Rating change probabilities: An empirical analysis of sovereign ratings


Bachelor Thesis, 2012

48 Pages, Grade: 1,0


Excerpt

Contents

List of Figures

List of Tables

1 Introduction

2 Previous Literature

3 Agencies’ rating assessment
3.1 Rating Definitions and Methodology
3.2 Rating Transition Matrices

4 Ordered Probit Model
4.1 Mathematical Framework
4.2 Model parameters
4.3 Marginal effects

5 Data
5.1 Data summary
5.2 Independent Variables and direction of change
5.3 Choice of Variables

6 Estimation
6.1 Simple Ordered Probit Model
6.2 Random Effects Ordered Probit Model
6.3 Marginal Effects

7 Predicted Probabilities and Transition matrices
7.1 Predicted Probabilities
7.2 Unconditional Transition Matrices
7.3 Conditional Transition Matrices

8 Conclusion

Appendix

References

List of Figures

1 Moody’s Credit Rating for Each Country in July 2012

2 Observation of Ratings per Rating Category

3 Distribution of Coarse Ratings per Year

4 Distribution of Coarse Rating Changes per Year

5 Histogram for Rating Changes and Standard Normal Distribution

6 Dot Plot On Rating Change Probabilities for all Countries

List of Tables

1 Moody’s Credit Rating Definitions

2 Explanation of Independent Variables, considered in the Ordered Probit

Model

3 Frequencies of Investment and Speculative Grade Ratings in my Data Set,

Rated by Moody’s

4 Frequencies of Investment and Speculative Grade Ratings in the Data Set

of Hu et al. (2002), Rated by S&P

5 Frequencies of Fine and Coarse Rating Changes, Classified by the Change

Direction and Notches Moved

6 Regression Table for the Ordered Probit Estimation in the Fine and Coarse

Rating Categories

7 Regression Table for the Random Effects Ordered Probit Model

8 Marginal Effects on the Probabilities of Rating Changes, Calculated as

Average of Marginal Effects (AME)

9 Marginal Effects on the Probabilities of Rating Changes, Calculated at

Means of the Independent Variables (MEM)

10 Comparison of Probabilities at Mean Values of the Independent Variables

and for Average Probabilities

11 Frequencies of Rating Changes in the Fine Rating Category

12 Frequencies of Rating Changes in the Coarse Rating Category

13 Estimated Transition Matrix for all Countries, Calculated with Average Probabilities

14 Estimated Transition Matrix for Non-Developing Countries, Calculated with Average Probabilities

15 Estimated Transition Matrix for Non-Developing Countries in Business Cy­cle Peak, Calculated with Average Probabilities

16 Estimated Transition Matrix for Non-Developing Countries in Business Cy­cle Trough, Calculated with Average Probabilities

1 Introduction

Sovereign Ratings are in the center of public attention, in times of sovereign debt crisis. They are important for countries as they determine their cost of capital (Bissoondoyal- Bheenick, 2005). And therefore their movements in either direction are observed critically. Whereas an upgrade can cause a decrease in borrowing cost, a downgrade can imply a burden and can cause the dependency of external support in case of default. It is also essential for institutional investors to know the probability of a rating change as it describes the potential gain or loss in their portfolio.

After Moody’s recent announcement of changing the outlook from stable to negative on long-term ratings of Germany, the Netherlands and Luxemburg, it seems that even countries which were thought to keep their Aaa credit standing in the sovereign debt crisis are affected of potential rating changes (Moody’s, 2012b). In the same announcement Moody’s affirm that ’’Finland’s unique credit standing” assures the country an ongoing stable rating outlook. In this case four countries all belonging to the Euro area and having the same Aaa credit rating, have different exposures to potential rating changes, according to Moody’s.

Standard and Poor’s Ratings Services however decided to keep Germany’s long-term rating outlook stable (S & P, 2012). S & P did not change Germany’s rating outlook because they emphasize ”its modern, highly diversified, and competitive economy [...] and expenditure discipline”. They also argue that Germany’s dealing with historic ’economic and financial shocks, such as the reunification of West Germany with East Germany in the 1990s and the global recession in 2009” allow them to have a stable outlook. In the same time, they changed the outlook of the other AAA rated countries Luxemburg, the Netherlands and Finland to negative. Even though S & P and Moody’s have different rating assessments in this case, their general assessment remains similiar. Differences in ratings can appear, depending on the agencies’ priorities and their assessment of risk.

The exemplary announcements of changes in rating outlooks raise the question which factors drive a change in the rating outlook or which determinants cause a real credit rating change. A lot of research has been done on the determinants of ratings. Less research has focused on the determinants of credit rating changes and the probabilities of rating transitions.

In my Bachelor thesis I will analyze the determinants of rating changes and the vari­ables’ marginal effects on rating change probabilities. Based on my results, I will present transition matrices by computing transition probabilities. Furthermore, I will analyze subsamples of my data set, conditional on the business cycle and the economic strength of a country, by using interaction effects. I thereby verify whether or how the transition matrices change by including interaction effects.

I apply a latent variable approach, using an ordered probit model, to calculate the effects of different variables on the probabilities of rating changes. I therefore make use of the discreteness of rating changes, classifing them by a number which indicates their change direction and the number of notches moved. I assign an order to the possible rating changes with eight classes, when considering the coarse rating category. This method is proposed by Wooldridge to deal with the underlying ordered nature of the ordered probit model (Wooldridge, 2001, p.506). The non-linearity, assumed by the model, also allows to treat the difference between the rating changes differently. A linear model would not account for the ordinal nature of rating changes, but treat the difference between the rating changes equally (Borooah, 2001, p.5). A further implication of the ordered probit model is the normal distribution of the error term . This is in contrast to the ordered logit model, where one assumes a logistic distribution of the error term tit . Both models are alternatives to each other and only differ in the tails of their graphs (Borooah, 2001, p.9). According to Greene (2012, p.729), one can justify the choice between the ordered probit or logit model with regard to ’’mathematical convienience”, but it is difficult to justify it with regard to the distributional assumptions. As the ordered probit model satisfies my mathematical requirements, I choose it for my calculations.

My analysis adds value to most prior work on sovereign rating changes because they are dated back ten years or more when sovereign ratings were less available, especially on a global scale. In my Bachelor thesis I account for the increase in availability of sovereign ratings. In my data set I use the maximum amount of countries that are both included in the World Bank data set and rated by Moody’s. This allows for better legitimacy of the ordered probit model as the ’ordered probit asymptotic properties do not generalise for small samples” (Afonso et al., 2011). For example the increase in rated countries since the paper of Hu et al. (2002), who also worked on rating changes, and my paper can be seen in Tables 3 and 4 in the Appendix. Whereas Hu et al. worked with 487 ratings in total, I work with 1,837 ratings in total. In 2011, which is my last year of observation, with a total number of 112 out of 113, nearly every country was rated. In addition to the extended sample, there is also an increase in rating changes due to the bust of the international financial crisis and the following sovereign debt crisis which enables a better estimation of rating changes and therefore allows a better analysis of rating changes.

The remainder of my Bachelor thesis is organized as follows. Chapter two presents an overview of the existing literature on rating changes. Chapter three illustrates the definitions of the rating categories and Moody’s approach to rating transitions. Chapter four introduces the ordered probit model. Chapter five explains my data choice and gives an analysis of the data set. Chapter six presents the estimation results of the determinants of rating changes. Chapter seven predicts rating change probabilities and presents the transition matrices with interaction effects. Chapter eight concludes.

2 Previous Literature

The research on the determinants of sovereign rating changes is based on the analysis of the determinants of sovereign ratings. As found in the influential work of Cantor and Packer (1996), using an OLS regression, ratings can be determined mainly on the basis of six variables, which are per capita income, GDP growth, inflation, external debt, level of economic development, and default history. Afonso et al. (2007) modified the analysis of underlying fundamentals of sovereign ratings by using panel estimation and random effects ordered probit approaches in order to account for the ordered nature of ratings. They find that under the ordered probit model there are the following variables that determine ratings: GDP per capita, GDP growth, government debt, government effectiveness indicators, external debt, external reserves, and default history.

It seems reasonable to suppose that changes in the above mentioned variables will lead to sovereign rating changes. However not every change in those variables will cause a sovereign rating change, especially not on long-term ratings, as in my case. Sovereign rating changes occur when certain thresholds in the fundamental values are crossed. The aim of my Bachelor thesis is to find out the determining variables which are responsible for the effective rating changes and to estimate those thresholds.

The literature of the determinants of sovereign rating changes often deals with different methodologies to assess precise rating change probabilities and the creation of precise transition matrices, given the constraint of low availability of sovereign ratings.

Most studies have been made with reference to corporate ratings because they allow to have more data and longer rating histories. Even though corporate ratings differ in their nature from sovereign ratings, some methodologies of their analysis can be helpful and sometimes be applied, in a different manner, to sovereign ratings. The results from corporate rating research can give some insights in the nature of rating changes.

Nickell et al. (2000) have estimated transition matrices for 6534 corporate obligors during the time 1970-1997, using an ordered probit approach. They examined how in­dustry, country and stage of the business cycle influence the change in ratings. Nickell et al. (2000) and Bangia et al. (2002) find out that upgrades are more likely in booms and downgrades are more likely in recessions. Altman and Kao (1992) investigated the risk of corporate bonds regarding the aging effect, that is the implication of the length of time since issue. They find that default risk increases in the first three or four years of an issue’s life and disappears afterwards. Hamilton and Cantor (2004) show that obligors that have been downgraded are nearly 11 times more likely to default than those that have been upgraded. Other analysis is based on rating heterogeneity within countries, like serial correlation, which is known as rating drift. According to Altman and Kao (1992), Kavvathas et al. (1993) ratings possess a memory. This means that future changes not only depend on the present credit standing but also on prior rating changes: ”The prob­ability of a downgrade following a downgrade within one year significantly exceeds that of an upgrade following a downgrade and vice versa.” (Eisenkopf, 2007).

Some studies about rating changes also refer to sovereign rating changes. Hu et al. (2002) search for determinants of rating transitions and find a method to create sovereign transition matrices in the case of a short rating history. They analyze both industrial and emerging market sovereign borrowers, even though the latter category has only few if any ratings available. Therefore they propose a simultaneous ordered probit model of credit ratings and default history. The default history adds to the ratings as a further source of information, in order to prevent biased maximum likelihood estimators. They find that previous year default, debt to GNP, reserves to imports, inflation, economic development, lagged debt service to exports and lagged inflation explain rating changes.

Moreover Al-Sakka and ap Gwilym (2009) apply a random effects ordered probit model and show that watchlist status, existing rating and issuers domicile region are useful determinants in modeling sovereign rating changes in emerging economies. Al-Sakka and ap Gwilym (2010) add to their model that the ’’estimation of sovereign rating migrations can be improved by considering the sources of heterogeneity, such as rating history, rating duration and Watchlist status, and cross-section error (country-specific heterogeneity)”. The idea is to make use of the random effects ordered probit model because the cross­section error term captures factors for developing countries like ’geopolitical uncertainty, political risk and social tensions” which are mostly time-invariant features and cannot be quantified (Al-Sakka and ap Gwilym, 2010). Also Afonso et al. (2009) find that the random effects ordered probit model slightly outperforms the simple ordered probit model.

Even though the approach of random effects ordered probit model by Al-Sakka and ap Gwilym (2010) adds information to the simple ordered probit model, they do not find out the true macroeconomic determinants of rating transitions. The watchlist status and the existing rating are part of the rating agencies’ assessment of rating transitions. Therefore they do not search for the sources of rating changes like Hu et al. (2002) do and like it is part of my research. My analysis will show which variables of Hu et al.’s selection also hold for my data set or which additional variables influence rating changes in the case of longer sovereign rating histories.

3 Agencies’ rating assessment

Rating agencies emphasize that ratings are based on both quantitative and qualitative elements. The qualitative approach is supposed to add additional information which is not incorporated in the quantitative data. Rating agencies have multiple instruments to disclose obligors’ creditworthiness to investors. Besides rating, also outlook and review serve as additional sources of information for investors. The outlook usually precedes rating reviews and serves as an indicator as to the direction of the credit assessment in the medium term. There are four categories ranging from positive, negative, stable and developing (Moody’s, 2012a). ”A review indicates that a rating is under consideration for a change in the near term” ranging from upgrade (UPG), downgrade (DNG) to uncertain direction (UNC). This section explains the basic rating definitions and introduces the agencies’ methodology to construct rating transition matrices.

3.1 Rating Definitions and Methodology

’’Sovereign debt ratings are forward-looking qualitative measures of the probability of default, given in the form of a code” (Afonso et al., 2009). There are both short- and long-term ratings. In this Bachelor thesis I will discuss only long-term ratings because they are better known and because they allow a broader ranking of letter rating categories. There are 9 coarse rating categories and 21 fine rating categories for long-term ratings, whereas there are only four short term ratings which are P-1, P-2, P-3 and not prime.

Table 1 in the Appendix shows the order of Moody’s ratings with the respective eval­uation of a corresponding country’s credit standing (Moody’s, 2012a). Next to Moody’s credit codes, I also added the codes that I use for my analysis, in order to be able to calculate with them.

Moody’s have a three-stage process to determine a sovereign rating (Moody’s, 2008). In the first step, they assess the country’s economic resiliency. In the second step, they assess the government’s financial robustness. Finally, in the third step they determine the rating within the fine rating category.

The first step is described by two factors. The first factor is the economic strength, which is given by the quantitative values of GDP per capita and the volatility of GDP. The second factor is the institutional strength of the country, such as ’property right, trans­parency, the efficiency and predictability of government action, the degree of consensus on the key goals of political action” (Moody’s, 2008).

The second step is again described by two factors. The first factor is the financial strength of the government, such as the sort of debt and the government’s ability to ”raise taxes, cut spending, sell assets, obtain foreign currency,...”(Moody’s, 2008). The second factor is the sensibility to event risk. This factor accounts for the country’s ability to resist to the ’’occurrence of adverse economic, financial or political events” (Moody’s, 2008).

Due to the last two mentioned steps, Moody’s have placed the country in one of the nine coarse rating categories. The third step defines the fine category from the 21 ratings by ’adjusting the degree of resiliency to the degree of financial robustness”.

3.2 Rating Transition Matrices

According to Moody’s, a rating change represents a ’variation in the intrinsic relative position of issuers and their obligations” due to ’alteration in creditworthiness, or that the previous rating did not fully reflect the quality of the bond as now seen” (Moody’s, 2012c). Even though long-term ratings are supposed to reflect the country’s long-term rating standing, there are cases, in which Moody’s change its rating assessment annually or quarterly. This especially happens in times of economic turmoils like the current sovereign debt crisis in Europe.

The big three rating agencies S & P, Moody’s and Fitch create one and five year credit transition matrices. Credit transition matrices describe migration probabilities of firms and countries to change from any initial credit standing to any terminal rating in a future time. Moody’s make use of survival or duration modeling, called ”multiple destination proportional hazards model” to calculate the transition probabilities (Moody’s, 2011). Duration models simulate how long the rating stays the same and whether there will be an upgrade, a downgrade or a default. The model uses both macroeconomic and is­suer specific variables to determine the transition probabilities. Moody’s have identified two macroeconomic factors to have general predictive power for the creditworthiness of countries, which are the unemployment rate forecast and the forecast of the high yield spread over Treasuries. The unemployment rate is meant to describe the ”macroeconomic health”. The high yield spread over Treasuries is meant to incorporate the ”market’s per­ception of credit quality and hence credit availability”. Additionally the credit transition matrix takes into account today’s and historic issuer-specific elements: ”the current rat­ing, whether the issuer was upgraded or downgraded into its current rating, how long the issuer has maintained its current rating, how long the issuer has consecutively maintained any credit rating, and the issuers current outlook or watchlist status” (Moody’s, 2011).

4 Ordered Probit Model

My model is similiar to the model of Nickell et al. (2000) and Hu et al. (2002) and is theoretically based on Long and Freese (2001), chapter 5, Greene (2012), chapter 18 and Borooah (2001). As the notations in the ordered probit model differ in Greene and Borooah with regard to the intercept and the first cut-off point, I will try to combine the explanations of both approaches. Moreover using Stata 12.0, this implies different assumptions with regard to the intercept and the cut-off points. I will explain the role of the cut-off points in the following section. According to Long and Freese (2001) the ordered probit model has too ’’many free parameters” and cannot estimate both the cut­off points and the constant. Therefore one has to decide for estimating either the constant and assume the first cut-off value to be zero or to omit the constant and to estimate all cut-off points (Long and Freese, 2001, p.187). Stata does not include an intercept term in its output because it is absorbed into the cut-off points, whereas Greene does include it and sets the first cut-off point to zero (Borooah, 2001, p.10). I will stick to the notation of Borooah in order to be congruent with the Stata output.

In contrast to Nickell et al. (2000) and Hu et al. (2001), I use credit rating changes as outcomes in the ordered probit model, instead of rating levels. The ordered probit model is useful in the observation of rating changes because I assign an order to the different rating changes by notches moved, where 8 upgrades is the highest outcome and 8 downgrades is the lowest outcome. The ordering is the following: +8, +7, +6, +5, +4, +3, +2, +1, 0, -1, -2, -3, -4, -5, -6, -7, -8, where a plus symbolizes an upgrade and a minus a downgrade. Also negative outcomes are possible in the ordered probit model, as long as ’larger values are assumed to correspond to ’higher outcomes” (Long and Freese, 2001, p.188).

4.1 Mathematical Framework

The sovereign rating from one to another period can be an upgrade, a downgrade or no change. In the coarse rating category, as there are 9 rating levels, there are always 8 possible rating changes. Considering the two extremes Aaa and C, the lowest level of a possible rating change is achieved by a downgrade by eight notches and the highest level of a possible rating change is reached by an upgrade by 8 notches. When also taking into account the possibility of no rating change, like shown above, this gives J =17 possible outcomes of rating changes in total.

Suppose that a country i ’s credit rating change in time t is determined by the realization of a linear unobserved latent variable y*t. The latent variable is determined by the sum of K (k=1,...,K) independent variables multiplied with the corresponding estimate for every country in the time horizon 1990-2011 (Borooah, 2001, p.8):

Abbildung in dieser Leseprobe nicht enthalten

Abbildung in dieser Leseprobe nicht enthalten

A country is allocated to a rating change outcome yit, if the latent variable y*t falls into the interval [^i_i,^i] (Hu et al., 2002). The p’s are the cut-off points which are unknown parameters that are to be estimated like 3’s using maximum likelihood estimation. After the estimation of the parameters, it is possible to assign the fitted values into the intervals between the J-1 = 16 cut-off points:

Abbildung in dieser Leseprobe nicht enthalten

The probability of a rating change is given by the standing of the latent variable in the normal cumulative distribution function. The observed outcome for a given value of yit is the area under the curve between a pair of J-1 cut-off points p. The cumulative distribution of a standard normal variate X is (Borooah, 2001, p.12):

Abbildung in dieser Leseprobe nicht enthalten

In the ordered probit model, $ is the standard normal cumulative distribution function. Moreover $ follows a standard normal distribution with

Abbildung in dieser Leseprobe nicht enthalten

If assuming that every possible rating change also occurs, the probabilities for every outcome are given by the following equations (Borooah, 2001, p.10):

Abbildung in dieser Leseprobe nicht enthalten

For the right assignment of the probabilities, we must have an increasing order for the cut-off points y:

Abbildung in dieser Leseprobe nicht enthalten

In my model the cut-off points are extended by negative values because the downgrades are defined to be negative and upgrades are defined to be positive. But as the increasing order is still given, my model satisfies the requirement.

4.2 Model parameters

In the ordered probit model, the estimated coefficient Z corresponds to the predicted probability of the direction of a rating change (Borooah, 2001, p.24). A positive sign of the coefficient indicates for dummy variables that all other variables constant, countries with this feature have a higher probability of being upgraded and a lower probability of being downgraded than countries that do not have this feature (Borooah, 2001, p.24). A negative sign suggests that, ceteris paribus, countries with this feature have a lower probability of being upgraded and a higher probability of being downgraded than countries that do not have this feature. However only the direction of change in the probabilities of the extreme outcomes can be inferred and not of the intermediate ones (Borooah, 2001, p.24). The interpretation of the estimates needs caution. However for interpretation purposes, I will try to interprete the coefficients, as far as this appears reasonable. The additionally estimated cut-off points p are given underneath the /A-coefficients, in contrast to Stata outputs in OLS regressions.

4.3 Marginal effects

The marginal effect gives the change in the probability of the outcome if the value of one of the independent variables changes. An important feature of the ordered probit model is that the marginal effects are not given by the estimated Z- coefficients. In the following, I therefore derive the marginal effects.

Marginal Effects of Continuous Variables:

If the independent variable is continuous, the marginal effect can be calculated by the derivative of the probability of an outcome, using the chain rule:

Abbildung in dieser Leseprobe nicht enthalten

As one can see from the marginal effects, again only the influence on the probabilities from the two extreme outcomes will be unambiguously determined (Greene, 2012, p.829). For the remaining outcomes, marginal effects do not have a clear sign and therefore do not allow a conclusion about the direction of the change. As a practical analysis one could investigate how the probability of a rating transition changes, if GDP growth increases. From the requirement that the probabilities add up to 1, results that the sum of the marginal effects is zero (Greene, 2012, p.830).

Marginal Effects of Dummy Variables:

If however the independent variable is a dummy variable, then the effect has to be determined differently. In this case, the effect has to be calculated as the difference in probabilities between the two variables which are to be compared. So the difference is calculated between the probabilities of a dummy variable taking the value one and a dummy variable taking the value zero, keeping all other variables at their means (Long and Freese, 2001, p.213):

Abbildung in dieser Leseprobe nicht enthalten

Like in the case of continuous variables, the interpretation only holds for the extreme outcomes. A practical example would be the analysis how the probability of a sovereign rating transition changes if the concerning country is a non-developing country.

5 Data

5.1 Data summary

My dataset contains 113 countries rated by Moody’s in the period 1990 until 2011. I chose all countries that Moody’s rated until 2011. The ratings were observed on the 31st of December of each year. Furthermore I chose the time period because of high data availability at the World Bank. The sovereign ratings refer to the foreign long-term issuer ratings, which means that it deals with maturities of one year or greater. I chose Moody’s sovereign ratings, representatively for all credit rating agencies. Cantor and Packer (1996) also state in their work that there is a high correlation between ratings of Moody’s and S & P with marginal differences: ”Of the forty-nine countries rated by both Moodys and Standard and Poors in September 1995, twenty-eight received the same rating from the two agencies, twelve were rated higher by Standard and Poors, and nine were rated higher by Moodys. When the agencies disagreed, their ratings in most cases differed by one notch on the scale, although for seven countries their ratings differed by two notches.” Furthermore my aim is not to investigate the difference between the rating agencies, but to analyze rating changes, given the agency’s decision on a rating change. I assume that the results do not improve for higher divergencies in opinions about ratings, given the low differences in ratings from different agencies.

As far as there are withdrawn ratings, I do not consider them for the period that they are withdrawn.

In the following I will present several tables and figures that are supposed to introduce my data set and its structure. I especially want to highlight how rating changes appear in my data set in general and over time. Tables 3 and 5 show that the 113 countries over the 22 years add up to a total number of 1,837 rating observations with 317 rating transitions in the fine rating category and 157 rating transitions in the coarse rating category. I have 1,188 investment grade (Aaa-Baa) rating observations and 649 speculative grade (Ba-C) ratings. Concerning the fine rating category, there were 178 rating changes within investment grade and 140 rating changes within speculative grade. Tabel 4 shows that in the dataset of Hu et al. (2002), there are only 487 sovereign rating observations of which 158 have speculative grade observations over the period 1981-1998. That means that the analysis based on my data set will provide more general results.

Figure 1 illustrates the distribution of Moody’s sovereign ratings for July 2012, colored by credit quality (Chartsbin, 2012). Especially it shows the strikingly low availability of sovereign ratings in Africa.

Figure 2 shows the distribution of sovereign ratings by rating category in my data set. It reveals the dominance of investment grade ratings (Aaa-Baa). Moreover there is a low presence of Caa and Ca rated countries and no C ratings. Figure 3 additionally shows the evolution of the rating distribution along the considered time period.

[...]

Excerpt out of 48 pages

Details

Title
Rating change probabilities: An empirical analysis of sovereign ratings
College
University of Mannheim
Grade
1,0
Author
Year
2012
Pages
48
Catalog Number
V231459
ISBN (eBook)
9783656470809
ISBN (Book)
9783656470823
File size
1026 KB
Language
English
Notes
Die Bachelorarbeit wurde mit dem 1. Platz beim Karrierepreis der DZ Bank Gruppe (2013) ausgezeichnet. Dies ist der höchstdotierte Hochschulpreis der Deutschen Wirtschaft für Akademische Abschlussarbeiten im Bereich "Banking &amp, Finance": www.karriere-preis.de
Tags
Economics, Rating, Ordered Probit Model, Volkswirtschaftslehre, VWL, Finance, Econometrics, Statistik, Markov Chain, Transition Matrix, Ökonometrie, Banking
Quote paper
Alex Bergen (Author), 2012, Rating change probabilities: An empirical analysis of sovereign ratings, Munich, GRIN Verlag, https://www.grin.com/document/231459

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