Chapter 9. Conditional Credit Migrations - Adjustments and Forecasts
9.1. Overview
In Chapter 6 we illustrated methods for detecting significant differences between transition matrices. While Bangia et al. (2002) examined the stability of migration matrices of a major agency, we found that transition matrices of an internal rating system also could not be considered as being time homogeneous or a first-order Markov chain. We further investigated the substantial effects of changes in migration behavior on expected loss, VaR, and especially on confidence intervals for PDs. One finding was that, especially in times of an economic downturn, the risk of a credit portfolio can be several times higher than during an expansion of the economy. The findings are similar to some other studies in the field. Helwege and Kleiman (1997) as well as Alessandrini (1999) have shown, respectively, that default rates and credit spreads clearly depend on the stage of the business cycle. Belkin et al. (1998b) developed a simple model for adjustment of transition matrices to the economy, while Nickell et al. (2000) have shown that probability transition matrices of bond ratings depend on business cycles. By separating the economy into two states or regimes, expansion and contraction, and conditioning the migration matrix on these states, Bangia et al. (2002) showed significant differences in the loss distribution of credit portfolios.
Still, despite the obvious importance of recognizing the impact of business cycles on rating transitions, the literature is rather sparse on this issue. The first model developed to explicitly link business cycles to rating transitions was in the 1997 CreditPortfolioView (CPV) by Wilson (1997a,b) and McKinsey & Company. Belkin et al. (1998b) developed a univariate model whereby ratings respond to business cycle shifts. Nickell et al. (2000) proposed an ordered probit model which permits migration matrices to be conditioned on the industry, the country domicile, and the business cycle. In this chapter we will first review some of the approaches on adjusting migration matrices to the business cycle mentioned above. Then we will illustate the adjustment methods suggested in Lando (2000), as we will use them later for our own adjustment procedure. The methods suggested there were actually not introduced for linking transition matrices to the business cycle but to obtain risk-neutral migration matrices being in line with market credit spreads. However we will show that they can also be used for the purpose of linking macroeconomic variables to changes in migration matrices.
9.2. The CreditPortfolioView Approach
In the so-called macro simulation approach by Wilson (1997b), a time series model for the macroeconomic situation is used to forecast an index Yj,t for each rating class j at time t. This index is then used in a logit model to determine the conditional default probability pj,t in period t:
The index Yj,t is derived from a multifactor time-series model of the form
According to the model the index Yj,t is dependent on economic variables Xj,k with k = 1, … , m using the coefficients βj. Further, vj,t represents an error term. In the CPV model the error term vj,t is interpreted as the index innovation vector and assumed to be independent of the Xj,k and identically normally distributed. Thus, we get vj,t ~ N(0, σj) and vj ~ N(0, ∑v). Hence, ∑v denotes the variance/covariance matrix of the index innovations.
The macroeconomic factors Xj,k are assumed to follow an autoregressive process of order 2 AR(2):
Here Xj,i,t−1 and Xj,i,t−2 denote the lagged values of variable Xj,k, and ej,k,t denotes an error term that is assumed to be iid, i.e.,
where ∑e is the covariance matrix of the error terms. The author points out that a better strategy might have been an ARM A(p, q) or a vector autoregressive moving average model. However, the model with two independent AR(2) processes was chosen due to its simplicity.
Combining equations (9.1), (9.2), and (9.3), we have to solve a system of equations to calibrate the model using the following assumptions:
After the estimation of equations (9.1) and (9.2), and (9.3), simulations are used to calculate a macroeconomic index and, thus, conditional default probabilities. Then the unconditional or average migration matrix has to be adjusted to simulate portfolio migrations and default behavior.
We will now illustrate the adjustment procedure for the transition matrices by a simplified example using a discrete approach
taken from Allen (2002). Let us consider a transition matrix with only four rating categories A, B, C, D. Let’s further assume that the unconditional or average default probability for C-rated debt is 
, while, e.g., the migration probability from state C to state B is 
and from state C to state A is 
, as denoted in Table 9.1.
Now suppose that based on current macroeconomic conditions the estimated conditional value of the default probability for a C-rated bond given by the model equations (9.1)–(9.3) is pCD,t = 0.174. In this case without adjusting the transition matrix we were likely to underestimate the VaR of a loan portfolio and especially the default probability of a C-rated loan. Hence, we have to adjust the transition matrix according to this estimate for pCD,t.
With 
a so-called diffusion term or shift parameter is determined. This parameter is then used to change the respective row in
the unconditional transition matrix to obtain the conditional transition matrix. Clearly the shift in transition probabilities
must be diffused throughout the row in a way ensuring that the sum of all probabilities equals one.[1] The procedure aims ∆pCC = −0.0204, ∆pCB = −0.006 and ∆pCA = +0.0024 and the obtained row of the conditional migration matrix as is denoted in Table 9.2.
1 For a more detailed description of the procedure, we refer to Allen (2002). Unfortunately, the described procedure contains some mistakes and does not give correct insight into how the final result is obtained.
TABLE 9.1. Unconditional or Average Transition Probabilities for Rating Category C
| A | B | C | D | |
|---|---|---|---|---|
| A | … | … | … | … |
| B | … | … | … | … |
| C | 0.01 | 0.04 | 0.80 | 0.15 |
TABLE 9.2. Conditional Transition Matrix
| A | B | C | D | |
|---|---|---|---|---|
| A | … | … | … | |
| B | … | … | … | … |
| C | 0.0124 | 0.034 | 0.7796 | 0.174 |
Saunders and Allen (2002) point out that to determine the complete transition matrix, this procedure is repeated for each row of the unconditional transition matrix.
In the documentation of CreditPortfolioView, a continuous-time approach using generator matrices is applied. To describe the complete adjustment procedure, Wilson uses a so-called shift operator that redistributes the probability mass within each row of the unconditional migration matrix. The shift operator is then written in terms of a matrix S = {Sij} and the shift procedure is accomplished by
Thus, the unconditional average transition matrix is multiplied by a matrix that consists of the identity matrix plus the shift matrix multiplied by a factor τ ≥ 0. According to Wilson the factor τ that determines the amplitude of the shift in segment j is calculated according to the following rule:
where pjD is the unconditional default probability for the jth segment (taken from the unconditional migration matrix). Thus, the amplitude is ensured to be ≥ 1. Obviously, the shift operator (or the shift matrix) should satisfy the following conditions for the adjusted migration matrix.
- It should preserve the sum of the migration probabilities in each row to be 1; thus
for i = 1, … , K.
- The shift operator should ensure that the new migration and default probabilities are all greater than or equal to zero and
less than 1; thus 0 ≤
for i,j = 1, … , K.
This leads, according to Wilson, to the following shift operator restrictions, given
for j = 1,…, KObviously, the conditions imposed to the shift operator matrix make it look very similar to a generator matrix. Wilson states that the conditions on S ensure that the conditional matrix Pcond is a valid migration matrix as long as Puncond is a valid migration matrix. For the proof we refer to the technical document of CreditPortfolioView (CreditPortfolioView, 1998). The problem, however, in dealing with discrete matrices is that the boundedness condition on τ is almost impossible to guarantee for any arbitrary series of speculative default rates—especially not for historical speculative default series which have a relatively high standard deviation-to-mean ratio. Under τ violating the boundedness condition, the resulting matrix could contain negative probabilities or also probabilities greater than one. Therefore, Wilson changes from a discrete shift operator to a continuous shift operator:
Equation (9.9) leads to the differential equation
Since we already defined how the amplitude of the shift operator is calculated, the remaining task is how to determine the shift matrix S. A right shift operator can be considered as a matrix shifting probability mass in the direction of increased downgrades and defaults. Alternatively, a left shift operator can be considered as a matrix shifting probability mass in economic expansion in the direction of higher rating grades.
Wilson defines a possible systematic right shift operator according to the following equation:
If the amplitude of the shift operator changes, the new migration probability equals the original migration probability plus a proportion from the higher class j − 1 minus the mass that is shifted to the lower class j + 1. Then the systematic right shift operator has the following form:
The systematic left shift operator is defined along the lines of the right shift operator. Clearly the relation that should be expressed is that as the speculative default rate increases, credit downgrades are more likely, while upgrades have lower probabilities and vice versa. Wilson suggests defining sj = α for downgrades (j ≥ r) and sj = β for upgrades (j < r) with r being the rating class to which S relates. But in further documentation, the systematic shift operator is restricted according to s = α = β to govern the form of the left- and right-shift operators. This restriction is supposed to ensure that in the absence of macroeconomic shocks, the mean of the simulated cumulative migration matrix equals the unconditional cumulative migration matrix; see CreditPortfolioView (1998).
In addition to calibrating expected defaults by the systematic shift operator according to CreditPortfolioView, it is also important to calibrate the ratio of expected to unexpected default rates. Since investment grade segments tend to be less sensitive to cyclical movements, the amount of volatility of default rates which can be described by the systematic risk models is lower for investment grade counterparties. Thus, while one can expect defaults to vary over the cycle in a more or less predictable manner for noninvestment grade categories, default events for highly rated counterparties have to be considered as more unsystematic and surprising. Hence, in addition to the systematic shift operator, there is also added a source of uncertainty which is independent of the state of the economy. It is called the unsystematic shift operator. It affects the higher rated companies more than the lower rating categories. The probability mass is directly moved from the default entry to each entry in a row of the migration matrix or vice versa. The unsystematic right shift matrix U is of the form
However, the task is not only to determine the form of S and U but also its values. Unfortunately, this part of the adjustment is not available in the published documentation of CreditPortfolioView, which has to be considered a major drawback of the model. For a simpified example of the adjustment procedure, see, e.g., Saunders and Allen (2002). In an empirical study, Wehrspohn (2004) examines the estimated long-term default probabilities by CreditPortfolioView and compares them to Standard & Poor’s cumulated default probabilities. To investigate the effect of the adjustment procedure in the Wilson model, he tested model forecasts both under an average macroeconomic situation and a recession scenario. His findings are rather disenchanting. The estimated default probabilities under the average macroeconomic scenario are on average 5 times higher than the cumulative 10-year default probabilities of Standard & Poor’s; for rating class A the estimated default probability is approximately 10%, and thus more than 8 times higher than the numbers provided by the rating agency. Hence, CreditPortfolioView is not able to estimate long-term default probabilities similar to the market. Further, the difference between simulated long-term default probabilities under the conditional recession and the average macroeconomic scenario are comparatively small. For the considered 10-year horizon, Wehrspohn (2004) obtained less than 10% difference, which is negligible compared to the differences in the deviations from cumulated default probabilities by the rating agencies. He concludes that the model has some deficiencies in representing market cumulative default probabilities, especially for longer time horizons, and should be refined in several ways.
9.3. Adjustment Based on Factor Model Representations
Kim (1999) developed a model for estimating conditional transition matrices. In his model he adopts a one-factor model to incorporate credit cycle dynamics into the transition matrix. Similar to CreditPortfolioView, the main idea is to improve the accuracy of credit loss simulation based on the technique of conditional transition matrices. He also points out that another goal is to yield an efficient method for stress testing according to the analyst’s view of the future economic state.
To implement the technique, in a first step one builds a credit cycle index, which indicates the credit state of the financial market as a whole. The model of the credit cycle index needs to include the most relevant macroeconomic and financial series, such that the forecasted credit cycle index will represent the credit state well. Then in a next step the transition matrix is conditioned on the forecasted credit cycle index. Unlike in the one-factor default mode model, the model of conditioning the transition matrix should cover events that lead to upgrading and downgrading, as well as default. Furthermore, in the face of the animadversion on the CreditPortfolioView model, the estimated results should be stable enough to apply to forecasting or stress testing of the transition matrix.
9.3.1. Deriving an Index for the Credit Cycle
The so-called credit cycle index Zt defines the credit state based on macroeconomic conditions shared by all obligors during period t. The index is designed to be positive in good days and to be negative in bad days. A positive index implies a lower downgrading and default probability and a higher upgrading probability and vice versa. To calibrate the index one uses the default probabilities of speculative grade bonds, since, similar to Wilson, Kim (1999) points out that highly rated bonds have very low default probabilities that are rather insensitive to the economic state.
Further, Zt is supposed to follow a standard Gaussian distribution and is standardized according to
where SDPt is the speculative grade default probability of period t; μ and σ denote the historical average and the standard deviation of the inverse normal transformation of the speculative grade default probability. Since the SDP is restricted to lie between 0 and 1, a simple regression model cannot be used and a transformation is needed. Thus, the relationship between the business cycle and SDPt is derived similarly to the CreditPortfolioView model. However, instead of the logit model suggested by Wilson (1997b), a probit model is suggested. Following CreditMetrics he assumes that the underlying, continuous credit-change indicator has a standard normal distribution:
with Xt−1 denoting a set of macroeconomic variables of the previous period and ∈t a random error term with Et−1(∈t). After estimation of the coefficients 
, the forecast for the inverse normal CDF of the speculative grade default probability is
Kim points out that the probit model allows an unbiased forecast of the inverse normal CDF of SDP to be created, given recent information about the economic state and the estimated coefficient.
After testing several macroeconomic variables, the author chose the spread between Aaa and Baa bonds, the yield of 10-year treasury bonds, the quarterly CPI inflation, and the quarterly growth of GDP for X. For the estimated model all coefficients showed the signs one would expect, and in backtesting, using mean absolute error as performance criteria provided better forecasts for average SDP than simply using the average speculative grade default probabilities as a forecast.
9.3.2. Conditioning of the Migration Matrix
Similar to the CreditPortfolioView model, the second step is to adjust the transition matrix according to estimated or forecasted values of the credit cycle index. Following the one-factor model suggested by Belkin et al. (1998b) we described in the previous section, it is assumed that ratings transitions reflect an underlying, continuous credit-change indicator Y following a standard normal distribution. The credit-change indicator Yt is assumed to have a linear relationship with the systematic credit cycle index Zt and an idiosyncratic error term ∈t, so we get the one-factor model parameterization:
Since both Zt and ∈t are scaled to the standard normal distribution with the weights chosen to be γ and 
, we get Yt also to be standard normal. Recall that γ2 represents the correlation between the credit change indicator Yt and the systematic credit cycle index Zt.
Figures 9.1 and 9.2 illustrate the effect of the shift of the credit-change indicator Yt depending on the outcome of the credit cycle index Zt. On average days we obtain Zt = 0 for the systematic risk index and the credit-change indicator Yt follows a standard normal distribution. If the assumed default event threshold is −2, the unconditional default probability
is equal to the probability that the idiosyncratic risk factor ∈t is less than 
. Therefore, we obtain for the unconditional PD
Let’s now assume that the correlation γ between Yt and Zt is 0.3. Hence, a positive outcome of the credit cycle index Zt = 1.5 shifts the credit-change indicator to the right side by γ·Zt. The conditional default probability dependent on Zt = 1.5 is equal to
The lower conditional PD is illustrated by Figure 9.1.
FIGURE 9.1. Average and conditional credit-change indicator for expansion scenario (Zt = 1.5).
In the case of a bad outcome of the systematic credit cycle index, the distribution moves to the left side. For example, assume that Zt = −1.5, so the distribution is shifted to the left by 0.5. Therefore, we get for the conditional PD
The effect on the conditional distribution for the PDs is illustrated in Figure 9.2.
To apply the above scheme to a multirating system, the author follows a procedure suggested by Belkin et al. (1998b). Following the CreditMetrics approach by Gupton et al. (1997) described in Section 4.3, it is assumed that, conditional on an initial credit rating i at the beginning of a year, one partitions values of the credit change indicator Y into a set of disjoint bins. According to Belkin et al. the bins are defined in a way that the probability that Yt falls within a given interval equals the corresponding historical average transition rate. The mapping procedure is illustrated in Figure 9.3. The methodology can be understood as mapping a firm’s future asset returns to possible ratings. The underlying assumption is that higher returns correspond to higher ratings, and vice versa. It should be noted that to calculate the scores, any meaningful statistical distribution could be used for the mapping. However, given the absence of preference for a particular distribution, for ease of calculation and estimation, the author chose the Gaussian distribution.
FIGURE 9.2. Average and conditional credit-change indicator for recession scenario (Zt = −1.5).
The mapping procedure is straightforward. Since the row sum in a transition matrix is always 1, one could, for each rating
class in the average transition matrix, construct a sequence of joint bins covering the domain of the Gaussian variable. This
can be done simply by inverting the cumulative normal distribution function starting from the default column. To illustrate
the procedure, we will consider an issuer, one in the speculative grade rating class Ba. For Ba-rated issuers, we have the
average transition probabilities given in Table 9.3. A default probability of 0.0141 corresponds to 
. Hence, the first bin is (−∞, −2.1945).For the next entry, summing 0.0141 and 0.0111 gives us the total probability that
the new rating is either C or a migration to default. The corresponding score is 
, and the next bin is (−2.1945, −1.9566]. Repeating this procedure gives the other scores, and finally the last bin corresponding
to a transition to Aaa is (3.5402, ∞).
When one uses the bins calculated from the average transition matrix, it is then straightforward to calculate the conditional
transition probability on the credit cycle index. In any year, the observed transition rates will deviate from the average migration matrix. It is
possible to find a value of Z so that the probabilities associated with the bins defined above best approximate the given year’s observed transition rates.
Thus, Zt is determined so as to minimize the weighted, mean-squared discrepancies between the model transition probabilities and the
observed transition probabilities. The conditional transition probability 
for rating state i to another rating state j has the ordered probit model:
FIGURE 9.3. Corresponding credit scores to transition probabilities for a company with BBB rating [compare Belkin et al. (1998b)].
TABLE 9.3. Average One-Year Transition Probabilities (TP) and Corresponding Scores for an Issuer with Rating Baa
| Aaa | Aa | A | Baa | Ba | B | C | D | |
|---|---|---|---|---|---|---|---|---|
| p(Ba, i) | 0.0002 | 0.0011 | 0.0052 | 0.0712 | 0.8229 | 0.0742 | 0.0111 | 0.0141 |
| (Score)Ba | − | 3.5402 | 3.0115 | 2.4838 | 1.4207 | −1.2850 | −1.9566 | −2.1945 |
The estimation problem then results in minimizing the following expression:
where nt,j denotes the number of transitions from initial grade i to j in the year t. Further observations are weighted by the inverses of the approximate sample variances 
. For the procedure of estimating the ordered probit model, we refer to Maddala (1983). In an empirical study by Kim, estimating equation (10.5) is done by using the transition matrix and the credit cycle index of 56 quarters from 1984 to 1998. The method for estimating
the ordered (multicategorical) probit model in (equation 9.3) is the same except that the ordered probit model uses the bin of credit rating thresholds as intercepts of the equation.
The estimated parameter γ is γinv = 0.0537 for the investment grade and γspec = 0.3384 for the speculative grade. To illustrate the adjustment procedure, we consider again the transition probabilities
and related scores of a Ba-rated issuer given in Table 9.3. Suppose that the outcome of the credit cycle index for year t* is Zt* = 1.5. Since the estimated correlation for speculative grade issuers is γspec = 0.3384, we get a shift of the credit-change indicator distribution by 0.3384 · 1.5 = 0.5076.
9.3.3. A Multifactor Model Extension
Wei (2003) extends the factor model representation by a multifactor, Markov chain model for rating migrations and credit spreads. The model allows transition matrices to be time varying and further driven by rating-specific latent variables. These variables can encompass a variety of economic factors including business cycles.
Similar to Wilson and Kim, Wei starts with the assumption that there exists an average transition matrix similar to 
, whose fixed entries represent average, per-period transition probabilities across all credit cycles. Further he assumes
that the entries in a transition matrix for a particular year will deviate from the averages, and the size of the deviations
is dependent on the condition of the economy. A further assumption is that the size of the deviations can be different for
different rating categories.
Since in his model the author works with several variables that drive the time-variations of the transition probabilities, he defines a set of average credit scores corresponding to the average transition matrix. To reflect the period-specific transition matrices, in the following, the movement of these credit scores is modeled—not the movement of the transition probabilities. Using the same procedure as Belkin et al. (1998b) or Kim (1999), this is done by partitioning the domain of a standard normal variable by a series of z-scores. The transition matrix can then be represented as a z-score matrix. Since the upper limit of rating Aa is equivalent to the lower limit of the highest rating Aaa and it doesn't make sense to model the absorbing default state, the z-score matrix will be of dimension (K − 1) × (K − 1). Alternatively, given a z-score matrix, a corresponding transition matrix can also be obtained. Obviously for a given rating, a downward shift in the credit scores leads to an increase in probabilities of transitting to ratings higher than or equal to the rating in question, while an upward shift in the z-scores leads to the opposite. Table 9.4 gives the matrix of z-scores corresponding to the average one-year migration matrix for Moody’s corporate bond ratings from 1982–2001, as reported in Table 1.1.
TABLE 9.4. Corresponding Z-Scores Matrix to Average One-Year Migration Matrix for Moody’s Corporate Bond Ratings Period 1982–2001
| Aa | A | Baa | Ba | B | C | D | |
|---|---|---|---|---|---|---|---|
| Aaa | −1.4583 | −2.4981 | −3.0327 | −3.4062 | −∞ | −∞ | −∞ |
| Aa | 2.4890 | −1.4160 | −2.3931 | −2.8485 | −2.9926 | −3.4035 | −3.7643 |
| A | 3.1777 | 1.9987 | −1.5262 | −2.3688 | −2.7364 | −3.1758 | −3.3046 |
| Baa | 3.3057 | 2.7128 | 1.5692 | −1.5031 | −2.1488 | −2.5773 | −2.7624 |
| Ba | 3.5872 | 3.0107 | 2.4813 | 1.4201 | −1.2857 | −1.9567 | −2.1936 |
| B | ∞ | 3.0784 | 2.6116 | 2.3564 | 1.4908 | −1.2835 | −1.5448 |
| C | 3.0384 | 3.0384 | 2.6432 | 2.3492 | 1.9583 | 1.0928 | −0.7098 |
The next step is then to model deviations from the scores of the average transition matrix. As an extension of the model suggested by Kim, Wei assumes that the deviations are driven by K mutually independent, Gaussian distributed factors. Hence, his multifactor credit migration model is of the form
with rating classes i = 1,…, K − 1 and j = 1,…, K. The first variable x denotes the common factor for all ratings, and the xi denote rating class specific factors, and εij represents the idiosyncratic factor. Similar to the described one-factor model, here the factors x, xi and εij are also scaled to a standard normal distribution. The factors x and xi encompass the impacts of all economic variables relevant to rating changes. Further the correlation between any two rating classes is
where i ≠ k. According to Wei, for an average year the realized deviations for all rating classes should be close to zero.
Trying to find the fitted transition matrix for each year, Wei suggests the following procedure:
In a first step the historical average transition matrix is calculated and converted into a z-score matrix. Then for each
period t and for each row, the shift of the z-score matrix that minimizes the sum of deviations ∆zit is sought. Therefore, a key assumption of the procedure is the equal magnitude of shifts in z-scores for a particular rating
class. This procedure yields a time series of z-score deviations for all rating classes and all periods ∆zit for all t = 1, … , T and i = 1,…, K − 1. To improve the estimation results for each row, one weighs the square of deviations by the inverse of the approximate
sample variance of each entry’s probability estimate. Then the average of the seven shifts for each year is calculated, denoting
the systematic shift for all rating classes 
for period t. In the next step the variance 
of the systematic shift time series is calculated. The estimator for the α is then 
. Then for each period t the common shift
is calculated. In the next step for period t and each rating class i, the rating-specific deviation is calculated. Finally, the fitted transition matrix for each period is calculated by using the average historical matrix and the z-score adjustments or deviations estimated in the previous steps.
The author points out that in the univariate model such as that of Belkin et al. (1998b), where there is no rating-specific shift, the same procedure is applied to the whole matrix for a particular year to find the common shift ∆zit. Then the parameter α is estimated in a similar fashion.
9.4. Other Methods
In the following we will briefly review two additional methods suggested for estimation of conditional migration matrices including ordered probit models (Nickell et al., 2000; Hu et al., 2002) and a regime-switching approach by Bangia et al. (2002). Note that both approaches will not be investigated in the empirical part.
Nickell et al. (2000) and Hu et al. (2002) propose the use of Bayesian methods in combination with an ordered probit model for conditioning credit migration matrices. The idea is to combine information from the historical average transition matrix estimate and results from other exogenous variables. The techniques are related to Bayesian methods for estimating cell probabilities in contingency tables. The transition matrix is smoothed via a function of covariates. In the first step a so-called appropriate prior is specified and then updated with a new estimator based on the observed data:
Here 
denotes some average historical transition matrix, Qt is the estimator for the transition matrix in period t obtained by an ordered probit model, and λ a weighting coefficient. Since the matrix 
is itself an estimator of the true transition matrix, updating this using other information actually corresponds to a pseudo
(or empirical) Bayes approach. Clearly, the problem, next to the estimation of Qt, is how to find an appropriate value for λ. For further explanation of the model, we refer to original articles by Nickell et al. (2000) and Hu et al. (2002). Wei (2003) points out that a large quantity of data is needed to estimate reliable parameters. Note that not only the model parameters
for the probit model and λ have to be determined, but also parameters for modeling of the business cycle as a Markov chain.
A similar approach to estimate conditional migrations is suggested by Lenk (2008). Exploring sources of heterogeneity in rating migration behavior, they adopt a Bayesian estimation procedure to estimate for each issuer profile its own continuous time Markov chain generator. While Nickell et al. (2000) employ a probit framework to compute conditional transition probabilities in a discrete-time model, Kadam and Lenk (2008) use a continuous-time model where the state durations are exponential and transition probabilities are logistic functions. Using Moody’s corporate bond default database, the authors further identify significant country and industry effects with respect to rating migration volatility, default intensity, and conditional transition probabilities. They further show that other characteristics, such as how long the issuer has been in existence, may also affect the rating migration behavior.
Bangia et al. (2002) link business cycle effects and transition matrices by a regime-switching model. The authors estimate a regime-switching model for quarterly expansion and contraction classifications. Further, average expansion and contraction transition matrices are determined. For applications it is straightforward to link the regime-switching and the estimated migration matrices. Based on estimated probabilities for being either in an expansion or contraction of the economy, using the regime-switching process one-period ahead forecasts for migration matrices can be obtained. However, simulating rating distributions based on their approach, the authors find no significantly different results for short-term migration and default behavior compared to using an average migration matrix (Bangia et al., 2002).
A more advanced application of Markov mixture models can be found in Frydman and Schuermann (2008). The authors propose a parsimonious model that is a mixture of (two) Markov chains. Hereby, the mixing is on the rate of movement among credit ratings. The estimation of the model is performed using credit rating histories and an algorithm originally suggested in Frydman (2005). The authors further provide evidence that the mixture model statistically dominates the simple Markov model and that the differences between two models can be economically meaningful. Therefore, Frydman and Schuermann (2008) find further evidence for the fact that the future distribution of a firm’s ratings depends not only on its current rating but also on its rating history in the past. This also confirms the results by Lando and Skúdeberg (2002), Krüger et al. (2005), or in Chapter 5 of this book on the Markov property and rating drifts where migration behavior was found to exhibit higher order Markov behavior.
Of course, it is also possible to apply the adjustment methods that were reviewed in Chapter 8. Obviously, the methods were initially designed to match transition matrices with default probabilities implied in bond prices observed in the market. However, given estimates for conditional default probabilities based on the macroeconomic situation, they can also be used to adjust transition matrices subject to anticipated changes in the business cycle. Hereby, both methods implementing the adjustment based on a discrete (Jarrow et al., 1997; Kijima and Komoribayashi, 1998) or continuous-time (Lando, 2000) transition matrix can be used. Further it is also possible to carry out the adjustments using the method suggested in Lando and Mortensen (2005). In the next section an empirical analysis will be conducted that actually uses the numerical adjustment techniques originally suggested in Lando (2000) using conditional default probabilities based on a macroeconomic index.
9.5. An Empirical Study on Different Forecasting Methods
This section will provide an empirical analysis on forecasting credit migration matrices based on a business cycle credit index.[2] Hereby, we compare the in-sample and out-of-sample performance of different adjustment methods for forecasting credit migration matrices. We consider Moody’s credit migration matrices for the U.S. market from 1984–1999. The in-sample period includes a history of 10 years from 1984–1993, while we use a six-year period from 1994 to 1999 to evaluate the out-of-sample forecasting ability of our models. The compared approaches include one-factor models based on the approach by Belkin et al. (1998a) and Kim (1999) as they were described in the previous section, and numerical adjustment procedures following Lando (2000). As benchmark results, we will also use the average historical migration matrices and the transition matrix of the previous period as forecasts for next year’s migration matrix.
2 Results of this section were originally published in Trueck (2008).
To determine one-period ahead forecasts of conditional PDs and the credit cycle index, we use a multiple regression model of the form
The process dynamic is influenced by the vector Xt−1 of d exogenous macroeconomic variables of the previous period. Using equation (11.3) and (9.3), we can then calculate forecasts for the one-period ahead credit cycle index Zt and the conditional default probabilities 
for each rating class. Table 9.5 displays the included variables in the multiple regression model. Both a variety of macroeconomic variables as well as credit
spreads and differences between long-term and short-term treasury bonds were considered. Having only 10 observations from 1984–1993 for both default probabilities and macroeconomic variables, to avoid
overfitting, not more than five exogenous variables were permitted in the regression model. In the following we will now describe
the procedure of model estimation and conditioning of the migration matrices.
TABLE 9.5. Included Variables for the Multiple Regression Model for Credit Cycle Indices
| Variable | Notation |
|---|---|
| Change in consumer price index | CPIt−1 |
| Change in GDP growth | GDPt−1 |
| Change in annual savings | SAVt−1 |
| Change in manufacturing & sales | MANt−1 |
| Change in working output per hour | OUTt−1 |
| Change in consumption expenditures | CONt−1 |
| Change in unemployment rate | UNt−1 |
| Treasury Yields 10, 5, 3 and 1 year | TY10t−1 etc. |
| Spread between 10-y and 1-y treasury | STRt−1 |
| Spreads on investment grade bonds | SINVt−1 |
| Spreads on speculative grade bonds | SSPEt−1 |
9.5.1. Forecasts Using the Factor Model Approach
Following Kim (1999) the multiple regression model (11.3) is used for modeling and forecasting the continuous credit cycle index Zt. It is assumed that the index follows a standardized normal distribution. Thus, a probit model will allow us to create unbiased forecasts of the inverse normal CDF of Zt, given the recent information of the last period about the economic state and the estimated coefficients. Note that unlike Kim (1999), who uses only one credit cycle index based on speculative default probabilities, we will consider two credit cycle indices: one for speculative grade and one for investment grade issues. For the investment grade issues, we use cumulative defaults of issuers rated Aaa, Aa, A, and Baa, and Baa, while for the speculative grade issues, default probabilities from Ba to C were included. Figure 9.4 exemplarily reports the observed default frequencies for the noninvestment grade rating classes Ba, B, and C that were used for estimation of the speculative grade credit cycle index.
In a second step the forecasts of the credit cycle indices are used for determining conditional migration probabilities 
. The adjustment is conducted following the procedure described in Section 3.1. However, for finding the optimal weights for the systematic risk indices wInv and wSpec, minimizing the discrepancies between the forecasted conditional and the actually observed transition probabilities, we introduce
some model extensions. We allow for a more general weighting of the difference between forecasted and empirical observation for the transition
probability in each cell. Hence, the weights for each of the cells are assigned according to some function f:
FIGURE 9.4. Moody’s historical default rates for speculative rating classes Ba (dotted), B (dashed), and C (solid) for the period 1984–1999.
where the outcome of 
may be dependent on the row i and column j of the cell as well as on the forecasted and actually observed transition probabilities 
and pt(i, j).
To achieve a better interpretation of the results, we will also use risk-sensitive difference indices suggested in Chapter 7 as optimization criteria for the distance between forecasted and actual migration matrix. Recall that based on the estimated model, the parameter w and the shifts on the migrations according to some optimization criteria are determined. In fact, this a crucial point of the model as it comes to forecasting credit migration matrices. While Belkin et al. (1998b) suggest minimizing a weighted expression of the form, Wei (2003) uses the absolute percentage deviation based on the L1 norm or a pseudo R2 as goodness-of-fit criteria. As it was illustrated in Chapter 6, most of the distance measures suggested in the literature so far do not quantify differences between migration matrices adequately in terms of risk. However, forecasts for transition matrices will be especially used for determining credit VaR, portfolio management, and risk management purposes. Therefore, especially risk-sensitive difference indices may be a rewarding approach for measuring the difference between forecasted and observed matrices. Following the results in Chapter 7, we suggest that the difference between a migration matrix P = (pij) and Q = (qij) can be determined in a weighted cell-by-cell calculation. Following Trueck and Rachev (2007), we will include two risk-sensitive directed difference indices in our analysis as optimization criteria:
To compare the results with standard criteria we will consider the classic L1 and L2 metric
Further, the measure of so-called normalized squared differences NSDsymm
is included in the analysis. Note that these criteria can also be used to evaluate the distance between forecasted and observed transition matrices for the numerical adjustment methods and the chosen benchmark models.
9.5.2. Forecasts Using Numerical Adjustment Methods
The second approach involves the numerical adjustment method suggested by Jarrow et al. (1997) and Lando (2000) that were reviewed in Chapter 8. Again we use a multiple regression model of the form (11.3.1). However, since the method needs estimates 
for the individual rating classes, for each speculative rating grade Ba, B, C as well as for the rating class Baa, a separate model is estimated. For the investment grade rating classes Aaa, Aa,A, we have to follow a different approach. Considering Moody’s historical default frequencies in several years, we could observe
no default for the three rating classes. To develop a regression model with only 10 observations, among them several with
PD, we should avoid zero. Hence, for rating grades Aaa, Aa and A, we decided to use default probabilities from the average historical migration matrix 
as estimators for the next period’s PDs in these rating classes. Based on these assumptions for each year, we can estimate
the vector for next year’s default probabilities in the individual rating classes:
In this section we will provide in-sample and out-of-sample results for the described models and compare them to benchmark results. Hereby, we will evaluate the performance of the chosen models against the standard approach of using historical average transitions or last year’s migration matrix for the calculation of credit VaR.
9.5.3. Regression Models
In a first step, the regression models for the credit cycle index and default probability scores are estimated. Recall that in order to avoid overfitting, at the most, five explanatory variables were permitted in each regression model. The in-sample period comprised the empirical default frequencies and the suggested macroeconomic variables for the period from 1984–1993. Among the tested models, the best results for the speculative grade credit cycle index were obtained using the macroeconomic variables change in GDP growth GDPt−1, change in annual savings SAVt−1, the change in consumption expenditures CONt−1, the change in unemployment rate UNt−1, and the spread between a 10-year and 1-year treasury STRt−1 bond. The model gave a coefficient of determination of R2 = 0.98, an F-statistic of 43.38 and a corresponding p-value of 0.0001, so it was highly significant. For the investment grade credit cycle index, the best results were obtained with a model including the macroeconomic variables change in the consumer price index CPIt−1, change in GDP growth GDPt−1, change of consumption expenditures CONt−1, and the change in unemployment rate UNt−1. The model gave an R2 statistic of 0.82, an F-statistic of 5.52, and a corresponding p-value of 0.045, so the model was still significant at the 5% level. Further, all regression coefficients were significant and showed the anticipated sign. Note that the fit for cumulated investment grade defaults was clearly worse, but it is generally accepted that investment grade defaults are less dependent on business cycle effects than speculative grade issuers (Nickell et al., 2000; Belkin et al., 1998a). The regression coefficients for speculative and investment grade credit cycle indices ZSpec, ZInv are displayed in Table 9.6. Estimation of the models for individual rating classes yield R2 statistics between 0.79 and 0.98. Further information on parameter estimates and statistics are available on request to the author.
TABLE 9.6. Parameter Estimates for the Multiple Regression Model (In-Sample Period from 1984–1993)
| Variable | Notation | ZInv | ZSpec |
|---|---|---|---|
| Constant | β0 | −0.3973 | −0.6182 |
| Change in consumer price index | CPIt−1 | 0.2187 | − |
| Change in GDP growth | GDPt−1 | −0.9338 | −0.2461 |
| Change in annual savings | SAVt−1 | − | 0.1838 |
| Change in consumption expenditures | CONt−1 | −1.3744 | −0.2509 |
| Change in unemployment rate | UNt−1 | 0.3616 | 0.2351 |
| Spread between 10-y and 1-y treasury | STRt−1 | − | −0.0057 |
9.5.4. In-Sample Results
After estimation of the regression model for the credit cycle index, in the next step we will determine conditional forecasts
for migration matrices based on the outcome of the credit cycle index. We first consider the results for the estimated weights
of the systematic credit cycle indices ZSpec and ZInv. Recall that in the chosen one-factor model approach, w is determined numerically in order to minimize the difference between the conditional forecast 
and empirically observed migrations pt(i, j) for all considered transition matrices in the in-sample period. The shift in the credit change indicator and, hence, the
shift in transition and default probabilities, is then an outcome of the forecasted credit cycle index of the next period
and the estimated weight w for the systematic risk factor Z.
Table 9.7 provides the weights wInv and wSpec for investment grade and speculative grade ratings giving the minimal distance between forecasts and observed migration matrices for the in-sample period 1984–1993. Note that depending on the chosen distance measures, we obtain different outcomes for the weights. For the speculative grade model, we find significantly higher weights of the systematic credit cycle index than for the investment grade model for all optimization criteria. We observe the lowest estimate for the weight wNSD,Spec = 0.1698 for the NSD distance criterion, while for the other distance measures the weight for the speculative credit cycle index is estimated to be between 0.2115 and 0.2544. For investment grade issues the estimated weights range from 0.0318 to 0.1762. As mentioned previously, this is in line with previous results in the literature (Belkin et al.1998a;, Wilson, 1997b). Especially when the shift is conducted to minimize the distance according to the L1 and NSD distance measure, the influence of the systematic risk factor becomes very small, wL1,Inv = 0.0318 and wNSD,Inv = 0.0504, respectively. This means that for these criteria the systematic risk index gives very little explanation for changes in rating behavior. The highest estimate for the weight wD2,Inv = 0.1762 is obtained when the distance is minimized subject to the risk-sensitive D2 difference index criteria. It seems as if, according to D2 changes in investment grade, migration behavior also could be explained by the systematic credit cycle index to a certain degree.
TABLE 9.7. EstimatedWeights w for the Credit Cycle Index Z, Representing the Influence of Z on the Change Indicators Y. Results Refer to the In-Sample Period 1984–1993 and are Estimated Based on a One-Factor Approach for Investment Grade (Inv) and Speculative Grade Ratings (Spec)
| Optimization Criteria | L1 | L2 | NSD | D1 | D2 |
|---|---|---|---|---|---|
| wInv | 0.0504 | 0.1143 | 0.0318 | 0.1089 | 0.1762 |
| wSpec | 0.2176 | 0.2544 | 0.1698 | 0.2115 | 0.2288 |
We will now investigate the in-sample one-period ahead forecast results for the different approaches. Table 9.8 provides in-sample results for mean absolute forecast errors according to the applied difference measures. As mentioned above, next to a factor-model approach (Factor) and the numerical adjustment methods (Num I, Num II), two standard benchmark methods were included in the results: using the average migration matrix of the in-sample periods (Naive I) or the transition matrix of the previous period (Naive II) as a forecast for the next period’s migration matrix. Best results for each distance measure are highlighted in bold. Note that the mean error or standard deviation of the errors for different indices within the columns cannot be compared due to a different scale. However, the results in the rows can be compared and provide the forecasting performance in comparison to other approaches. For each of the considered distance criteria, the one-factor model outperforms all other approaches including the numerical adjustment procedures. In contrast to these results, the numerical adjustment methods fail to provide better results than the naive approach for the criteria L1, L2, and NSD. Especially Num II that was applied in the seminal work by Jarrow et al. (1997) gives rather bad one-year ahead forecasts based on the estimated default probabilities with the credit cycle index. Considering these results and the relevance of the approach in the literature, we recommend a more thorough investigation on how migration probabilities are changed by these methods in the future.
It is not surprising that the best in-sample results are obtained for the one-factor model approach. Based on the optimization procedure in (9.5) that chooses the weight for the systematic risk factor in order to minimize the distance between the forecasted and empirical transition probabilities, these results could be expected. However, it is interesting to investigate how much the results improved subject to the considered optimality criteria. For the L1, L2 metric and the NSD difference index, we observe a reduction in the mean absolute error (MAE) by a fraction between 10% up to 50% compared to the naive approaches. The reduction for the risk-adjusted difference indices D1 and D2 are clearly higher. Comparing mean absolute errors between conditional and unconditional estimates for the D1 and D2 criteria, we find that according to the chosen criteria, the improvement is highly significant. Forecasting errors for the naive approaches are approximately 4–5 times higher; e.g., using naive approaches, the MAE for the risk-sensitive D2 criterion are approximately D2,NaiveI = 9.6931 and D2,NaiveII = 10.89, while for the one-factor model, we obtain an MAE of D2,Factor = 1.99. For these criteria also the numerical adjustment methods Num I and Num II give better results. Since more weight is allocated in the default column, the additional information of PD forecasts for the next period improves the results. Overall, in comparison to the one-factor model, for the numerical adjustment techniques, the forecast errors are still significantly higher.
TABLE 9.8. In-Sample Results for Mean Forecast Errors According to Applied Di3erence Measures and Adjustment Techniques. The Estimation Period Included 10 Years from 1984–1993. (Best results for each distance measure are highlighted in bold)
| Method | Factor | Num I | Num II | Naive I | Naive II | |
|---|---|---|---|---|---|---|
| Dist. | Distance Statistics
|
|||||
| L1 | MAE | 0.8058 | 1.3461 | 1.7092 | 0.8809 | 1.0245 |
| Std | (0.2665) | (0.1415) | (0.2966) | (0.2414) | (0.3056) | |
| L2 | MAE | 0.0580 | 0.1802 | 0.2947 | 0.0725 | 0.1022 |
| Std | (0.0483) | (0.0503) | (0.0885) | (0.0510) | (0.0948) | |
| NSD | MAE | 0.2956 | 0.4773 | 0.8774 | 0.3483 | 0.5911 |
| Std | (0.1363) | (0.1356) | (0.2544) | (0.1800) | (0.4556) | |
| D1 | MAE | 0.3218 | 0.7592 | 1.1505 | 1.2417 | 1.4969 |
| Std | (0.2189) | (0.3765) | (0.9526) | (0.8903) | (0.7522) | |
| D2 | MAE | 1.9926 | 6.2919 | 9.4428 | 9.6931 | 10.8898 |
| Std | (1.3993) | (3.2154) | (4.8858) | (7.1228) | (5.5701) | |
We also investigated whether the improvement of the forecasting results of the one-factor model was mainly due to the speculative or investment grade rating classes of the migration matrix. Table 9.9 provides the results of the one-factor model and the naive approaches separately for initial speculative and investment grade ratings. We find that especially for the risk-sensitive evaluation criteria the improvement using a credit cycle index comes from better forecasts for the speculative grade default probabilities and rating changes. For the rating classes Ba–C the forecast error is reduced up to 80% when the risk-sensitive measures D1 or D2 are applied. Further, as it is indicated by Table 9.9, the large deviations from actual observed migration matrices take place in the speculative grade area of the matrix where more variation can be observed.
TABLE 9.9. In-Sample Results (Mean Absolute Errors) Separately for Speculative Grade (Ratings Ba, B, and C) and Investment Grade ( Aaa, Aa, A, and Baa) Ratings
| Speculative Grade | Investment Grade | |||||
|---|---|---|---|---|---|---|
| Dist. | Factor | Naive I | Naive II | Factor | Naive I | Naive II |
| L1 | 0.5060 | 0.5756 | 0.6558 | 0.2997 | 0.3052 | 0.3687 |
| L2 | 0.0423 | 0.0557 | 0.080 | 0.0157 | 0.0168 | 0.0222 |
| NSD | 0.1913 | 0.2425 | 0.3548 | 0.1043 | 0.1058 | 0.2363 |
| D1 | 0.2770 | 1.1620 | 1.3375 | 0.0449 | 0.1197 | 0.1628 |
| D2 | 1.6544 | 8.8322 | 8.8505 | 0.3382 | 0.8609 | 1.0393 |
At this point we should also emphasize the advantage of the directed difference indices D1 and D2 as a measure for the goodness-of-fit. It concerns the question of interpretation of the results. Obviously, an MAE of 0.8058 for the L1 norm cannot be interpreted in terms of risk. Though, using the risk-sensitive difference indices, we are able to give an interpretation of the results from a risk perspective. Trueck and Rachev (2007) show that for credit portfolios, differences between migration matrices are highly correlated with the estimated credit VaR. Using Moody’s historical migration matrices, for an exemplary credit portfolio, a relationship between credit VaR and the deviation of a transition matrix from Moody’s average historical migration matrix is derived. Setting the recovery rates to a constant, the relationship between VaR for the exemplary loan portfolio and D2 is then approximately expressed by (in Mill. Euro)
The estimated regression model yields an R2 > 0.9 (Trueck and Rachev, 2007). Hence, due to the very high correlations between the directed difference index and credit VaR, for exemplary loan portfolios
we would be able to measure our errors on migration matrix forecasts in terms of risk. This means that the mean error of 9.6931
from Table 9.8, using the average migration matrix 
as an estimator, could be interpreted for the exemplary portfolio as an average misspecification of VaR of approximately
4.7110 · 9.6931 ≈ 45 Mill. Euro per year. For using the migration matrix of the previous year, we obtain an approximate error of 51 Mill. Euro. When one uses the one-factor model in order
to condition migration matrices to business cycle effects, the mean absolute error is reduced to 1.9926, yielding an average
error on one-year VaR forecasts of 9.38 Mill. Euro for the exemplary portfolio. It is important to point out that these are
just approximate numbers for an exemplary portfolio, ignoring variations in LGD figures and other components. However, as
a general result, we argue that using the risk-sensitive difference indices as a goodness-of-fit measure, the forecast error
may also be quantified in terms of risk. We point out that further research on this issue will be needed, especially on the
sensitivity of the difference indices. Overall, the advantage of an index giving a strong interpretation in terms of risk
is obvious.
9.5.5. Out-of-Sample Forecasts
Finally, we used the developed models for out-of-sample forecasting of rating migration behavior. The considered period was the subsequent years from 1994–1999. Based on a yearly re-estimation of the regression model and the weights of the systematic credit cycle index in the chosen one-factor model, and conditional PD estimates, forecasts for the migration matrix of the following year were calculated. Hereby, the in-sample estimation period was increased each year from 1984–1993 to 1984–1994, 1984–1995, … , 1984–1998. Results for the yearly re-estimated weights wInv and wSpec of the systematic credit cycle index using D2 are described in Table 9.10. Results for the other distance measures are available on request from the author. As it could be expected, the weights change through time and vary between 0.234 and 0.191 for the speculative grade issuers and between 0.168 and 0.149 for the investment grade credit cycle index. Generally, the weight of the systematic credit cycle index decreases for both investment categories through time.
Table 9.11 finally investigates the out-of sample performance of the considered models. We find that especially for the conditional approaches (Factor, Num I, and Num II) that use the credit cycle index, the results are not as good as for the in-sample period. This can be explained by the decreasing influence of the systematic credit cycle index, for the out-of-sample period that was reported in Table 9.11. Still the factor model significantly outperforms the numerical adjustment methods Num I and Num II and the benchmark models Naive I and Naive II. Except for the years 1996 and 1999, it gives the best forecasts of next year’s migration matrix in each year. Further, clearly the lowest average forecast error is obtained using the factor model approach. However, compared to the in-sample estimation, the average forecasting error is higher, and we obtain a mean absolute error of 4.1740. For the naive models the results for in-sample and out-of-sample periods are similar, while for the numerical adjustment methods, the error increased as well. Note that Num II provides the worst results of all models and is outperformed even by the naive approaches.
TABLE 9.10. Re-Estimated Weights of the Credit Cycle Index for Investment and Speculative Grade Ratings in the One-Factor Model Approach Using the Distance Index D2
| Year | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 |
|---|---|---|---|---|---|---|
| Speculative Grade | ||||||
| D2 | 0.2340 | 0.2058 | 0.2005 | 0.1914 | 0.1961 | 0.1964 |
| Investment Grade | ||||||
| D2 | 0.1663 | 0.1681 | 0.1579 | 0.1493 | 0.1529 | 0.1489 |
TABLE 9.11. Out-of-Sample Results for Mean Absolute Forecast Errors According to Applied Difference Measures D2 and Adjustment Techniques. The Out-of-Sample Period Included Six Years from 1994–1999. (Best results for each year are highlighted in bold)
| Method | Factor | Num I | Num II | Naive I | Naive II |
|---|---|---|---|---|---|
| Year | Distance Statistics
|
||||
| 1994 | 3.2563 | 6.9856 | 9.2745 | 8.7058 | 4.4462 |
| 1995 | 1.9834 | 3.5110 | 4.2809 | 1.0289 | 7.6769 |
| 1996 | 8.8284 | 14.3459 | 21.7397 | 18.9697 | 17.9408 |
| 1997 | 5.2049 | 10.8375 | 14.5619 | 12.4049 | 6.5649 |
| 1998 | 2.5774 | 8.9729 | 8.0307 | 5.5436 | 17.9484 |
| 1999 | 3.1936 | 7.2204 | 9.7712 | 8.5408 | 2.9973 |
| Average | 4.1740 | 8.6456 | 11.2765 | 9.1989 | 9.5958 |
We point out that the results could have been improved by changing the variables of the macroeconomic forecasting model for the credit cycle index. However, in order to guarantee a genuine out-of-sample test of the model, we choose the macroeconomic variables to be the same for in-sample and out-of-sample model evaluation. Still we conclude that a regular re-estimation of the model for the credit cycle indices may be recommendable.