (2002): “Dynamic Credit Risk Modeling”.D.P. 34

Lando, D. (1998): “On Cox Processes and Credit Risky Securities”, Review of Derivatives Research, 2, 99-120. Li, D. (2000): “On Default Correlation: A Copula Function Approach”, The Journal of Fixed Income; 9,4; 43-54. Lucas, D. (1995): “Default Correlation and Credit Analysis”, The Journal of Fixed Income; 76-87. Merton, R. (1974): “On the Pricing of Corporate Debt: the Risk Structure of Interest Rates”, Journal of Finance, 19(2), 449-470. Nagpal,K., and R., Bahar (1999): “An Analytical Approach For Credit Risk Analysis Under Correlated Default”, Credit metrics monitor, 51-79, April. Nagpal,K., and R., Bahar (2001)a: “Credit Risk In Presence of Correlations, Part 1: Historical Data For US Corporate”, Risk. Nagpal,K., and R., Bahar (2001)b: “Measuring Default Correlation”, Risk, 14,129-132. Nickell P., Perraudin, W., and S., Varotto (2000): “Stability of Rating Transitions”, Journal of Banking & Finance, 24, 203-227. Stevenson, B., and M., Fadil (1995): “Modern Portfolio Theory: Can It Work for Commercial Loan ?”, Commercial Lending Review, 10, 9-12. Yu, F. (2002): “Correlated Defaults In Reduced Form Models”, Working paper, University of California, Irvine.

35

Table 16: Estimates of cutting points by ML Year 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 2000 2001 2002

1.3661 1.2328 0.8836 0.4762 1.4730 1.3981 1.6781 1.7516 1.6096 1.7523 1.7528 1.3120 1.7714 1.3740 1.7346 1.4842 1.7430 1.7367 1.7455 1.7523 1.7521

4.0453 5.4060 4.9459 4.5752 5.6924 6.1111 3.0130 6.3559 6.2486 6.3558 6.3558 6.3407 6.4200 5.6667 6.4124 5.9733 6.3529 6.3444 6.2718 6.3540 6.3554

6.7862 5.7112 5.3035 5.0654 5.9981 6.8695 3.4471 6.5843 6.4738 6.5841 6.5841 6.8013 6.6501 6.1096 6.7819 6.3003 6.5814 6.5732 6.4996 6.5822 6.5837

7.9719 11.1906 14.4934 21.7740 6.3915 7.4043 12.5313 13.1165 6.3230 7.3473 14.3599 14.5940 6.0260 7.0337 13.5765 13.8405 6.6785 7.6913 12.8185 13.4035 8.2936 9.3165 15.6508 16.2461 4.9318 5.3918 10.2598 10.6971 7.4906 8.5181 12.4404 12.8253 7.3797 8.4072 12.3099 12.6922 7.4904 8.5179 12.4398 12.8247 7.4904 8.5178 12.4403 12.8253 7.4614 8.4754 12.4825 13.0590 7.5566 8.5842 12.5176 12.8985 6.8535 7.8650 12.6246 13.1608 7.6652 8.7394 12.6521 12.9990 7.0845 8.0982 12.3535 12.9919 7.4876 8.5151 12.4344 12.8191 7.4797 8.5071 12.4302 12.8151 7.4063 8.4337 12.3694 12.7540 7.4885 8.5160 12.4418 12.8268 7.4900 8.5175 12.4392 12.8243

36

Table 17: Estimates of latent means by ML Year 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

2.9240 4.8464 4.3121 3.9321 5.2212 4.2245 2.4434 5.8423 5.5406 5.7538 5.6897 5.7174 5.6975 5.0663 5.7881 5.6740 5.6163 5.6073 5.6149 5.7456 5.6990 5.9045

5.5025 5.6891 5.2506 4.9777 5.7261 6.1843 3.3273 6.5490 6.4430 6.5526 6.5506 6.5560 6.6095 5.8367 6.7248 6.2695 6.5420 6.5370 6.4652 6.5506 6.5483 6.5550

7.3725 10.4454 11.2920 14.5371 6.1221 6.9520 10.2301 12.9337 5.8132 6.8925 11.0118 14.4256 5.5373 6.4559 10.1781 13.8242 6.4060 7.2337 10.5059 12.9768 7.6656 8.7729 13.5479 16.1361 4.2742 5.1666 7.8905 10.4983 7.0491 8.0426 10.5814 12.6925 6.9350 7.7662 10.3837 12.6760 7.0769 8.0415 10.9218 12.7546 7.0729 8.0420 11.0193 12.7536 7.1403 7.8629 10.5612 12.8517 7.1416 8.0309 10.0311 12.6133 6.4616 7.3098 10.2538 12.9480 7.1926 8.1587 10.6233 12.8957 6.6451 7.5289 10.0106 12.6047 7.0434 7.9396 10.3595 12.6089 7.0784 8.0356 10.5839 12.6892 6.9496 7.9675 10.8201 12.6900 7.0677 8.0415 10.8981 12.7421 7.0740 8.0409 11.2624 12.7868 7.1595 8.0407 11.0484 12.7882

37

Table 18: Estimates of latent standard deviations by ML Year 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

0.7560 0.4357 0.4133 0.4167 0.4360 1.4233 0.3647 0.4667 0.4655 0.4668 0.4668 0.4551 0.4650 0.4259 0.4298 0.1866 0.4670 0.4665 0.4665 0.4668 0.4669 0.4668

0.8638 0.0172 0.0318 0.0483 0.0172 0.0468 0.0017 0.0224 0.0204 0.0224 0.0224 0.0178 0.0225 0.0205 0.0313 0.0148 0.0224 0.0225 0.0218 0.0224 0.0224 0.0224

0.3619 0.2267 0.3698 0.4454 0.2272 0.5999 0.5493 0.2960 0.2957 0.2960 0.2960 0.2457 0.2887 0.2093 0.2647 0.2276 0.2963 0.2963 0.2974 0.2960 0.2955 0.2960

38

1.5557 0.3773 0.4718 0.3881 0.3774 0.3542 0.1703 0.3593 0.3701 0.3593 0.3593 0.3470 0.3657 0.3103 0.3636 0.3624 0.3593 0.3591 0.3596 0.3593 0.3595 0.3594

0.0624 1.6596 2.1354 1.9095 1.6596 2.2799 1.5875 1.3576 1.3576 1.3577 1.3577 1.6591 1.3589 1.5746 1.3586 1.4301 1.3582 1.3585 1.3607 1.3576 1.3576 1.3577

0.0341 0.2889 0.0598 0.0169 0.2888 0.1247 0.1695 0.2073 0.2132 0.2072 0.2072 0.2905 0.2068 0.2509 0.1959 0.2482 0.2071 0.2075 0.2046 0.2072 0.2074 0.2073

Table 19: * -goodness of fit by ML under three constraints Year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

Case 1 Case 2 Case 3 0.030 0.032 0.028 0.027 0.031 0.023 0.046 0.037 0.154 0.067 0.067 0.117 0.013 0.014 0.033 0.011 0.006 0.012 0.052 0.034 0.121 0.004 0.005 0.087 0.113 0.122 0.012 0.013 0.009 0.012 0.014 0.012 0.011 0.032 0.030 0.034 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 c.p. m e a n s

m e a n s

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

1 1.6497 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

2 6.2765 0.433 5.7248 5.6636 5.6705 5.6101 5.6519 5.6429 5.5652 5.5719 5.5827 5.5991 5.7165 5.6751 5.8703

3 6.5597 0.022 6.5291 6.5271 6.5206 6.5203 6.5215 6.5202 6.5144 6.5215 6.5245 6.5254 6.5275 6.5266 6.5315

Note: c.p. stands for “cutting point”.

39

4 7.4233 0.28 7.0309 7.0242 7.0042 7.0259 6.9756 6.9593 6.9489 7.0002 7.0415 6.9858 7.0214 7.0283 7.1108

5 8.4603 0.343 7.9798 7.9362 7.8307 7.9093 7.8951 7.8950 7.8718 7.8775 7.9467 7.9901 7.9708 7.9909 7.9834

6 12.5324 1.410 10.9591 11.0634 10.5696 9.9879 10.4830 10.4332 10.2800 10.3869 10.6204 10.9386 10.9364 11.3175 11.0926

7 12.9579 0.218 12.8786 12.8781 12.8042 12.6414 12.7901 12.8302 12.7038 12.7262 12.8053 12.8863 12.8643 12.9118 12.9162

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 0.024 O 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 1 2 3 4 5 6

2 0.004 0.000 0.001 0.001 0.004 0.001

3 0.000 0.000 0.000 0.000 0.001 0.000

4 0.001 0.000 0.001 0.001 0.002 0.000

40

5 0.001 0.000 0.001 0.002 0.009 0.002

6 0.004 0.001 0.002 0.009 0.073 0.016

7 0.001 0.000 0.000 0.002 0.016 0.004

Table 24: Eigenvalues and eigenvectors of the difference of matrices Values V E C T O R

1 2 3 4 5 0.074 0.004 0.000 -0.001 -0.021 0.057 0.950 0.307 0.005 -0.010 0.010 0.022 -0.053 -0.998 -0.003 0.029 0.304 -0.950 0.058 -0.023 0.093 0.011 -0.018 -0.001 0.995 0.990 -0.065 0.012 0.008 -0.092 0.078 0.003 0.003 0.000 0.003

41

6 -0.127 -0.008 -0.001 -0.001 -0.010 -0.077 0.997

Table 25: The prediction of transition matrix, & " Horizon

1 year

2 year

5 year

10 year

AAA AA A BBB BB B CCC AAA AA A BBB BB B CCC AAA AA A BBB BB B CCC AAA AA A BBB BB B CCC

AAA

AA

A

BBB

BB

B

CCC

D

0.9505 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9034 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.7758 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.6019 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0495 0.9141 0.0334 0.0001 0.0000 0.0001 0.0000 0.0923 0.8381 0.0550 0.0038 0.0000 0.0002 0.0000 0.1884 0.6598 0.1037 0.0283 0.0015 0.0006 0.0001 0.2736 0.4636 0.1298 0.0652 0.0082 0.0019 0.0005

0.0000 0.0847 0.8192 0.1191 0.0000 0.0007 0.0000 0.0042 0.1477 0.7060 0.1993 0.0060 0.0017 0.0001 0.0307 0.2563 0.4968 0.3069 0.0319 0.0060 0.0012 0.0886 0.3207 0.3861 0.3266 0.0589 0.0124 0.0036

0.0000 0.0012 0.1472 0.8432 0.0487 0.0053 0.0000 0.0001 0.0141 0.2336 0.7295 0.0753 0.0096 0.0011 0.0050 0.0802 0.3627 0.5355 0.1085 0.0185 0.0048 0.0332 0.1906 0.3885 0.4114 0.1040 0.0241 0.0080

0.0000 0.0000 0.0002 0.0375 0.7141 0.0177 0.0000 0.0000 0.0000 0.0054 0.0585 0.5281 0.0268 0.0036 0.0001 0.0033 0.0278 0.0764 0.2205 0.0325 0.0105 0.0019 0.0159 0.0475 0.0673 0.0721 0.0232 0.0093

0.0000 0.0000 0.0000 0.0000 0.2372 0.7803 0.2014 0.0000 0.0000 0.0000 0.0088 0.3463 0.6419 0.2418 0.0000 0.0005 0.0080 0.0422 0.4029 0.4040 0.1805 0.0006 0.0071 0.0330 0.0741 0.2760 0.2023 0.0906

0.0000 0.0000 0.0000 0.0000 0.0000 0.1441 0.4125 0.0000 0.0000 0.0000 0.0000 0.0326 0.1719 0.2057 0.0000 0.0000 0.0007 0.0059 0.0936 0.1279 0.0634 0.0001 0.0010 0.0059 0.0162 0.0789 0.0629 0.0285

0.0000 0.0000 0.0000 0.0000 0.0000 0.0519 0.3860 0.0000 0.0000 0.0000 0.0000 0.0117 0.1479 0.5476 0.0000 0.0000 0.0004 0.0048 0.1411 0.4105 0.7395 0.0001 0.0011 0.0093 0.0393 0.4018 0.6732 0.8596

42

Table 26: The key cells in & " & " & " for 2-year horizon co-movement

up 1 bucket

unchanged

down 1 bucket

AAA AA A BBB BB B CCC AAA AA A BBB BB B CCC AAA AA A BBB BB B CCC

AAA — — — — — — — 0.0001 0.0000 0.0000 -0.0012 0.0930 -0.0006 0.0001 0.0000 0.0000 0.0000 0.0003 -0.0037 0.0000 -0.0001

AA — 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 -0.0014 -0.0013 0.0856 -0.0003 -0.0003 0.0000 0.0009 -0.0070 0.0007 -0.0107 0.0008 -0.0024

A — 0.0000 0.0445 -0.0009 0.0292 0.0001 0.0260 0.0000 -0.0014 0.1678 -0.0129 0.1915 0.0006 0.0513 0.0000 -0.0070 0.1425 -0.0005 0.0907 -0.0053 0.0821

43

BBB — 0.0000 -0.0009 0.0023 -0.0099 0.0005 0.0037 -0.0012 -0.0013 -0.0129 -0.0002 0.0655 -0.0020 -0.0039 0.0003 0.0007 -0.0005 0.0004 -0.0021 0.0009 0.0017

BB — 0.0000 0.0292 -0.0099 0.0187 -0.0011 0.0230 0.0930 0.0856 0.1915 0.0655 0.2104 0.0678 0.0861 -0.0037 -0.0107 0.0907 -0.0021 0.0672 -0.0131 0.0817

B — 0.0000 0.0001 0.0005 -0.0011 0.0001 0.0009 -0.0006 -0.0003 0.0006 -0.0020 0.0678 0.0018 -0.0021 0.0000 0.0008 -0.0053 0.0009 -0.0131 0.0028 -0.0050

CCC — 0.0000 0.0260 0.0037 0.0230 0.0009 0.1008 0.0001 -0.0003 0.0513 -0.0039 0.0861 -0.0021 0.0672 -0.0001 -0.0024 0.0821 0.0017 0.0817 -0.0050 0.1685


up 1 bucket

unchanged

down 1 bucket

AAA AAA — AA — A — BBB — BB — B — CCC — AAA 0.0000 AA 0.0000 A 0.0000 BBB -0.0045 BB 0.0623 B -0.0011 CCC 0.0000 AAA 0.0000 AA -0.0002 A -0.0004 BBB 0.0021 BB 0.0005 B -0.0002 CCC -0.0001

AA — 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0036 -0.0070 -0.0045 0.0508 0.0013 -0.0002 -0.0002 0.0113 -0.0258 0.0053 -0.0090 0.0033 -0.0039

A — 0.0000 0.0643 -0.0080 0.0287 0.0002 0.0174 0.0000 -0.0070 0.1413 -0.0385 0.1110 0.0015 0.0113 -0.0004 -0.0258 0.1216 -0.0007 0.0531 -0.0087 0.0385

44

BBB — 0.0000 -0.0080 0.0191 -0.0172 0.0034 0.0030 -0.0045 -0.0045 -0.0385 0.0094 0.0249 -0.0052 -0.0035 0.0021 0.0053 -0.0007 0.0024 0.0023 0.0028 0.0076

BB — 0.0000 0.0287 -0.0172 0.0124 -0.0006 0.0124 0.0623 0.0508 0.1110 0.0249 0.0813 0.0363 0.0177 0.0005 -0.0090 0.0531 0.0023 0.0404 -0.0062 0.0360

B — 0.0000 0.0002 0.0034 -0.0006 0.0005 0.0018 -0.0011 0.0013 0.0015 -0.0052 0.0363 0.0065 0.0006 -0.0002 0.0033 -0.0087 0.0028 -0.0062 0.0046 -0.0048

CCC — 0.0000 0.0174 0.0030 0.0124 0.0018 0.0454 0.0000 -0.0002 0.0113 -0.0035 0.0177 0.0006 0.0079 -0.0001 -0.0039 0.0385 0.0076 0.0360 -0.0048 0.0893


up 1 bucket

unchanged

down 1 bucket

AAA AAA — AA — A — BBB — BB — B — CCC — AAA 0.0000 AAA 0.0000 AA -0.0007 A -0.0055 BBB 0.0175 BB 0.0005 B 0.0000 CCC 0.0004 AA -0.0013 A -0.0025 BBB 0.0036 BB 0.0086 B -0.0001 CCC -0.0002

AA — 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0125 -0.0100 -0.0083 0.0123 0.0030 0.0000 -0.0013 0.0308 -0.0351 0.0083 0.0068 0.0026 -0.0018

A — 0.0000 0.0531 -0.0097 0.0145 0.0007 0.0070 -0.0007 -0.0100 0.0763 -0.0422 0.0293 0.0013 0.0020 -0.0025 -0.0351 0.0677 0.0021 0.0220 -0.0039 0.0056

45

BBB — 0.0000 -0.0097 0.0354 -0.0166 0.0041 0.0002 -0.0055 -0.0083 -0.0422 0.0232 0.0033 -0.0030 -0.0014 0.0036 0.0083 0.0021 0.0026 0.0051 0.0016 0.0119

BB — 0.0000 0.0145 -0.0166 0.0072 0.0000 0.0052 0.0175 0.0123 0.0293 0.0033 0.0114 0.0076 0.0024 0.0086 0.0068 0.0220 0.0051 0.0260 0.0011 0.0289

B — 0.0000 0.0007 0.0041 0.0000 0.0005 0.0009 0.0005 0.0030 0.0013 -0.0030 0.0076 0.0047 0.0006 -0.0001 0.0026 -0.0039 0.0016 0.0011 0.0014 -0.0017

CCC — 0.0000 0.0070 0.0002 0.0052 0.0009 0.0122 0.0000 0.0000 0.0020 -0.0014 0.0024 0.0006 0.0017 -0.0002 -0.0018 0.0056 0.0119 0.0289 -0.0017 0.0282

6

Figure 1a Estimates of the mean

•

•

• • • • •

• • • • • • •

•

•

•

• • • •

• • •

•

6

•

5

•

•

7

•

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•

•

•

• •

•

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m3 6

m2 5

m1 4

•

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•

• • •

• •

•

• • • • •

•

•

•

•

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•

85

90

95

00

•

85

90

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85

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90

95

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9

12

15

13

10

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16

3

4

5

•

•

•

• • • • • • •

•

• •

•

•

•

•

• • • •

• •

• •

• •

•

• • • • • • • • • • • • • • •

•

12

7

10

•

• • •

• •

•

m6

•

•

•

13

• •

•

m5 11

m4 8

•

•

14

•

8

11

6

9

•

•

•

5

• 85

90

95

00

85

90

46

95

00

85

90

95

00

Figure 1b : Estimates of standard deviation •

•

0.8

1.4

0.6

Estimates of standard deviation •

1.0

0.6

0.5

1.2

•

•

st.d3 0.4

st.d2 0.4

st.d1 0.8

•

•

0.4

0.2

• • • • • •

• • • •

• • • • • •

• •

0.3

0.6

•

• • • •

• • • • • •

•

•

•

00

85

90

95

•

•

• 85

•

90

95

00

•

•

2.0

0.25

•

st.d4 0.8 1.0

1.5

•

•

•

• •

•

•

1.0

•

• •

•

• •

• • • • • •

• • • • • •

•

st.d6 0.15

•

st.d5

• • • •

•

0.20

1.2

• •

0.10

•

0.6

• • • • • •

•

• • • • • • • •

0.0

• 85

0.05

• • •

0.2

•

0.5

•

0.4

•

•

•

00

0.30

95

1.4

1.6

90

0.2

• 85

• • • • • • • • • • • • • • • • • • • • •

0.0

0.2

•

90

95

00

•

• • •

85

90

47

95

00

85

90

95

00

Figure 1c :Estimated of cutting points 7

1.8

Estimates of cutting points ••••••

•

•

1.6

•

••

•

•••

• •

• •

• •

•

••

•

•

•

••

•

•••

•

•

•

•

•

•

•

•

•

•

••••

••

•

5

a3

a2

•

5

•

• ••

6

•

•

90

95

00

•

85

90

95

00

85

22

85

•

•

•

18 • •

•

•• •••• ••••••• •

14

•

•

•

12

••

7

•

a7 16

8

•

a6 13

9 a5

••••••

•

•

• •

•

• 90

95

00

• •••••• ••••••••

12

6

11

•

85

00

•

•

••••••

95

•

14

•

90

20

10

15

11

•

3

•

5

0.6

4

0.8

4

•

•••

•

•

1.0

a1

•

•

•

7

• •

1.2

•

•

6

1.4

••••

6

• • •

•

•

8

••

•

a4

•

•

85

• 90

95

00

85

48

90

95

00

• 85

90

95

00

Figure 2 Estimated means, constant cutting points 13 m

2 m 3 m 4 m 5 m 6 m 7

12

11

estimates

10

9

8

7

6

5

89

91

93

95

97 Year

49

99

01

03

Figure 3 Estimated standard deviations, constant cutting points 1.5 σ 2 σ3 σ 4 σ 5 σ6 σ 7

estimates

1

0.5

0

89

91

93

95

97 Year

50

99

01

03

Figure 4 Smoothed latent variable

1.0

1.5

•

• • •

0.5

•

0.0

•

-0.5

•

•

• •

-1.0

•

-1.5

•

• 1990

1992

1994

1996 Year

51

1998

2000

2002

Figure 5 Evolution of downgrade probabilities A

BBB

0.04

0.08 Probability 0.06 0.04

0.02

0.02

0.06

0.04

0.08

Probability 0.10 0.12

Probability 0.06 0.08

0.14

0.16

0.10

0.10

0.18

AA

1990

1992

1994

1996 time

1998

2000

2002

1990

1992

1994

1996 time

1998

2000

2002

1990

1992

1994

B

1996 time

1998

2000

2002

1998

2000

2002

CCC

Probability 0.3 0.2

Probability 0.05

0.05

0.1

0.10

Probability 0.10

0.15

0.15

0.4

0.20

BB

1990

1992

1994

1996 time

1998

2000

2002

1990

1992

1994

1996 time

1998

2000

2002

1990

1992

1994

1996 time

Figure 6 Means and predicted means AA

5.9

A Means Predictions

6.54

Means Predictions

5.85

6.535

5.8

BBB Means Predictions

7.15 7.1

6.53 7.05

5.75 6.525 5.7

7 6.52

5.65

5.55

6.95

6.515

5.6 89

94

99

04

6.51

89

94

Year BB

8

99

04

6.9

89

94

Year B

11.5

11

7.9

10.5

04

CCC

12.95

Means Predictions

12.9 7.95

99 Year

12.85 12.8 12.75

7.85

7.8

89

94

99 Year

12.7

99

Means Predictions 04

9.5

Means Predictions 89

94

99 Year

52

12.65 04

12.6

89

94

99 Year

04

Figure 7 Smoothed Factor Values

•

1.0

• •

•

•

Smoothed -0.5 0.0

0.5

•

•

• •

-1.0

•

•

-2.0

-1.5

•

• 1990

1992

1994

1996 Year

53

1998

2000

2002