Testing Time-Homogeneity of Rating Transitions After Origination

Noname manuscript No.(will be inserted by the editor)

Testing Time-Homogeneity of Rating TransitionsAfter Origination of Debt

Rafael Weißbach1?, Patrick Tschiersch2, Claudia Lawrenz2

1 Institut fur Wirtschafts- und Sozialstatistik, Technische Universitat Dortmund, Dortmund,Germany e-mail:[email protected]

2 Credit Risk Management, WestLB AG, Herzogstr. 15, 40217 Dusseldorf, Germany

The date of receipt and acceptance will be inserted by the editor

Abstract When modelling rating transitions as continuous-time Markov pro-cesses, in practice, time-homogeneity is a common assumption, yet restrictive, inorder to reduce the complexity of the model. This paper investigates whether rat-ing transition probabilities change after the origination of debt. Accordingly, wedevelop a likelihood-ratio test for the hypothesis of time-homogeneity. The alter-native is a step function of transition intensities. The test rejects time-homogeneityfor rating transitions observed over seven years in a real corporate portfolio. Espe-cially one-year transition probabilities increase over the first year after origination.This time effect suggests that banks should manage their credit portfolios withrespect to the age of debt.

Key words Portfolio credit risk, rating transitions, Markov model, time-homo-geneity, likelihood ratio

? JEL subject classifications. C51, G11, G18, G33Correspondence to: Rafael Weißbach, Institut fur Wirtschafts- und Sozialstatistik, Technis-che Universitat Dortmund, 44221 Dortmund, GermanyPhone: +49 231 755 5419, Fax: +49 231 755 5284

2 Rafael Weißbach et al.

1 Introduction

The interest shown by banks in the estimation of rating transition matrices is ris-ing. Their objective is to improve several applications in risk management, such asportfolio models for internal bank steering or the pricing of (credit) risky financialproducts like CDS and CDOs. Additionally, the Basel II accord (c.f. Basel Com-mittee on Banking Supervision (2004)) is stimulating methodological advances inmeasuring credit risk. In planning to calculate regulatory capital with the inter-nal ratings-based approach (IRB) within Basel II, banks have developed internalrating systems. By so doing, banks are provided with continuous-time observa-tions of rating transitions of their obligors, leading to new facilities in analysingrating transitions. This data allows for a more precise estimation of rating transi-tion matrices than discrete-time rating transition information published by externalrating agencies like Standard & Poor’s. For the estimation of transition matricesin discrete time, the common ‘cohort method’ is used (cf. Jafry and Schuermann(2004)). Assume that there areYh obligors in rating classh at the beginning of acertain year, i.e. at risk of migrating to one of the other rating classes. By countingthe nhj obligors that start in Ratingh and end up in Ratingj at the end of theyear, the estimator for the one-year transition probability isnhj/Yh. This estima-tor is the ‘best’, in the sense that it has minimal variance under all unbiased andlinear estimators. However, there are various problems associated with this estima-tor. Some are technical, e.g. how to incorporate multiple rating transitions withina period, how to account for obligors that are not exposed to transition over theentire year, or how to distinguish between an obligor that migrates very early inthe year and one that migrates very late.

Suppose we observe rating transitions, not over one, but over two (successive)years, with sayY first year

h andY second yearh obligors under transition risk in class

h, andnfirst yearhj andnsecond year

hj transitions to classj out of the obligors at risk.We may combine the events and estimate the unconditional one-year transitionprobability by

nfirst yearhj + nsecond year

hj

Y first yearh + Y second year

h

.

The main disadvantage is now that in the denominator, we are double-countingthe obligors. Many obligors that we follow over two years will remain in the samerating classh over both years and will then be part ofY first year

h as well as ofY second year

h . We cannot assume observingY first yearh + Y second year

h indepen-dentBernoulli experiments. Fortunately, there is one condition under which thisdouble-counting is admissible, namely if the obligor ‘forgets’ what has happenedin the last year. The transition stochastic is then denoted as ‘free of memory’. Theformalisation is that all pairwise transition intensities are constant. This is the defi-nition of a time-homogeneous process. On the basis of transition data over a longerperiod, we generalise the above formula (for two years) by counting obligors thatreside at a certain point in timet in a rating classh, and denoting them asYh•(t),and counting the rating transitions from classh to classj from the beginning of

Testing Time-Homogeneity of Rating Transitions After Origination of Debt 3

the study until timet, and denoting them byNhj•(t). Studying those continuous-time counting processes is not only a traditional field of statistics, but has alsobeen used recently for financial modelling, for instance, in portfolio credit risk byDelloye et al. (2006).

Furthermore, Lando and Skødeberg (2002) emphasise the strong advantagethat the continuous-time model permits the testing of common assumptions in rat-ing models. (They focus on the Markov assumption for rating transitions, mea-sured in calender time, which they reject in favour of a rating drift.) We wishto assess the assumption of time-homogeneity for transitions measured since theorigination of debt. Our primary goal is to test whether rating transition behaviourtends to change when debt matures. Time since origination of the debt is a naturalchoice for rating transitions, and is used by Calem and LaCour-Little (2004), forexample to assess portfolio credit risk. If the test does not reject the null of time-homogeneity, we recommend that in this case, rating transitions can be treated astime-homogeneous Markovian processes within time periods that are of practicalrelevance. This implies that loans prior to maturity can be substituted by new loanswith the same rating and the rating quality can be expected to develop in a similarmanner. The same holds for secondary market loans and credit arising from credit-derivatives exposure, such as short positions in CDS contracts - these exposurescan be expected to behave like new loans. Instead, if time-homogeneity is rejected,the bank can assume that there is an aging effect of debt, as long as it is ensuredthat the sample rating transitions from all observed business cycle stages are dis-tributed constantly across the entire time horizon since origination. In such cases,business cycle effects will not lead to substantially time-varying transition proba-bilities with respect to time since origination. This is the reason why any (other)influence of the transition probabilities by covariates - such as business cycle, in-dustry or domicile of the obligor (see Nickell et al. (2000), Lando and Skødeberg(2002)) - is ruled out. Hence, we need not assume a specific regression model.

We address the two related topics: (i) We test rating transitions for time-ho-mogeneity, because, in a time-homogeneous Markov model, the effort requiredfor the estimation of the transition matrices is reduced, compared to the complexinhomogeneous model. As a by-product, discrete data are easily aggregated asdescribed above. (ii) We compare estimates of the one and two-year transitionprobabilities in the two situations, both when homogeneity is assumed and whenit is not.

It is important to note that the additional effort of assigning rating transitionsnot only to the year, but also to the day in that year, does indeed change the esti-mation. Jafry and Schuermann (2004) show that the cohort method yields ratingtransition matrices that are substantially different to those matrices estimated bythe transition intensities approach for time-continuous transitions. The differencesare both statistically significant and economically relevant, since they lead to largedifferences in economic capital calculated with credit portfolio models.

Kiefer and Larson (2006) develop a similar test for time-homogeneity, butwithin the framework of discrete Markov chains, as opposed to our time-continuousMarkov model, since they work with discrete time transition data. Additionally,they assume that rating transitions cannot be observed directly at the obligor level


and at each discrete time along the Markov chain. Instead, their approach is basedon transitionratesover different time horizons. Basically, our approach to testingtime-homogeneity is suitable for internal rating data, since it uses the full informa-tion, while Kiefer and Larson (2006) designed their test for rating data as publishedby rating agencies. Another major difference is that different time scales are used.While Kiefer and Larson (2006) use classical calendar time, we use the time sinceorigination of debt.

With respect to our data base: access to an internal rating system with8 rat-ing classes was allowed by WestLB. This data covers the rating histories of3, 699obligors, mostly corporates, over seven years. The2, 743 observed rating transi-tions lead to a clear rejection of homogeneity. We compare the estimation of tran-sition matrices for the homogeneous, reduced model, to non-parametric estimationin the full model and find substantial differences.

This result is contrary to the finding of Jafry and Schuermann (2004), that when“relaxing the time homogeneity assumption [...] the two methods yield statisticallyindistinguishable migration matrices.” On the other hand, the inhomogeneous be-haviour coincides to some extent with the empirical analysis in Kiefer and Larson(2006), who state that corporate rating transitions by Standard & Poor’s are notcaptured appropriately by the time-homogeneous Markov Chain, whereas Com-mercial Papers (CP), sovereign debt and municipal bonds can be modelled quitewell with this model. However, the main difference is that the time-homogeneousMarkov model for external ratings breaks after three one-year steps, while our in-ternal data yields the most substantial deviations from the homogeneous Markovmodel in the first year. This may arise due to the different time scale, but couldalso indicate that our internal ratings are designed as point-in-time ratings, whileexternal ratings are through-the-cycle ratings (cf. Altman and Rijken (2004)). Fur-ther research is necessary to establish whether internal and external ratings reallyexhibit different behaviour for rating transition and, if so, what is the reason. Inthe future, when banks have collected longer transition histories, practitioners canapply the test to various asset classes and determine whether or not an aging effectof debt is present in particular classes.

The remainder of paper is structured as follows: Section 2 reviews the Markovmodel and estimation techniques needed in the following sections. In Section 3,the test for homogeneity is constructed and statistical properties assessed by sim-ulation. In Section 4, real rating transitions are analysed using the derived results.Section 5 concludes the paper with suggestions for further research.

2 A continuous-time model for rating transitions

We model rating transitions of obligors as time-continuous Markov processes, de-noted byX = Xt, t ∈ [0, T ], defined on a probability space(Ω,F , P ). Xt

gives us the rating of the obligor at timet ∈ [0, T ] after the origination of the debt.The possible rating classes constitute the finite state spaceK = 1, . . . , k, whereClass1 represents the highest rating and Classk bankruptcy. A Markov process is


determined by its transition matrix

P (s, t) = (phj(s, t))h,j=1,...,k ∈ Rk×k; s, t ∈ [0, T ], s ≤ t, (1)

where the transition probabilitiesphj(s, t) = P (Xt = j | Xs = h) ∀h, j ∈ Kgive the probability for a transition from Ratingh to j within the time periods tillt. The infinitesimal generator of the process is defined by the transition intensities

qhj(t) = limu→0+

phj(t, t + u)u

, t ∈ [0, T ]; h, j ∈ K.

If the Markov process is time-homogeneous, the transition probabilities onlydepend on the lagu = t − s, resulting in constant intensitiesqhj(t) ≡ qhj ∈R+

0 ∀t ∈ [0, T ];h, j ∈ K. We denote the infinitesimal generator of a homogeneousprocess byQ = (qhj)h,j=1,...,k ∈ Rk×k. The transition probabilities can then beexpressed as a function ofQ,

P (u) = exp(uQ) =∞∑

n=0

(uQ)n

n!∀u ∈ [0, T ]. (2)

The relationship between intensities and transition matrices (2) can be gener-alised to the inhomogeneous Markov process.P (s, t) may be expressed in termsof the cumulative transition intensitiesAhj(t) =

∫ t

0qhj(s)ds; h, j ∈ K : h 6= j,

Ahh(t) = −∑j 6=h Ahj(t) by

P (s, t) = limmaxi=1,...,n |ti−ti−1|→0

∏

i

(Ik + A(ti)−A(ti−1)), (3)

wheres = t0 ≤ t1 ≤ t2 ≤ . . . ≤ tn = t is a partition of the finite time interval[s, t], A(t) = (Ahj(t))h,j=1,...,k ∈ Rk×k collects the cumulative intensities ina matrix, andIk is thek-dimensional identity matrix (see Andersen et al., 1993,pg.93).

Next, we consider the estimation of the transition matrices for a homogeneousprocess, i.e. ofP (u), and for an inhomogeneous process, i.e. ofP (s, t) usingcounting processes (see e.g. Fleming and Harrington (1991)). The data are ratingtransition historiesXi = Xi

t , t ∈ [0, T ] for each of thei = 1, . . . , n obligors ina portfolio. Compared to the use of all transition historiesX1, . . . ,Xn, there is noloss of information when using the vector of initial ratingsX1

0 , . . . , Xn0 together

with the processes

Nhj•(t) =n∑

i=1

Nhji(t), t ∈ [0, T ], h 6= j, (4)

counting the number of transitions from Ratingh to j until time t in the wholeportfolio. Nhji(t) =

∑ni=1(]s ∈ [0, t] : Xi

s− = h,Xis = j) counts the transi-

tions for obligori. For large portfolios, this is a clear reduction in the number ofrandom processes.


According to (2) each transition matrixP (u) can be estimated consistently byestimating of the constant generator. Albert (1962) derives

qnhj =

Nhj•(T )∫ T

0Yh•(s)ds

, h 6= j, (5)

as the maximum-likelihood (ML) estimator for the time-invariant transition in-tensitiesqhj , h 6= j, if

∫ T

0Yh•(s)ds > 0 and setsqn

hj = 0, otherwise. HereYh•(t) =

∑ni=1 Yhi(t) ≤ n, with Yhi(t) = IXi

t−=h, gives the number of oblig-ors in Ratingh just before timet.

For the estimation of the inhomogeneous transition probabilities one shouldnote that the individual counting processesNhji(t) are related to the cumula-tive transition intensitiesAhj(s). An estimator for the matrixP (s, t) can be con-structed via equation (3) by estimating the cumulative transition intensitiesAhj(t),j 6= h. This is done by the Nelson-Aalen-estimator (see Nelson (1969); Aalen(1978))

Ahj(t) =∫ t

0

IYh•(s)>0Yh•(s)

dNhj•(s), j 6= h, Ahh(t) = −∑

j 6=h

Ahj(t). (6)

This results in the non-parametric maximum likelihood estimator forP (s, t), theAalen-Johansen estimator,

P (s, t) =∏

t(i)∈[s,t]

(Ik + A(t(i))− A(t(i)−)), (7)

with A(t) = (Ahj(t))h,j=1,...,k ∈ Rk×k. Here,t(i) denotes the ordered time forthei-th of any rating transitions in the portfolio occurring within[s, t]. The struc-ture of thek× k-dimensional matricesA(t(i))− A(t(i)−) is described in detail byLando and Skødeberg (2002).

3 Likelihood ratio test for time-homogeneity

We will now construct a likelihood ratio test for time-continuous rating transitions.Because the distribution is only known asymptotically, we study the finite sampleproperties thereafter by means of a Monte Carlo simulation.

3.1 Test construction

Our objective is to determine whether rating transition histories can be modelledadequately by a homogeneous Markov model and transition probabilities, thus, ex-hibiting time-invariant behaviour. Therefore, the null hypothesis of time-homogeneitycan be stated as

H0 : ∀h, j ∈ K : ∃qhj ∈ R+ : qhj(t) ≡ qhj∀t ∈ [0, T ]. (8)


We are interested in the alternative that transition probabilities are time-dependent,which can be approximated by structural breaks in the transition intensities, i.e.

H1 : ∃h, j ∈ K : qhj(t) =b∑

i=1

qhjiI[ti−1,ti)(t), qhji ∈ R+

: ∃i1, i2 ∈ 1, . . . , b : qhji1 6= qhji2 , (9)

where0 = t0 ≤ t1 ≤ t1 ≤ . . . ≤ tb = T is a partition of[0, T ] consistingof b intervals. A popular principle for constructing such a test is the likelihoodratio. The likelihood of the observations, assuming a particular model (the hypoth-esis), is compared with the likelihood of the observations assuming another model(the alternative). If the ratio deviates substantially from one, the hypothesis is re-jected. For a time-discrete model, the likelihood ratio for time-homogeneity wasdeveloped by Anderson and Goodman (1957) for Markov chains and explored byKiefer and Larson (2006) to assess rating transitions.

For time-continuous transitions, the transition data for the portfolio are givenby observations of the counting processesNhj•(t) andYj•(t) defined in (4) andthereafter. Equation (3) allows expressing the model of a general Markov processby means of the cumulative transition intensitiesAhj(t) from the preceding sec-tion. We compare the maximised likelihood under the null of time-homogeneitywith the likelihood when maximising under the alternative of structural breaks inqhj(t):

LR =supqhj ,j 6=h LH0(qhj |Nhj•(t), j 6= h;h, j ∈ K; t ∈ [0, T ])

supqhji,i=1,...,b;j 6=h L(Ahj(t)|Nhj•(t), j 6= h;h, j ∈ K; t ∈ [0, T ])(10)

The compared models imply two probability measures and the likelihood ratio isa ratio of the densities corresponding to these two measures. Technically speak-ing, the likelihood ratio is a Radon-Nikodym derivative. For our observation ofthe counting processesNhj•(t), the density evolves by repeated (conditional) bi-variate distributions. The first distribution is that of the first transition time and thedistribution for the target rating class of the jump. The second distribution, con-ditional on the state of the process after the first transition, is that of the secondtransition time together with the jump distribution, and so on. The likelihood ratiofor a multivariate counting process was described in detail by Jacod (1975). An-dersen et al. (1993)[equation 2.7.4’] allows us to display the likelihoods in (10)by

L (Ahj(t)|Nhj•(t)) =

∏

t∈[0,T ],∃h,j;h6=j,

Nhj•(t−)6=Nhj•(t)

∏

h∈K

∏

j 6=h

(Yh•(t)qhj(t))∆Nhj•(t)

exp

−

∑

j 6=h

∫ T

0

Yh•(t)qhj(s)ds

. (11)


Here∆Nhj•(t) = Nhj•(t)−Nhj•(t−) ist only not zero ifNhj•(t) jumps. Thesejumps are of height one, since the transition times are continuous. The actualvalue of the likelihood ratioLR in (10) can be calculated by inserting the ML-estimates for the transition intensitiesqhj(t) in the likelihood in (11), under thenull in the numerator and under the alternative in the denominator. Assuming time-homogeneity, the ML-estimator for the constant transition intensities isqn

hj in (5)and thus determines the numerator ofLR.

With respect to the fact that the transition intensities in (9) are potentially piece-wise constant under the alternative, we can also apply this estimator to derivethe ML-estimator forqhji. This is accomplished by confining the time intervalfrom [0, T ] to the i-th time interval[ti−1, ti). Accordingly, the ML-estimator ofqhji, i = 1, . . . , b, is given by

qnhji =

Nhj•(ti−)−Nhj•(ti−1)∫ ti−ti−1

Yh•(t)dt, for

∫ ti−

ti−1

Yh•(t)dt > 0.

The estimator has the same consistency properties asqnhj .

After rearranging the terms, it is readily seen thatLR in (10) can be calculatedefficiently by

LR =b∏

i=1

∏

t∈[ti−1,ti],∃h,j,h6=j,

Nhj•(t−)6=Nhj•(t)

∏

h∈K

∏

j 6=k

(qnhj

qnhji

)∆Nhj•(t)

. (12)

The latter expression is well defined, because the estimateqnhji is only zero when

∆Nhj• is zero.By restricting the class of alternatives to structural breaks in transition inten-

sities, the power of the test for homogeneity has been increased. We are able toget a better grip of this special term structure, which captures many other possiblefunctional forms, even when only a few rating transitions are observed, as is usualin practice. Conversely, testing against a broad class of inhomogeneous models asalternative would require a large number of observed rating transitions.

The test statistic of the likelihood ratio test for our test problem is

Φ(X1, . . . ,Xn) = −2 ln(LR). (13)

Using standard theorems of likelihood testing, as laid out in (Serfling, 1980, pg.158), for instance,Φ in (13) converges in distribution, under the null, as the samplesizen goes to infinity - but with fixed time periodT - to aχ2-distribution withddegrees of freedom. The degrees of freedom are given by the excess dimensionsof the parameter space under the alternative over those of under the null. It iscommon practice to assume that an obligor, having once defaulted, cannot recoverhis business activity once again. Thus, defaultk is modelled as an absorbing state,which impliesqkj = 0 ∀j ∈ K, so that the degrees of freedom resolve tod =(b− 1)(k − 1)2.


3.2 Finite sample properties

Because the test for time-homogeneity of rating transitions presented in Section 3is asymptotical, we carry out a Monte Carlo simulation to assess its finite sampleproperties. Therefore, we determine the probabilities for the type I and type II er-rors for this test, based on the test statisticΦ defined in (13). In order to reducecomplexity, we simulate from a Markov model in a binary state spaceK = 1, 2,where1 denotes any non-default rating and2 denotes bankruptcy, thus only ac-counting for the defaults of obligors. When modelling an absorbing default stateq21(t) = 0 ∀t ∈ [0, T ] holds and the default model is parametrised by only onetransition intensity,q12(t).

The null of time-homogeneity becomesH0 : q12(t) = q1 ∈ R+∀t ∈ [0, T ].First, it should be tested against the specific alternative thatq12(t) exhibits a struc-tural break atT/2,

H1 : q12(t) =

q1, 0 ≤ t < T2

q2 6= q1,T2 ≤ t ≤ T

; q1, q2 ∈ R+.

We use the following technique to simulate a portfolio ofn obligors whose transi-tion histories are assumed to be generated by the same Markov processX on thestate spaceK with given transition intensitiesq12(t). It is sufficient to know thedefault timesτi ∈ [0, T ] of each obligori = 1, . . . , n, because that is the pointin time whenXi jumps to two, whereas before, it was constantly one. We have tosimulate a sampleτ = (τ1, . . . , τn) of default times with the distribution function

F (t) = 1− exp(−

∫ t

0

q12(s)ds

). (14)

In order to analyse the actual size of the homogeneity test based onΦ, we simu-latensim = 20, 000 such samplesτ of default times under the null of homoge-neous rating transition histories corresponding to a one-year probability of defaultPD1 = F (1) of 1 percent – orq12 ≡ − ln(0.99) –, i.e. a mediocre rating class of‘BB’ in the Standard & Poor’s system.

Altogether, the power of theLR-test is studied by simulatingnsim = 20, 000such samplesτ under three alternatives. The first transition intensity, given byqI12(t) = − ln(0.99)I[0,T/2)(t)− 2 ln(0.99)I[T/2,T ](t), doubles half way and per-

fectly matches the alternative specification (9). The second model, given byqII12(t) =

−2 ln(0.99)tI[0,T ](t), is linear and hence allows us to assess the power subject toa miss-specified alternative. Here, the one-yearPD1 in Model II is the same asin Model I. The third model, given byqIII

12 (t) = − 32 ln(0.99)tI[0,T ](t), is again

linear, but now with the two-yearPD2 as in ModelI. Two further situations ofmiss-specification are conceivable. Let us assume the intensity of Model I, but witha miss-specified partition of[0, T ], namely intob, equaling four (not two) intervals[0, T/4)-[3T/4, T ]. We denote this situation as ModelIV and in ModelV , weinvestigate the power of the intensity in ModelIII again withb = 4.

Table 1 shows the actual size under the null and the power under the alterna-tives I-V for different sample sizesn andT = 2 years. In each iteration step of


Table 1 Simulation of actual sizeα∗ and powerβ for LR homogeneity test

Sample size Size Powerβ under Modeln α∗ I II III IV V

30 0.006 0.017 0.043 0.023 0.005 0.008300 0.089 0.204 0.495 0.383 0.581 0.695

3,000 0.053 0.895 1.000 0.999 1.000 1.00030,000 0.050 1.000 1.000 1.000 1.000 1.000

the simulation, the homogeneity test at significance levelα = 0.05 is carried outby calculating theLR-test statisticΦ as defined in (13), using representation (12)for LR.

TheLR-test shows, to some extent anti-conservative behaviour, because of thevalues of the size which are slightly higher than the significance level0.05. Thismeans that theLR-test tends to reject the null of time-homogeneity a bit too often.However, when the sample size increases,α∗ converges toα. Moreover, the test isconsistent against the tested alternatives, because apparently, the power increaseswhen the sample sizen increases. Already in the case ofn = 3, 000 obligors, thenull of time-homogeneity is rejected correctly in at least90 percent of simulationsfor all models, even though the true intensities reveal only somewhat moderatedeviations from the null, as there are only one or three structural breaks and corre-sponding linear increases. Also bear in mind that a one-yearPD1 of one percentand a two-yearPD2 = 1 − exp(−3 ln(0.99)) = 0.03 is definitely not unrealisticbut is a valid value for ratings ranging from mediocre to poor. Throughout, it is ev-ident that a miss-specification of the alternative does not cause test inconsistency,nor is the asymptotic distorted. Clearly, as the comparison of Models II and IIIshows, the greater the slope of the intensity, the higher the power.

Summing up, this simulation study clearly provides no evidence which un-dermines the appropriateness of the asymptoticLR-test for time-homogeneity oftime-continuous rating transition histories. These results can be transferred to ageneral model for rating transitions with more than one parameter being testedwhen there are enough transition intensities which are not constant, what is a likelyscenario.

4 Empirical analysis of rating transitions after origination of debt

We now analyse time-continuous rating transitions as observed by banks, withintheir internal rating systems, using the methodological results derived in Sections2 and 3. Our main objective is to accurately estimate rating transition matrices.Therefore, we answer the question of whether rating transitions could be mod-elled by a time-homogeneous Markov model, or if dependence on the time sinceorigination of debt has to be taken into account, by applying theLR-test for ho-mogeneity presented in Section 3 to our data. After selecting the ‘right’ model, wepresent the results of both estimation procedure described in Section 3. On the one


0 1 2 3 4 5 8

Transitions

0

10

20

30

40

50

%

Fig. 1 Empirical distribution of the number of rating transitions per obligor during theindividual rating history over a maximum of seven years

hand, we estimate the generatorQ which parametrises the less complex homo-geneous model and, on the other hand, we estimate the time-dependent transitionmatricesP (s, t) of the general inhomogeneous model directly by means of theAalen-Johansen-estimator. We then compare both estimation procedure.

4.1 The data

WestLB AG granted us access to an internal system of credit-ratings with8 rat-ing classes, where1 denotes the top internal rating and8 the poorest rating of anobligor, and9 a default class. Rating histories were observed over seven years from1.1.1997 until 31.12.2003. In order to use the time since origination, we restrictedour analysis to a portfolio comprising3, 699 global obligors of WestLB that en-tered the portfolio after the commencement date of the study. In this portfolio,2, 743 rating transitions, including transitions to default, occurred within that pe-riod. When a rating change took place, exact dates were recorded. Figure 1 showsto what extent obligors tend to change their ratings.

This statistic underpins the fact that rating transitions are rare credit events. Thecreditworthiness of even a half of the analysed counterparties does not change atall. Each year, an obligor changes his rating on average only2, 743/10, 641 = 0.26times.


Table 2 Likelihood ratio test for homogeneity of internal rating for three data models: Fordata including continuous observations, the test statisticΦ is calculated. For the discretemodel, the Anderson-Goodman test statistic (A−G) is applicable. The number ofb rangesbetween2 and7

b 2 3 4 5 6 7

Data recorded continuouslyΦ 93.9 125.9 289.3 345.8 447.3 626.2

p-value 0.009 0.535 < 0.001 < 0.001 < 0.001 < 0.001

Defaults recorded continuously, other transitions quarterlyΦ 101.52 157.45 294.73 381.93 543.62 708.99

p-value 0.0020 0.0395 < 0.0001 < 0.0001 < 0.0001 < 0.0001

Transitions recorded quarterlyA−G 137.22 181.81 218.23 266.33 273.47 564.87p-value < 0.0001 0.0013 0.0941 0.3156 0.9719 < 0.0001

4.2 Results of test for time-homogeneity

We now focus on the question of whether a time-homogeneous Markov modelis suitable for the rating transition histories of the 3,699 obligors in the analysedportfolio. We are interested in testing the null of time-homogeneity set out in (8)at the significance levelα = 0.05 against the alternative of transition intensitieswith structural breaks, which are given in (9). We consider different equidistantpartitions0 = t0 ≤ t1 ≤ t2 ≤ . . . ≤ tb = 7 of the time interval[0, 7] containingall times since origination, when ratings of any obligor in the portfolio changed.The maximum number of breaks is set to six, yielding seven one-year intervalswhere transition intensities could vary.

So far, we have assumed that transitions are observed and recorded on a con-tinuous time scale, as is the case in our data set. Consequently, we useΦ from(13), for which we have studied the finite sample properties in the simulation ofSection 3. However, apart from the continuous model, one complication and onesimplification are conceivable. The internal rating of non-defaulted obligors is of-ten re-assessed in intervals, of say a quarter of a year. Therefore, our second, hy-brid, data model is that of discrete, namely quartely data for non-defaulted oblig-ors and continuous time information for defaults. Here, we assume that multipletransitions within an interval are recorded at the end of the interval and we applyour LR-testΦ to these data. Thirdly, we study a model for purely discrete data,e.g. quarterly transition observations, where information from multiple transitionswithin an interval is lost. The likelihood ratio test for this data model was devel-oped by Anderson and Goodman (1957) and we implemented the test along withour proposal. The test results for the three data models are given in Table 2.

For the continuous time model, regarding the striking smallp-values, exceptthe test results forb = 3, it is proven that the time since origination does influ-ence rating transition probabilities significantly. More granular partitions do notnecessarily yield smallerp-values, as the number of parameters to be estimated


increases, but the gain in precision in the approximation of continuous transi-tion intensities declines. The construction of the test (12) implies that local in-homogeneity within an interval of the alternative cannot be discovered by the test.The largep-value forb = 3 indicates this situation and a possible reason is thenon-monotonicity of some of the intensities. In a simplified situation, Weißbachand Dette (2007) proposed a test that will detect any alternative, i.e. is globallyconsistent. From a practical point of view, this exception is accounted for hereby processing our test on different partitions. The second model results in sim-ilar p-values, and the loss in information does not seem to reduce the power ofthe test. However, it must be emphasised, that the theoretical properties are notwell-known and are difficult to study, because of their hybrid character. Apply-ing the Anderson-Goodman test to the third, discrete, data model, yields morenon-significant partitions. The aggregation of the information seems to result ina lower power of the test for partitions with more than three time intervals andapparently, testing with continuous data is superior to discrete testing, especiallyfor the practically relevant case of one and two-year breaks.1 Further numbersbof time steps were analysed, namelyb = 10, 14 (seminal steps),21, 28, 42, 56, 70and84 (monthly steps), homogeneity is rejected for all steps.

In order to refute doubts that can result from inferential statistics being anunconditional decision, and also to investigate the indication for non-monotonicityof intensities, we proceed with descriptive statistics. If rating transition histories ofthe WestLB-portfolio really are homogeneous, the cumulative transition intensitiesAhj(t), and hence its Nelson-Aalen-estimateAhj(t), h 6= j of (6) would show alinear relationship with time since origination. Accordingly, in Figure 2, we checkgraphically for deviations ofAhj(t), h 6= j from a characteristic linear curve.

It is evident that patterns of cumulative transition intensities differ substantiallywhen analysing transitions from ratings with different qualities, here Ratings2, 4,and7. The more pronounced slope for transitions in neighbouring ratings, such asfor transitions from Rating2 to 3, means that the risk of migrating instantly to avicinal rating is all the time much higher than migrating to distant ratings. With2, 090 transitions, the vast majority of a total2, 743 transitions are indeed targetedat a direct neighbour rating.

There seems to be no global trend for the transition intensities, since theAhj(t),h 6= j do not have a gradient which increases over the whole period[0, 7]. Rather,there are local different slopes for all analysed cumulative rating transition inten-sities. Especially the flat curve at the beginning points to a reduced risk of transi-tion, since only a few transitions occur up to one year after the origination, eventhough the number of obligors during the starting phase is higher than in other pe-riods. Thus, the rejection of homogeneous transitions as a result of theLR-test isconfirmed by the non-linear term structure of the estimated cumulative transitionintensities.

This term-structure explains thep-value of0.009 whenb = 2, which is some-what greater than forb ≥ 4 and that the test forb = 3, i.e. for two breaks, fails

1 It is noteworthy that the insignificant partitions are disjunct with those for the first twodata models. However, it may again indicate non-monotone intensities.


Fig. 2 Nelson-Aalen-estimatorAhj(t), h 6= j for cumulative transition intensities plottedagainst timet since origination of debt for selected rating combinations:2 → 3 (black solidline),4 → 3 (black long-short broken line),7 → 8 (grey broken line),7 → 6 (black brokenline), 2 → 7 (black short-broken line)

even to reject the null (p-value= 0.535). These partitions are too coarse to detectsuch deviations from the null of homogeneity, because the average of transitionintensities over longer periods smoothes the different levels.

These findings indicate the main drawback of theLR-test for detecting struc-tural breaks: the test results depend on the choice of the numberb of structuralbreaks. Since the number of parametersqhji, i = 1, . . . , b under the alternative in-creases byk(k−1)−(k−1) = (k−1)2, here by64, each time[0, T ] is devided byone more interval, we recommend choosingb in terms of the size of the portfolioand length of the partition intervals. Because the number of transitions could fallbelow the number of parameters to estimate under the alternative, very quickly,resulting in highly inefficient estimatorsqn

hji for the parameters of the transition

intensitiesqhj(t) =∑b

i=1 qhjiI[ti,ti+1)(t), h 6= j.In our example, we find it questionable to estimate more than588 parameters

with 2, 743 observed rating transitions, since transitions into ratings that vary sub-stantially from the current rating, say from Rating1 to default, are expected to befairly unlikely. Moreover, dividing the observation horizon[0, T ] into seven one-year intervals seems to be appropriate, as we intend to analyse long-term changesin the transition probabilities instead of short-term variations. That is, we considerhow much the one-year probability for a transition from Ratingh to j changes ifan obligor has already been in our portfolio three years, in contrast to the periodimmediately after origination.

It seems reasonable to argue that the influence of the time since origination ofdebt stated here is in fact induced by other covariables such as the domicile or in-dustry of the obligor as established in other empirical studies such as Nickell et al.


(2000). Indeed, a bias could appear if the structure of the portfolio with respectto these variables changes over time. Suppose, for instance, that the proportion ofobligors with a certain rating in the telecommunications industry is small in theperiod up to one year after origination and then rises, because the deals of thebank with these obligors last longer than in other industries. If there really is anindustry effect which causes different transition risks over the same time horizonin different industries this could induce time-varying transition intensities.

In order to prevent this undesirable effect, we checked the composition of theportfolio every year after origination with respect to the covariables domicile andindustry and found no substantial imbalance that would undermine the analysisand results. In the first six years after origination, covering the last rating transi-tion for more than 95 percent of all obligors in our sample, we observed stableproportions with respect to region and industry distribution. Specifically, for 8 outof 15 regions, the variation coefficient which measures the variance of the region’sportfolio share over time relative to its mean, is not larger than 10 percent, whilethree other regions are around 30 percent. The two remaining regions have varia-tion coefficients of around 120 percent, due to some outliers for the time period ofsix years, which arise because only 15 percent of the obligors in the sample havesuch long rating transition histories. This is only of minor relevance for the analy-sis, since only 3 percent of all transitions occurred after five years. The industrydistribution turned out to be even more stable over time. The variation coefficentsof the industry’s portfolio share over the entire time horizon of seven years doesnot exceed 10 percent for 25 of 35 industries. For the remaining 10 industries,they do not exceed 25 percent. This is not true for variations in transition intensi-ties stemming from different macroeconomic conditions according to the businesscycle. This cannot be avoided, because we only observed rating transitions over atime period of seven years. Therefore, rating transitions occurring after longer timeperiods since origination in this sample, necessarily all took place after the eco-nomic downturn in 2001. Conversely, rating changes within a shorter period afterorigination up to three years, could also possibly be observed during the economicboom until 2001. Clearly, one would expect the transition intensity for downgradesto increase during the recession. We found that this is true for some ratings, but byno means for all downgrade rating moves. Altogether, the business cycle seems tohave an influence which differs for different rating-class combinations. However,we see no domination over the influence of time since origination, because the vastmajority (75 percent) of all rating transitions occurred within the first four yearsafter origination. Up to that time, the sample consisted of at least 20 percent oblig-ors, either in the economic boom before 2001 or thereafter. Clearly, in the firsttwo years after origination, the obligors in the boom phase dominated the sam-ple, whereas for longer time periods, the contrary holds. Summing up, a potentialbusiness-cycle effect seems to be moderate, since, for most of the transitions, theportfolio shows no strikingly different composition.


Fig. 3 One-year transition probabilitiesphj(s, s + 1) for various rating combinationsh, jplotted against times since origination of debt -Line description as in Figure 2-

4.3 Analysing rating transition probabilities

The clear rejection of the null of homogeneous rating transitions indicates the ne-cessity to estimate transition matrices by the Aalen-Johansen-estimator to accountfor the term structure of transition probabilities. This shows up when analysingtransition probabilities for a fixed period, say one year, as shown in Figure 3.

Variations of one-year transition probabilities,P (s, s + 1), are more or lesspronounced when times since origination evolves. What all analysed transitionshave in common, is that probabilities rise sharply during the first year. For instance,probabilities for transitions from Rating2 to3 almost double from10 to18 percent.However, their levels depend heavily on the rating and the distance between startand end-rating.

Furthermore, there are only local, and no global trends. This is in line with theterm structure of the cumulative transition intensities in Figure 2. Nevertheless,the transition probabilityp23(s, s + 1) from Rating2 to 3 tends to increase up tos = 5 years and then drops immediately from a high level of30 percent. The largejumps in the probabilities for transitions from Rating7 derives from the fact thatthere are less obligors in this rating. Similar results could be found for transitionprobabilities for other rating combinations and other time-horizons.

Tables 3 and 4 present the estimation results for the one-year transition matri-cesP (0, 1) andP (1, 2), using the Aalen-Johansen-estimatorP (0, 1) andP (1, 2)given in (7). Due to the security policy of WestLB, we had to omit thePD-columnand the diagonal-elements of the transition matrices. Nevertheless, the results ofthe analysis also apply to thePDs.

Most of the estimated probabilities for transitions within one year after time oforigination in Table 3 are positive, even though not every rating transition was ob-served directly, and neither did indirect rating changes by the same obligor neces-sarily take place. The Aalen-Johansen-estimatorP (s, t) captures transition risk for


Table 3 Aalen-Johansen-estimatorP (0, 1) for one-year rating transition matrixP (0, 1)

Rating 1 2 3 4 5 6 7 8

1 0.00 0.00 0.01 3.20 0.00 0.01 0.002 0.00 9.90 2.59 0.11 0.01 0.01 0.003 0.00 1.88 7.58 0.65 0.15 0.08 0.014 0.00 0.60 6.68 5.67 0.41 0.13 0.035 0.18 0.05 2.76 11.42 2.97 1.20 0.686 0.00 0.03 0.36 4.83 6.06 2.65 0.807 0.00 0.01 0.16 3.23 3.98 2.00 1.868 0.00 0.00 0.01 0.20 3.03 1.50 3.89

Table 4 Aalen-Johansen-estimatorP (1, 2) for one-year rating transition matrixP (1, 2)

Rating 1 2 3 4 5 6 7 8

1 0.15 3.81 4.69 2.71 0.13 0.18 0.052 0.33 15.93 5.56 2.93 0.75 0.81 0.063 0.21 3.04 14.23 3.30 0.49 0.52 0.094 0.32 0.82 11.12 8.83 1.18 1.30 0.455 0.42 0.54 2.72 14.97 3.16 4.39 0.736 0.06 0.20 0.52 6.67 13.05 11.13 1.287 0.05 2.26 1.91 6.37 4.98 4.55 5.078 0.01 0.04 0.04 0.24 2.18 2.17 2.33

transitions fromr1 to rm, as long as there is a sequence of Ratingsr1,r2,. . .,rm−1,rm, where at least one of then obligors migrates from the preceding to the sub-sequent rating within the time period considered[s, t], as Lando and Skødeberg(2002) mention. The concentration of zero-estimates for transitions into and outof Rating1 stems from the fact that there were no such transitions in the portfoliowithin only one year, i.e.Nh1(1) = 0 andN1j(1) = 0 for h, j = 2, 3, 6, 7, 8.

Since the transition probabilities almost never exceed10 percent and thePDs,for most ratings, could be expected to be substantially smaller, there is about an80 percent chance of retaining the same rating for one year after origination. If,however, a rating change does occur, it is very unlikely to migrate to a distantrating, since the probability of transition to a neighbour rating is highest, exceptfor Ratings1 and7. As the distance between current and target rating increases,the transition probabilities decrease very fast. For instance, obligors in Rating5have a chance of11.4 percent of being upgraded by one rating class, but only havea chance of2.8 percent of an upgrade to rating3. This structure of the one-yeartransition matrix reflects the rating policy of banks that adapt ratings in small stepswithin short time periods, rather than with one rating change of more categoriesafter a longer period of time.

Moreover, the quality of the current rating influences the direction of the transi-tion. We need to distinguish between a pronounced downgrade risk in the Ratings


Table 5 ML-estimationQ for the infinitesimal generatorQ

Rating 1 2 3 4 5 6 7 8

1 0.00 2.12 1.06 5.30 0.00 0.00 0.002 0.19 18.83 4.29 1.21 0.47 0.28 0.003 0.11 3.18 15.37 2.40 0.62 0.43 0.054 0.09 0.62 11.11 9.75 1.51 0.77 0.335 0.19 0.25 3.11 16.50 5.08 3.24 0.706 0.00 0.31 0.31 5.29 13.39 12.45 0.937 0.00 0.98 0.33 3.60 5.90 4.26 5.908 0.00 0.58 0.00 0.00 1.74 2.31 4.63

1, 2, 3 with high quality, and an upgrade tendency in the Ratings4, 5, and6. Al-ways, in these ratings, transitions into the next rating are most likely. WestLB’srating system is calibrated in such way that for the poor ratings of7 and8, defaultrisk is predominant.

Comparing the estimated one-year transition matricesP (0, 1) at the time oforigination in Table 3, andP (1, 2) after one year in the portfolio in Table 4,it is evident that the transition probabilities for changing the rating within oneyear, differ considerably when an obligor had already been in the portfolio for oneyear. In particular, the difference arises because nearly all transition probabilitiesphj(1, 2), h 6= j are remarkably much higher than the counterpartsphj(0, 1) atthe time of origination where the maximum difference of8.4 is observed for adowngrade from Rating6 to 7. This means that, in general, there is higher tran-sition activity in the second year after entering the portfolio. This leads to evenhigher downgrade risks in the high Ratings1, 2, and3 and, likewise, to more pro-nounced upgrade potential in the Ratings4, 5, and6, as transitions to more distantratings become more likely after obligors have spent one year in the portfolio.Yet, downgrade risk in Rating7 is no longer predominant. Moreover, no transitionprobability is estimated at zero.

We emphasise that it still makes a difference if one still presumes time-homo-geneity erroneously, since, the estimation results of homogeneous matrices via the

generator, differ substantially. The estimationQ =(qnhj

)h,j=1,...,k

of the genera-

tor Q is given in Table 5, where its components are estimated by (5).The resulting estimationP (1) of the homogeneous transition matrixP (1),

presented in Table 6 for ratings within one year at any time after origination,are calculated via the Taylor-type expansion mentioned in Section 2. All entriesqhj , h, j ∈ K of Q, estimated with zero, are caused by the fact that no obligor mi-grated from Ratingh to j over the entire observation period, i.e.Nhj(T ) = 0. Thisis the case even though all (but one) transition probabilities inP (1) are positive.

The homogeneous one-year transition matrix reveals differences from bothinhomogeneous matricesP (0, 1) andP (1, 2), which were analysed before. It isnot surprising that the homogeneous transition probabilitiesphj(1), h, j ∈ K arehigher than the probabilitiesphj(0, 1) for the same transitions at origination, and


Table 6 One-year transition matrixP (1) by estimated generatorQ

Rating 1 2 3 4 5 6 7 8

1 0.04 1.95 1.40 4.45 0.12 0.08 0.022 0.17 15.06 4.58 1.33 0.45 0.31 0.033 0.10 2.55 12.35 2.49 0.62 0.46 0.094 0.09 0.64 8.93 7.61 1.34 0.82 0.315 0.17 0.30 3.14 12.86 3.81 2.73 0.656 0.01 0.31 0.70 4.95 10.12 9.27 1.017 0.01 0.80 0.58 3.30 4.82 3.29 4.558 0.00 0.47 0.09 0.25 1.54 1.81 3.67

Fig. 4 Homogeneous transition probabilitiesphj(u) for various rating combinationsh, jplotted against time horizonu -Line description as in Figure 2-

also lower than the one-year probabilitiesphj(1, 2) at one year. SinceP (1) isbased on the generator, all transition information is incorporated in the estimation.As pointed out before, the generator averages the time-varying transition inten-sities, therefore generating a transition matrix which also neglects the discrepan-cies in time-dependant transition matrices for the same time-horizonu. Althoughthe level of transition probabilities is different, the structure of the matrixP (1),concerning the predominant up and downgrade tendencies in Ratings4, 5, 6 andRatings1, 2, 3 respectively, remains stable.

Finally, we analyse graphically how rating transition probabilities change whenconsidering different time-horizons within which rating changes occur. The ho-mogenous transition probabilitiesphj(u), h, j ∈ K, plotted in Figure 4, increaseas the time-horizonu is extended, at least up to a six-year horizon, and seems to beconcave in general. This empirical result is theoretically backed by the Chapman-Kolmogorov-equationP (s+ t) = P (s)P (t) = P (t)P (s)∀s, t ∈ T . Accordingly,


Fig. 5 Transition probabilitiesphj(s, s + u) for various rating combinationsh, j and start-ing timess = 0 (black lines) ands = 1 (grey lines) after time of origination plotted againsttime horizonu

with a rising time-horizon, it becomes more likely that obligors arrive at a certainrating through a sequence of rating changes.

Again, probabilities for rating transitions into adjacent ratings, i.e.2 to 3, 4 to3, 7 to 8 and7 to 6, clearly dominate the probability for transitions into distantratings, for instance2 to 7. Within that group, transitions from2 into 3 and4 to3 are much more likely, since the probabilities for periods of more than one yearpermanently exceed10 percent and even rise up20 and30 percent, respectively.Conversely, probabilities for transitions from7 to 8 reach a maximum of only8.5percent foru = 4.9 years. Probabilities for transitions into distant ratings herenever exceed2.6 percent.

The stylised, smooth evolution comes from the calculation via the generatorQ being an ensemble of constant, and hence maximally smooth, intensities. Ad-ditionally, the grouping of the transition probabilities remains stable as the time-horizon increases.

Time-inhomogeneous transition probabilitiesphj(s, s + u) also increase witha rising time-horizonu, when the starting times is fixed, as seen in Figure 5. How-ever, there could be local downturns due to statistical variations in the estimations,since eachphj(s, s + u) is estimated separately for each pointt = s + u in time,using the rating transition which recently occurred.

The largest differences between transition probabilities at times = 0 ands =1 could be observed for short periods up to1.5 years where, for instance, theprobabilities for transitions from2 to 3 at s = 1 year in the portfolio increasesharply and reach the level of nearly8 percent for time-horizons ofu = 4 months,


whereas the same probabilities at origination are only that high for time horizonsof one year. For larger time-horizons, transition probabilities at different points intime behave fairly similarly.

An interesting result is that the inhomogeneous transition probabilities ap-proach the level of their homogeneous counterparts as the time-horizon increases.This is feasible, since both estimation techniques ultimately use the identical tran-sition data.

5 Conclusions

Homogeneity of rating transitions simplifies the application of transitions matricesto credit risk valuation. Moreover, estimating the parameters for the Markov modelis efficient. Statistical inference allows discrimination of time-homogeneity fromthe converse, on the basis of a representative historical sample. Our rating historyrejects the homogeneity hypothesis in favour of level changes of the (constant)transition intensities at predefined breaking times. In the homogeneous model, ob-serving the debt from the time origin is not necessary. The more fundamental rea-son is the lack of memory of the exponential distribution, which is the univariateprovenience of the homogeneous Markov process. Measuring time with respect toa calendar is possible. The inhomogeneous model calls for an origin of time. Wepropose measuring time since the origination of debt, e.g. the beginning of a loancontract in commercial lending and thus adopt the view of a portfolio owner, abank, forecasting its (portfolio) credit risk.

We find, for example, that transitions in the first year (of lending commit-ment) are less frequent than in the second, and that the one-year estimate basedon homogeneity ranges in between. Consequently, forecasting portfolio credit riskwith rating-transition probabilities for a one-year horizon with the homogeneityassumption overestimates the risk, e.g. the economic capital, for a portfolio of‘fresh’ exposures and underestimates the risk if all exposures originated one yearago. The portfolio composition with respect to the age of debt needs to be takeninto account, not only for the internal portfolio model, but also for regulatory capi-tal under Basel II. The study confirms a significant covariable for credit risk, whichwas also found in discrete modelling by Calem and LaCour-Little (2004).

However, identifying the right model for rating transitions is beyond the scopeof this study. Inhomogeneity is not sharp, and modelling the transition mechanismin accordance with the historical evidence requires a careful selection of transition-intensity models. Our non-parametric estimate is only the least restrictive approachand does, for example, not fit the need for forecasts of transitions in the distantfuture. One could in fact start from either of these extremes in order to achievea realistic model. On the one hand, a controlled step towards a more restrictivemodel, compared to the totally non-parametric approach, would be to assume thatthe transition intensities are, for instance, twice differentiable. Kernel estimatesfor the intensity as proposed in Weißbach (2006) may then be accumulated toform smooth estimates for the cumulative transition intensities needed to expressthe inhomogeneous transition matrix. On the other hand, one could ask whether


intensities are ‘almost’ constant. This hypothesis can be tested with the so-calledequivalence hypothesis, an idea introduced by Hodges and Lehmann (1954). Thehypothesis of constancy of intensities in all points in time can be relaxed to thehypothesis that intensities are constant, apart from some small tolerance. A generalapproach in this manner is described by Munk and Pfluger (1999).

AcknowledgementsWe are indebted to the editor Prof. B. Fitzenberger and two anony-mous referees for their detailed recommendations. Furthermore, we thank M. Gordy for hiscritical comments. The financial support of the Deutsche Forschungsgemeinschaft (DFG),SFB 475 “Reduction of Complexity in Multivariate Structures,” project B1, is gratefullyacknowledged. The views expressed here are those of the authors and do not necessarilyreflect the opinion of WestLB AG. All data analysis was performed in SAS, Carry, USA.

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Legend for Table 1.∗ using test statisticΦ, nsim = 20, 000 simulation runs andsignificance levelα = 0.05

Legend for Table 2.∗ using test statisticΦ and the WestLB-portfolio of3, 699obligors with rating histories observed within maximal seven years since origina-tion of debt, for the second data model, the test is only an approximate likelihoodratio test

Legend for Table 3.∗ at time since origination of debt, in percent

Legend for Table 4.∗ at one year since origination of debt, in percent

Legend for Table 5.∗ in percent

Legend for Table 6.∗ assuming homogeneity, in percent

Testing Time-Homogeneity of Rating Transitions After Origination

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