# Reconstruction Rating Model of Sovereign Debt by Logical Analysis of Data

Authors: Elnaz Gholipour; Béla Vizvári; Zoltán Lakner
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Sovereign debt ratings provided by rating agencies measure the solvency of a country, as gauged by a lender or an investor. It is an indication of the risk involved in investment and should be determined correctly and in a well-timed manner. The current system is lacking transparency of rating criteria and mechanism. The present study reconstructs sovereign debt ratings through logical analysis of data (LAD), which is based on the theory of Boolean functions. It organizes groups of countries according to 20 World Bank-defined variables for the period 2012–2015. The Fitch Rating Agency, one of the three big global rating agencies, is used as a case study. An approximate algorithm was crucial in exploring the rating method, in correcting the agency’s errors, and in determining the estimated rating of otherwise unrated countries. The outcome was a decision tree for each year. Each country was assigned a rating. On average, the algorithm reached almost 98% matched ratings in the training set and was verified by 84% in the test set.


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## Body

## 1. Introduction

The aim of the research reported in this paper was to understand sovereign debt rating procedure as rating affects the economies of countries and to develop a simple tool for determining the rating of the debts of sovereign countries. If a country is rated, then its rating can be obtained from the Internet. However, even in this case, it is possible that it is underrated or overrated. There are countries that are not rated at all. These countries need debts as well. An investor might be interested to the risk of borrowing money to such a country. In that case, a tool determining an approximate rating of the country can help in the decision. It is obvious that a tool that can be used by all investors cannot be complicated. For example, the journalWorld Bank Research Observer claims from the authors that “No equations, or if they have to be used, they should be comprehensible at about the seventh-grade level” [1]. The tool which has been developed is a simple decision tree, which satisfies this requirement. The method that is used to obtain the decision tree is logical analysis of data (LAD).


### 1.1. Sovereign Credit Ratings (SCRs)

Sovereign credit ratings (SCRs) refer to a country’s capability to repay the money that it has borrowed. Therefore, sovereign debt rating can be a metric to help potential investors, financial organizations, banks, and even other governments when making investment or making lending decisions regarding a particular country. Sovereign debt rating reflects the risk involved in doing so. Generally, governments look for credit ranking to simplify their access to international capital markets, where investors desire to pick rating securities over unrated ones even with the same credit risk [2]. With an SCR, the government is less reliant on the banks’ monetary policy, and it can join international markets. Moreover, SCR can lead to financial improvement by drawing in foreign investors [3].SCRs play an important role in the credit rating industry [4]. They can decrease the asymmetric information between investors and borrowers to increase the borrower’s willingness to access funds and lessen the credit risk from the lender`s point of view [5]. The rating class of Fitch Agency, as one of the CRAs, has been shown in Table 1.Table 1
Fitch ranking system and descriptions (source: [6]).
#DescriptionFitch rating scale1(Highest rank): the safest, least risky investmentsAAA2(Above average): safer and less riskyAAP3AA4AAM5(Average): average risk and safetyAP6A7AM8(Slightly below average): slightly risky and less safeBBBP9BBB10BBBM11(Well below average): riskier and less safeBBP12BB13BBM14(Lowest): the riskiest and least safeBP15B16BMThis process is carried out by credit rating agencies (CRAs) to reduce the information gap between lenders and borrowers. An SCR is supposed to reflect a country’s financial, economic, and political position [7]. In particular, since economic and political factors are taken into account by SCRs, it is not easy to measure qualitative variables in the rating procedure in terms of predicting sovereign ratings. Because governments may renege on their obligations or become less financially solvent due to a political decision, the inclusion of qualitative measures in the rating process is difficult. This is why CRAs give their opinions on the creditworthiness of the country and not an investment advice or assessments for obligation [8]. This fact was proved by Ferri et al. [9], who noted that CRAs were not able to foresee the East Asia crisis of 1999, and this influenced them to become sufficiently conservative to downgrade high-risk countries. Fund [10] explained the importance of political and economic factors in SCRs. From a different point of view, “conflicts of interest” has been used to justify the failures of CRAs [11]. Arguments among experts are abound in concerning the effectiveness of these variables on SCRs.


### 1.2. Credit Rating Agencies and Significant Factors

Documented proof is available of how credit rating agencies’ shortcomings contributed to the 2008 financial crisis—when they overrated and underrated some countries—and how they can take down governments and blow up capital markets [12]. At the time in question, they appeared to publish their SCRs in the manner of a black box, while hiding important issues from investors [4]. In the present study, the Fitch agency’s rating system has been chosen for analysis. There are a number of financial, political, and social variables that could have been studied when exploring its rating model. Research on the significant variables of CRAs is used as input, along with the corresponding data from the World Bank, so that the Fitch rating method—otherwise concealed—can be illuminated. Cantor and Packer [2] were the first researchers in SCR to observe that items such as GDP per capita, GDP growth, inflation, external debt, and default history are important variables. Many other papers have used the same variables while adding new ones. A number of variables being taken into account by different investigators time after time are unemployment, government debt, foreign reserves, fiscal balance, economic development, political stability, mobile phones, real interest rate, total debt, real exchange rate, and unit labor costs [8, 13, 14]. Spilioti and Vamvoukas [15] prove that there is a positive correlation on government debt and economic growth. Alexe et al. [16] proposed a model for the Standard and Poor agency’s rating system by selecting certain economic-financial and political parameters and by using multiple regression. In another study, fiscal uncertainty as a single determinant was used to explain the reason for changes in sovereign ratings during the financial crisis [17]. The application of the results of previous research requires expertise in statistics and is therefore not easy for noneconomists.Gültekin-Karakaş et al. [4] claimed that high-income countries tend to receive higher ratings than low-income ones. GDP per capita was an essential variable for the high-income countries’ rating evaluation. Unlikely, regression analysis has been applied in some previous investigations [4, 8, 13]. Routinely, the regression models that were employed involved different significant indicators that needed to have coefficients as a certain weight of the related variable to give a country’s rating. Therefore, the users of these methods would most probably have had a certain expertise. The present study introduces a different way by the application of LAD for the approximation of the credit rating of countries, which is understandable and applicable for every user, and it makes our technique distinctive and untried.The SCR process is still not understood completely. There are several recent papers connecting SCR with other factors. Bouri et al. [18] investigate the oil production of BRICS countries (Brazil, Russia, India, China, and South Africa) to their ratings. Bouri et al. [19] also connect oil production and sovereign ratings in general. They also logically analyze the relation of ratings with financial factors. Amstad et al. [20] analyze the effect of the exchange rates. Agiakloglou and Deligiannakis [21] investigate the effect of credit default swaps in eight important countries of European Union. Volz et al. [22] even connect the rating of sovereign debts and climate change.


### 1.3. Logical Analysis of Data

Logical analysis of data (LAD) is a way of studying datasets consisting of two classes. Generally, LAD classifies the outputs into two categories as true-false, positive-negative, yes-no, or 1-0 to explore the hidden classification processes. LAD consists of several main steps. First, a set of patterns is generated, and each pattern covers a subset of the desired observations. Mainly, the number of patterns is substantial. As a second main step, a small subset of the patterns is selected such that the target subset of the observations is covered in an accurate way. This subset of the selected patterns gives a disjunctive normal form in the sense of mathematical logic. There are different methods for the way of selection; thus, LAD has many approaches [23]. Besides, Hammer and Bonates[24] analyzed the application of LAD to medical problems, including the evolution of diagnostic and prognostic systems in cancer investigation. Mirzaei [25] applied the LAD approach to a one-year series of Moody’s ratings. Hammer et al. [26] and Hammer et al. [27] also analyzed sovereign debt rating by LAD with quite different approaches. There are also cases in natural sciences such that more than two classes must be distinguished [28]. LAD has been applied in several areas recently. Many engineering and medical applications are surveyed in [29]. Das et al. [30] detect cyber-attacks against large system such as power grid and water treatment plant. Another application in fault detection is discussed in Ragab et al. [31]. Jocelyna et al. [32] use LAD for the prevention of accidents caused by belt conveyors. LAD also has applications in computer science [33, 34]. Leujune et al. [29] and Ouyang & Chou [35] also report recent developments in the theory of LAD.To sum up, previous studies have revealed drawbacks and advantages in the application of this methodology to CRA policies. The most valuable contribution of the present study is that it offers instructions to all users who wish to rank countries, without them having to possess any special mathematical, financial, or political knowledge. The user must compare only the values of some significant variables of a country to some fixed values determineda priori by LAD to learn if the country in question has a certain rating. The rating classes are organized into a binary decision tree according to their strength as it is shown in Figure 1. This type of decision tree has no stochastic component.Figure 1
The decision tree of the ratings.A decision tree is a rooted directed graph in which every nonleaf vertex has two outgoing arcs and the leaf vertices are assigned to either 0 or 1. This well-known structure is applied for instance in [29]. In the case of the current study, each leaf represents a different credit rating class of the countries. The decision procedure by using the decision tree is as follows: starting from the root, a top-down approach is made in the tree. The procedure arrives in a leaf, i.e., the rating class of the country is identified as the rating class represented by the leaf, if the attributes of the country satisfy the constraints determined by LAD for that rating class (Figure 1).Incidentally, decision trees can be successful and competitive with other classifications models [36].


## 1.1. Sovereign Credit Ratings (SCRs)

Sovereign credit ratings (SCRs) refer to a country’s capability to repay the money that it has borrowed. Therefore, sovereign debt rating can be a metric to help potential investors, financial organizations, banks, and even other governments when making investment or making lending decisions regarding a particular country. Sovereign debt rating reflects the risk involved in doing so. Generally, governments look for credit ranking to simplify their access to international capital markets, where investors desire to pick rating securities over unrated ones even with the same credit risk [2]. With an SCR, the government is less reliant on the banks’ monetary policy, and it can join international markets. Moreover, SCR can lead to financial improvement by drawing in foreign investors [3].SCRs play an important role in the credit rating industry [4]. They can decrease the asymmetric information between investors and borrowers to increase the borrower’s willingness to access funds and lessen the credit risk from the lender`s point of view [5]. The rating class of Fitch Agency, as one of the CRAs, has been shown in Table 1.Table 1
Fitch ranking system and descriptions (source: [6]).
#DescriptionFitch rating scale1(Highest rank): the safest, least risky investmentsAAA2(Above average): safer and less riskyAAP3AA4AAM5(Average): average risk and safetyAP6A7AM8(Slightly below average): slightly risky and less safeBBBP9BBB10BBBM11(Well below average): riskier and less safeBBP12BB13BBM14(Lowest): the riskiest and least safeBP15B16BMThis process is carried out by credit rating agencies (CRAs) to reduce the information gap between lenders and borrowers. An SCR is supposed to reflect a country’s financial, economic, and political position [7]. In particular, since economic and political factors are taken into account by SCRs, it is not easy to measure qualitative variables in the rating procedure in terms of predicting sovereign ratings. Because governments may renege on their obligations or become less financially solvent due to a political decision, the inclusion of qualitative measures in the rating process is difficult. This is why CRAs give their opinions on the creditworthiness of the country and not an investment advice or assessments for obligation [8]. This fact was proved by Ferri et al. [9], who noted that CRAs were not able to foresee the East Asia crisis of 1999, and this influenced them to become sufficiently conservative to downgrade high-risk countries. Fund [10] explained the importance of political and economic factors in SCRs. From a different point of view, “conflicts of interest” has been used to justify the failures of CRAs [11]. Arguments among experts are abound in concerning the effectiveness of these variables on SCRs.


## 1.2. Credit Rating Agencies and Significant Factors

Documented proof is available of how credit rating agencies’ shortcomings contributed to the 2008 financial crisis—when they overrated and underrated some countries—and how they can take down governments and blow up capital markets [12]. At the time in question, they appeared to publish their SCRs in the manner of a black box, while hiding important issues from investors [4]. In the present study, the Fitch agency’s rating system has been chosen for analysis. There are a number of financial, political, and social variables that could have been studied when exploring its rating model. Research on the significant variables of CRAs is used as input, along with the corresponding data from the World Bank, so that the Fitch rating method—otherwise concealed—can be illuminated. Cantor and Packer [2] were the first researchers in SCR to observe that items such as GDP per capita, GDP growth, inflation, external debt, and default history are important variables. Many other papers have used the same variables while adding new ones. A number of variables being taken into account by different investigators time after time are unemployment, government debt, foreign reserves, fiscal balance, economic development, political stability, mobile phones, real interest rate, total debt, real exchange rate, and unit labor costs [8, 13, 14]. Spilioti and Vamvoukas [15] prove that there is a positive correlation on government debt and economic growth. Alexe et al. [16] proposed a model for the Standard and Poor agency’s rating system by selecting certain economic-financial and political parameters and by using multiple regression. In another study, fiscal uncertainty as a single determinant was used to explain the reason for changes in sovereign ratings during the financial crisis [17]. The application of the results of previous research requires expertise in statistics and is therefore not easy for noneconomists.Gültekin-Karakaş et al. [4] claimed that high-income countries tend to receive higher ratings than low-income ones. GDP per capita was an essential variable for the high-income countries’ rating evaluation. Unlikely, regression analysis has been applied in some previous investigations [4, 8, 13]. Routinely, the regression models that were employed involved different significant indicators that needed to have coefficients as a certain weight of the related variable to give a country’s rating. Therefore, the users of these methods would most probably have had a certain expertise. The present study introduces a different way by the application of LAD for the approximation of the credit rating of countries, which is understandable and applicable for every user, and it makes our technique distinctive and untried.The SCR process is still not understood completely. There are several recent papers connecting SCR with other factors. Bouri et al. [18] investigate the oil production of BRICS countries (Brazil, Russia, India, China, and South Africa) to their ratings. Bouri et al. [19] also connect oil production and sovereign ratings in general. They also logically analyze the relation of ratings with financial factors. Amstad et al. [20] analyze the effect of the exchange rates. Agiakloglou and Deligiannakis [21] investigate the effect of credit default swaps in eight important countries of European Union. Volz et al. [22] even connect the rating of sovereign debts and climate change.


## 1.3. Logical Analysis of Data

Logical analysis of data (LAD) is a way of studying datasets consisting of two classes. Generally, LAD classifies the outputs into two categories as true-false, positive-negative, yes-no, or 1-0 to explore the hidden classification processes. LAD consists of several main steps. First, a set of patterns is generated, and each pattern covers a subset of the desired observations. Mainly, the number of patterns is substantial. As a second main step, a small subset of the patterns is selected such that the target subset of the observations is covered in an accurate way. This subset of the selected patterns gives a disjunctive normal form in the sense of mathematical logic. There are different methods for the way of selection; thus, LAD has many approaches [23]. Besides, Hammer and Bonates[24] analyzed the application of LAD to medical problems, including the evolution of diagnostic and prognostic systems in cancer investigation. Mirzaei [25] applied the LAD approach to a one-year series of Moody’s ratings. Hammer et al. [26] and Hammer et al. [27] also analyzed sovereign debt rating by LAD with quite different approaches. There are also cases in natural sciences such that more than two classes must be distinguished [28]. LAD has been applied in several areas recently. Many engineering and medical applications are surveyed in [29]. Das et al. [30] detect cyber-attacks against large system such as power grid and water treatment plant. Another application in fault detection is discussed in Ragab et al. [31]. Jocelyna et al. [32] use LAD for the prevention of accidents caused by belt conveyors. LAD also has applications in computer science [33, 34]. Leujune et al. [29] and Ouyang & Chou [35] also report recent developments in the theory of LAD.To sum up, previous studies have revealed drawbacks and advantages in the application of this methodology to CRA policies. The most valuable contribution of the present study is that it offers instructions to all users who wish to rank countries, without them having to possess any special mathematical, financial, or political knowledge. The user must compare only the values of some significant variables of a country to some fixed values determineda priori by LAD to learn if the country in question has a certain rating. The rating classes are organized into a binary decision tree according to their strength as it is shown in Figure 1. This type of decision tree has no stochastic component.Figure 1
The decision tree of the ratings.A decision tree is a rooted directed graph in which every nonleaf vertex has two outgoing arcs and the leaf vertices are assigned to either 0 or 1. This well-known structure is applied for instance in [29]. In the case of the current study, each leaf represents a different credit rating class of the countries. The decision procedure by using the decision tree is as follows: starting from the root, a top-down approach is made in the tree. The procedure arrives in a leaf, i.e., the rating class of the country is identified as the rating class represented by the leaf, if the attributes of the country satisfy the constraints determined by LAD for that rating class (Figure 1).Incidentally, decision trees can be successful and competitive with other classifications models [36].


## 2. Material and Methods

The present study examined the Fitch agency rating system to reconstruct a rating model of sovereign debt with LAD, a classifying methodology based on optimization and Boolean logic. It uses binary data (i.e., 0 and 1) [37].


### 2.1. Logical Analysis of Data (LAD)

LAD was initiated by Peter L. Hammer. He demonstrated that the LAD classification system produces accurate, transparent, and generalizable results [38]. There is a set of objects that are similar to each other and are described by the same set of attributes. The objects can be very different according to the area of application, for example, patients in a hospital, customers who obtain a loan from a bank, or drilling locations in the oil industry. However, the nature of the objects is the same in an application. The objects are divided into two parts, for instance, patients who have a particular disease and those who do not. LAD is a method that creates a description for each of the two parts. It is a machine learning method with supervisor as the objects in the database are classified a priori. The descriptions of the two classes generated by LAD can be applied to new objects. It is supposed that all attributes of the objects are Boolean variables, that is, the value of each attribute is either true or false. If the values must be expressed numerically, then 1stands for true and 0 stands for false. It is also assumed that there is no contradiction in the database, i.e., there is no pair of objects such that the two objects have the same values in all attributes but belong to the two aforementioned different subsets of objects. The number of different objects is at most 2n, where n is the number of the binary attributes. However, even in the case when n has a moderate value such as 15 to 20, it is unlikely that all possible observations will have occurred. LAD aims to forecast which new (i.e., until now nonoccurring) observation belongs to which subset.The database can be considered as the description of an incomplete Boolean function ofn Boolean variables. The function is incomplete because its value is not known for all possible values of the attributes (variables), just for the observed values. What LAD must do is to find a complete Boolean function such that its value is the same as the value of the incomplete Boolean function. The database of the training set must contain a complete classification, that is, each object must be classified as either 1 or 0. LAD can describe any of the two subsets. The synonyms of class 1 and 0 are positive and negative, respectively. Numerical data can be approximated by Boolean attributes as it is illustrated in the example of the BBBM or better ratings as it is discussed later.The countries are described by economic data that are numeric and not Boolean. These data are transformed in the example to Boolean ones. For example, let us consider that GDP per capita is at least $5,436, and this divides countries into two classes. For some, the GDP per capita is $5,436 and above, and it is less in case of other countries. A Boolean variable is then introduced. If the condition is satisfied, that is, GDP per capita is at least $5,436, then it istrue—otherwise false value is obtained. It is possible to divide the countries into two groups by using the same economic variable in a different way. GDP per capita is used a second time in the example below, where it is at least $14,189. Again, if the condition is satisfied, then it is true—otherwise false value is obtained. The transformation of numerical data to Boolean variables is discussed in Section 2.3.If a country had a BBBM rating or higher in 2012, then it could be checked in three different ways. If any of the three groups gave a positive result, then the country belonged to that category. If none of the three options were satisfied, then its rating was BBP or worse.First group: the GDP per capita of the country is at least $5,436 AND the export of goods and services is at least 38.185 percentage of GDP AND the PPP conversion factor is at most 6.075%.Second group: the net cash surplus/deficit is at least -5.88 percentage of GDP, that is, the deficit is not too high, AND the total reserves are at least $17,824,012,000 AND the inflation rate of consumer prices is at most 9.165%.Third group: the expenses are not greater than $53.195 AND the male unemployment rate is at least 8.5% (where the proportion of male labor force is modelled on ILO estimates) AND the GDP per capita is at least $14,189.To have a rating BBBM or better, a country must satisfyall conditions for at least one of the three groups. It is not enough that it satisfies some conditions for every group. Assume, for example, that the GDP of a country called Nowhere is $8,000. The GDP per capita occurs twice, that is, in the 1st and 3rd groups. Nowhere satisfies the GDP constraint of the 1st group, but violates the similar constraints of the 3rd group, that is, overall. Therefore, if its exports are great enough and its PPP conversion factor is low enough, it still can be a BBBM country because it satisfies the criteria of the first group.


### 2.2. The Original Concept of LAD

The original concept of LAD is based on mathematical logic and Boolean variables. One basic theorem of mathematical logic is that every Boolean function can be obtained as a disjunctive normal form (DNF). The three groups of Section2.1 cover all the countries that have the BBBM or better ratings. This is an example of a DNF. The three ways consist of conjunctions of Boolean variables. Each of these conjunctions of (perhaps several) Boolean variables is called patterns in the context of LAD. The general form of the DNF is that there are several subsets of statements. The Boolean function (DNF) is true if and only if all statements of at least one subset are true. One statement or its opposite can be a part of several subsets. The subsets of statements may have a different number of elements. It is just by chance that each group in the example has three statements.


### 2.3. Transformation of Numerical Values to Boolean Attributes

Most real-life problems have numerical attributes, not Boolean ones. LAD is applicable only if the numerical data are “translated” into Boolean attributes. The example of the BBBM or better rating shows how this can be done. Each statement in the example has a numerical value that separates the countries. These numerical values are always between a BBBM or a better country and a country with a lower rating for instance, the two countries are closest with a GDP of $5,436 per capita; Azerbaijan belonged to the BBBM class in 2012, with a GDP per capita of $7,189. Meanwhile, Guatemala had a lower GDP per capita ($3,166) and was in the BBP class. The statement that the countryNowhere has a GDP greater than $5,436 has a Boolean value, i.e., it is either true or false. The name of any country can be substituted for Nowhere in this statement because every country has a Boolean value in this respect. LAD constructs the DNF from these Boolean attributes. The separating values are called cut-points. Thus, $5,436 is a cut-point in the example above.Mathematically, binarization can be achieved by introducing cut-points for each of the numerical variables in such a way that the resulting partitioning of the space should consist only of “pure intervals,” that is, intervals that do not contain both positive and negative points (see the different color points in Figure2). Minimizing sets of cut-points with corresponding variables was the optimization element of the present study. We explored and described the rating categories of the Fitch rating system by the minimum number of patterns for each year. Figure 2 visualizes the iterative procedure of generating decision trees by logical analysis of data. The countries of the rating category of the iteration and the countries of the better rating classes are assigned to LAD class 1. Any other country is assigned to LAD class 0 like in Figure 2(a). The LAD classes of the countries of the next rating category are changed from 0 to 1 in the next iteration. To go through all predefined rating categories, the countries are moved to LAD class 1 gradually except for rating category BM (Figures 2(b) and 2(c)). At the end, there may be regions such that countries in the region are not classified by LAD (Figure 2(d)). The iterative procedure starts with rating category AAA by Fitch rating agency. These countries are always in LAD class 1. In the first step of the algorithm, these AAA countries are separated from the other countries. Separation means the determination of groups like in the example of Section 2.1, which exactly cover the AAA countries. If a country is classified as AAA country in the first step, then this classification is never modified. In the second step, AAA and AAP countries are separated from the countries, which have rating AA or worse. The logic of the description of AAP countries is as follows: if a country is separated, and it is not an AAA country, then it is classified as an AAP country. AAA, AAP, and AA countries are separated from other countries in the third step, etc. The steps of the algorithm are represented in Figures 3 and 4.Figure 2
Selected cases as an output of LAD for each rating class.
(a)(b)(c)(d)Figure 3
The algorithm of rating.Figure 4
Order of the questions.R is the position of theory of rating class r in the decision tree.


## 2.1. Logical Analysis of Data (LAD)

LAD was initiated by Peter L. Hammer. He demonstrated that the LAD classification system produces accurate, transparent, and generalizable results [38]. There is a set of objects that are similar to each other and are described by the same set of attributes. The objects can be very different according to the area of application, for example, patients in a hospital, customers who obtain a loan from a bank, or drilling locations in the oil industry. However, the nature of the objects is the same in an application. The objects are divided into two parts, for instance, patients who have a particular disease and those who do not. LAD is a method that creates a description for each of the two parts. It is a machine learning method with supervisor as the objects in the database are classified a priori. The descriptions of the two classes generated by LAD can be applied to new objects. It is supposed that all attributes of the objects are Boolean variables, that is, the value of each attribute is either true or false. If the values must be expressed numerically, then 1stands for true and 0 stands for false. It is also assumed that there is no contradiction in the database, i.e., there is no pair of objects such that the two objects have the same values in all attributes but belong to the two aforementioned different subsets of objects. The number of different objects is at most 2n, where n is the number of the binary attributes. However, even in the case when n has a moderate value such as 15 to 20, it is unlikely that all possible observations will have occurred. LAD aims to forecast which new (i.e., until now nonoccurring) observation belongs to which subset.The database can be considered as the description of an incomplete Boolean function ofn Boolean variables. The function is incomplete because its value is not known for all possible values of the attributes (variables), just for the observed values. What LAD must do is to find a complete Boolean function such that its value is the same as the value of the incomplete Boolean function. The database of the training set must contain a complete classification, that is, each object must be classified as either 1 or 0. LAD can describe any of the two subsets. The synonyms of class 1 and 0 are positive and negative, respectively. Numerical data can be approximated by Boolean attributes as it is illustrated in the example of the BBBM or better ratings as it is discussed later.The countries are described by economic data that are numeric and not Boolean. These data are transformed in the example to Boolean ones. For example, let us consider that GDP per capita is at least $5,436, and this divides countries into two classes. For some, the GDP per capita is $5,436 and above, and it is less in case of other countries. A Boolean variable is then introduced. If the condition is satisfied, that is, GDP per capita is at least $5,436, then it istrue—otherwise false value is obtained. It is possible to divide the countries into two groups by using the same economic variable in a different way. GDP per capita is used a second time in the example below, where it is at least $14,189. Again, if the condition is satisfied, then it is true—otherwise false value is obtained. The transformation of numerical data to Boolean variables is discussed in Section 2.3.If a country had a BBBM rating or higher in 2012, then it could be checked in three different ways. If any of the three groups gave a positive result, then the country belonged to that category. If none of the three options were satisfied, then its rating was BBP or worse.First group: the GDP per capita of the country is at least $5,436 AND the export of goods and services is at least 38.185 percentage of GDP AND the PPP conversion factor is at most 6.075%.Second group: the net cash surplus/deficit is at least -5.88 percentage of GDP, that is, the deficit is not too high, AND the total reserves are at least $17,824,012,000 AND the inflation rate of consumer prices is at most 9.165%.Third group: the expenses are not greater than $53.195 AND the male unemployment rate is at least 8.5% (where the proportion of male labor force is modelled on ILO estimates) AND the GDP per capita is at least $14,189.To have a rating BBBM or better, a country must satisfyall conditions for at least one of the three groups. It is not enough that it satisfies some conditions for every group. Assume, for example, that the GDP of a country called Nowhere is $8,000. The GDP per capita occurs twice, that is, in the 1st and 3rd groups. Nowhere satisfies the GDP constraint of the 1st group, but violates the similar constraints of the 3rd group, that is, overall. Therefore, if its exports are great enough and its PPP conversion factor is low enough, it still can be a BBBM country because it satisfies the criteria of the first group.


## 2.2. The Original Concept of LAD

The original concept of LAD is based on mathematical logic and Boolean variables. One basic theorem of mathematical logic is that every Boolean function can be obtained as a disjunctive normal form (DNF). The three groups of Section2.1 cover all the countries that have the BBBM or better ratings. This is an example of a DNF. The three ways consist of conjunctions of Boolean variables. Each of these conjunctions of (perhaps several) Boolean variables is called patterns in the context of LAD. The general form of the DNF is that there are several subsets of statements. The Boolean function (DNF) is true if and only if all statements of at least one subset are true. One statement or its opposite can be a part of several subsets. The subsets of statements may have a different number of elements. It is just by chance that each group in the example has three statements.


## 2.3. Transformation of Numerical Values to Boolean Attributes

Most real-life problems have numerical attributes, not Boolean ones. LAD is applicable only if the numerical data are “translated” into Boolean attributes. The example of the BBBM or better rating shows how this can be done. Each statement in the example has a numerical value that separates the countries. These numerical values are always between a BBBM or a better country and a country with a lower rating for instance, the two countries are closest with a GDP of $5,436 per capita; Azerbaijan belonged to the BBBM class in 2012, with a GDP per capita of $7,189. Meanwhile, Guatemala had a lower GDP per capita ($3,166) and was in the BBP class. The statement that the countryNowhere has a GDP greater than $5,436 has a Boolean value, i.e., it is either true or false. The name of any country can be substituted for Nowhere in this statement because every country has a Boolean value in this respect. LAD constructs the DNF from these Boolean attributes. The separating values are called cut-points. Thus, $5,436 is a cut-point in the example above.Mathematically, binarization can be achieved by introducing cut-points for each of the numerical variables in such a way that the resulting partitioning of the space should consist only of “pure intervals,” that is, intervals that do not contain both positive and negative points (see the different color points in Figure2). Minimizing sets of cut-points with corresponding variables was the optimization element of the present study. We explored and described the rating categories of the Fitch rating system by the minimum number of patterns for each year. Figure 2 visualizes the iterative procedure of generating decision trees by logical analysis of data. The countries of the rating category of the iteration and the countries of the better rating classes are assigned to LAD class 1. Any other country is assigned to LAD class 0 like in Figure 2(a). The LAD classes of the countries of the next rating category are changed from 0 to 1 in the next iteration. To go through all predefined rating categories, the countries are moved to LAD class 1 gradually except for rating category BM (Figures 2(b) and 2(c)). At the end, there may be regions such that countries in the region are not classified by LAD (Figure 2(d)). The iterative procedure starts with rating category AAA by Fitch rating agency. These countries are always in LAD class 1. In the first step of the algorithm, these AAA countries are separated from the other countries. Separation means the determination of groups like in the example of Section 2.1, which exactly cover the AAA countries. If a country is classified as AAA country in the first step, then this classification is never modified. In the second step, AAA and AAP countries are separated from the countries, which have rating AA or worse. The logic of the description of AAP countries is as follows: if a country is separated, and it is not an AAA country, then it is classified as an AAP country. AAA, AAP, and AA countries are separated from other countries in the third step, etc. The steps of the algorithm are represented in Figures 3 and 4.Figure 2
Selected cases as an output of LAD for each rating class.
(a)(b)(c)(d)Figure 3
The algorithm of rating.Figure 4
Order of the questions.R is the position of theory of rating class r in the decision tree.


## 3. Results

### 3.1. How to Apply LAD to Sovereign Credit Rating?

LAD was designed to separate two classes from one another. Sovereign rating has many classes. These classes are ordered according to the risk represented by the countries of the classes. Thus, it is possible to decompose the multiclass rating into a sequence of binary classifications. Countries representing a certain level of risk consist of one class and all other countries with a higher risk are in the other. For example, countries with a rating from AAA to AAM are in the first class and countries having AP or lower are in the second. Every such separation creates a problem for LAD. The DNFs provided by LAD form a decision tree. If a country satisfies the DNF separating AAA countries from other countries, then it is an AAA country. Otherwise, if it satisfies the DNF separating AAA and AAP countries from the others, then it is an AAP country.


### 3.2. Parameters Used in LAD Calculation

To apply LAD, we made a number of assumptions. The extent of a pattern is the number of Boolean variables in a conjunction. The prevalence of a positive (negative) pattern is the ratio of positive (negative) observations covered by the pattern to the number of all positive (negative) observations in the dataset. The homogeneity of a positive (negative) pattern is the percentage of the positive (negative) observations covered by the pattern to the number of all observations covered by it. Every database may require a different value for the LAD parameters. The highest permitted degree for a pattern was 3. It means that a pattern may have three conditions only and not more. The prevalence was at least 70%. The homogeneity was claimed to be 100%. The decision trees for four years from 2012 to 2015 are shown in Tables2–5. The abbreviations of the selected variables that are used in the tables are listed in Table 6.Table 2
Decision tree of the year 2012.
Rating classYear 2012Decision treeAAA(U ≥ 73.715, G ≥ 52456.10), OR (U ≤ 90.02, C ≥ −9.54, G ≥ 40460.80)AAP(UN ≤ 3.60, GS ≥ 26.17.5, G ≥ 34424.90), OR (C ≥ −9.54, U ≤ 90.02, G ≥ 40460.80)AA(G ≥ 34424.90, CG ≤ 107.355)AAM(G ≥ 34424.90, CG ≤ 107.355), OR (GS ≥ 25.435, U ≥ 73.715, G ≥ 22801.90)AP(GS ≥ 23.5438, U ≥ 65.98, G ≥ 22293.20), OR (GS ≥ 12.64, U ≥ 73.03, G ≥ 34424.90), OR (GG ≥ 1.285, PA ≤ 16.985, G ≥ 16463.90), OR (SD ≤ 26.6415, PA ≥ 15.88, PA ≤ 18.755)A(GS ≥ 12.83, U ≥ 84.915, G ≥ 22293.20), OR (CG ≤ 99.96, I ≥ −0.32, G ≥ 16463.90), OR (PG ≥ 0.35, GS ≥ 22.085, PA ≤ 19.7.5)AM(GG ≥ −1.37, CG ≤ 109.17, G ≥ 16463.90), OR (CG ≤ 99.96, E ≥ 29.4973, U ≥ 73.03), OR (CG ≥ 42.545, UN ≤ 0.91, PA ≤ 17.785)BBBPGS ≥ 12.83, GG ≥ −1.37, G ≥ 15008.10), OR (I ≤ 5.60, R ≥ 26351500000, PPP ≤ 6.075), OR (GS ≥ 21.18, G ≥ 25449.20, U ≥ 74.11)BBB(EX ≥ 39.785, CG ≤ 54.94, G ≥ 8405), OR (GG ≤ 7.92, GS ≥ 12.83, G ≥ 15008.10), OR (GS ≥ 14.58, E ≥ 16.33, R ≥ 49114000000), OR (M ≥ 91.745, PPP6≤116.135, R ≥ 48396000000)BBBM(G ≥ 5435.98, EX ≥ 38.185, PPP ≤ 6.075), OR (C ≥ −5.88, R ≥ 17824000000, I ≤ 9.165), OR (E ≤ 53.195, UN ≤ 8.5, G ≥ 14189.30)BBP(R ≥ 47951500000, G ≥ 1357.30), OR (EX ≥ 30.42, GG ≤ 4.36, SD ≥ 8.625), OR (IV ≥ 19.845, R ≥ 3462500000, I ≤ 6.45), OR (PA ≤ 30.175, I ≤ 6.68, GS ≥ 14.04), OR (IV ≥ 21, M ≥ 106.835, G ≥ 13587.10)BBC ≥ −5.75, R ≥ 3462500000, I ≤ 9.165), OR (GS ≥ 14.715, M ≥ 96, G ≥ 13587.10), OR (GS ≥ 16.51, IV ≤ 31.075, M ≥ 96), OR (I ≤ 6.68, IV ≥ 19.555, PA ≤ 30.175)BBM(SD ≤ 23.1219, C ≥ 37.075), OR (GS ≥ 16.925, GG ≤ 4.125, GS ≤ 30.36), OR (IV ≥ 21.655, GG ≤ 5.035, IM ≥ 26.645), OR (I ≤ 9.165, IV ≥ 21.095, R ≥ 3462500000), OR (IV ≥ 21.655, PG ≤ 1.03, PA ≥ 14.14), OR (PA ≤ 30.175, I ≤ 7.905, GS ≥ 13.505)BP(EX ≥ 31.19, PG ≥ 0.175, RE ≥ 26.4706), OR (GG ≤ 5.675, GS ≥ 9.27, SD ≥ 10.075), OR (EX ≥ 23.73, GS ≥ 15.78, UN ≥ 0.325), OR (IV ≥ 21.655, PA ≤ 30.175, M ≤ 121.28))B(E ≤ 53.195, SD ≤ 23.1219), OR (GS ≥ 9.27, PPP ≥ 0.705), OR (GS ≥ 9.27, UN ≥ 0.325)BM(E ≤ 53.195)Table 3
Decision tree of the year 2013.
RatingYear 2013ClassDecision treeAAA(G ≥ 47058.30, PA ≤ 19.40), OR (G ≥ 42861.50, EX ≤ 68.985, GS ≥ 24.62)AAP(G ≥ 35760.90, CG ≤ 102.88, GG ≤ 1.515), OR (U ≥ 78.615, M ≥ 122.63, R ≥ 1443740000000)AA(CG ≤ 118.525, PPP ≤ 10.545, G ≥ 35760.90), OR (G ≥ 45717.70, U ≤ 90.575, CG ≤ 118.52)AAM(PA ≤ 20.81, CG ≤ 118.52, G ≥ 23758.60), OR (G ≥ 23029.80, CG ≤ 63.955, PA≥14.835)AP(G ≥ 15203.90, GG ≥ −0.96, PA ≤ 19.40), OR (IV ≥ 32.155, I ≥ 2.805, G ≥ 15172.90), OR (G ≥ 15172.90, PPP ≥ 0.81, R ≥ 3240000000), OR (UN ≥ 3.05, PA ≤ 21.59, IV ≥ 35.055)A(U ≥ 78.615, PPP ≥ 0.505, G ≥ 15203.90), OR (PA ≤ 19.40, GG ≥ −0.96, G ≥ 15203.90)AM(U ≥ 72.18, G ≥ 15203.90, PPP ≥ 0/685), OR (PA ≤ 19.40, GG ≥ 0.305, CG ≥ 51.565), OR (CG ≤ 58.835, IV ≥ 31.855, EX ≥ 54.065), OR (UN ≤ 13.55, I ≤ 4.02, EX ≥ 74.515), OR (SD ≤ 57.89, IV ≥ 31.85.5, PA ≤ 19.68)BBBP(I ≤ 4.24235, R ≥ 43160000000, G ≥ 5808.70), OR (CG ≤ 50.22, I ≤ 3.13, G ≥ 9139.48), OR (I ≤ 2.685, C ≥ −3.66, E ≥ 28.825)BBB(G ≥ 10738.50, I ≤ 5.125, UN ≤ 15.95), OR (IV ≤ 23.88, TD ≥ 11.425, CG ≤ 116.03), OR (IM ≤ 31.14, GG ≤ 1.46, EX ≤ 41.555)BBBM(PA ≤ 34.91, CG ≤ 67.12.71, SD ≤ 68.87), OR (PPP ≤ 6.405, EX ≥ 38.815, G ≥ 5669.61), OR (GS ≥ 21.57, GS ≤ 23.945, U ≤ 94.895)BBP(TD ≥ 6.66, GS ≥ 18.72, IV ≤ 32.025), OR (G ≥ 5669.61, IV ≤ 39.53, IV ≥ 20.625), OR (GS ≥ 17.485, PA ≤ 29.015, I ≤ 6.735), OR (GG ≤ 1.46, R ≥ 4830000000, C ≥ −3.37)BBNo caseBBM(IV ≥ 22.675, EX ≥ 26.235, GG ≤ 3.91), OR (IV ≥ 25.265, I ≤ 9.835, TD ≥ 10.365), OR (SD ≥ 7.53, I ≤ 5.09, PG ≥ −0.53), OR (PPP ≤ 266.50, PG ≤ 1.425, IV ≥ 25.265)BP(UN ≥ 4.35, GS ≥ 13.82, C ≤ 3.65), OR (R ≥ 2505000000, M ≥ 60.91, C ≥ −8.36)B(G ≥ 5808.70, GS ≥ 12.985), OR (IV ≤ 39.53, C ≥ −8.905)BMNo caseTable 4
Decision tree of the year 2014.
RatingYear 2014ClassDecision treeAAA(CG ≤ 97.51, G ≥ 44264.40)AAP(CG ≤ 97.14, G ≥ 44264.40), OR (IM ≥ 31.69, R ≥ 76865000000, G ≥ 37322.70)AA(GG ≤ 3.325, CG ≤ 102.37, G ≥ 37322.70), OR (G ≥ 23895.10, UN≤7.20, CG ≤ 76.265)AAM(PA ≤ 20.59, CG ≤ 117.265, G ≥ 24571)AP(C ≤ 0.365, G ≥ 15730.30, E ≤ 30.185), OR (C ≥ −4.20, CG ≤ 102.37, G ≥ 24512.50), OR (E ≤ 42.62, PG ≥ 0.465, PA ≤ 19.10)A(GG ≥ −0.725, G ≥ 15730.30, PA ≤ 18.245), OR (GS ≥ 22.90, U ≥ 66.17, GG ≤ 1.985), OR (M ≤ 138.415, I ≤ 2.72, R ≥ 65565000000)AM(GG ≥ −0.225, GG ≤ 1.545, I ≤ 3.66), OR (M ≤ 152.09, R ≥ 6285000000, G ≥ 13680.90), OR (GG ≥ −0.99, PA ≤ 19.29, G ≥ 15618.90), OR (GG ≤ 3.295, GS ≥ 22.90, U ≥ 72.715), OR (M ≤ 138.415, R ≥ 54060000000, I ≤ 2.805)BBBP(I ≤ 2.91, I ≥ 0.315, G ≥ 13680.90), OR (R ≥ 60320000000, UN ≤ 7.82, PPP ≤ 14.815), OR (M ≥ 83.455, U ≥ 77.645, GG ≤ 2.285)BBBR ≥ 565000000, G ≥ 13680.90, GS ≥ 17.065), OR (PG ≥ −0.365, R≥93800000000, I ≤ 5.525), OR (UN ≤ 9.90, U ≥ 66.505, I ≤ 2.91)BBBM(R≥565000000, G ≥ 13680.90, UN ≤ 15.10), OR (PG ≥ 0.15, R ≥ 32595000000, SD ≥ 9.635), OR (U ≥ 54.17, M ≥ 121.215, PA ≤ 27.475), OR (UN ≤ 7.82, PA ≤ 22.905, M ≥ 103.16)BBP(GG ≤ 3.265, TD ≥ 13.52, GS ≥ 20.40), OR (TD ≥ 7.595, GG ≥ 0.195, R ≥ 16430000000), OR (I ≤ 4.345, GG ≥ −0.03, R ≥ 6935000000), OR (G ≥ 22493.40, I ≤ 2.76, M ≥ 83.455)BB(TD ≤ 32.585, IM ≤ 81.065, R ≥ 12170000000, SD ≥ 18.68), OR (I ≥ 0.93, M ≤ 146.655, M ≥ 105.905, I5≤5.66)BBM(PG≤1.26, GS ≥ 13.755, GS ≤ 3.63), OR (SD ≥ 32.355, IV ≥ 23.23, GG ≤ 4.30), OR (IM ≤ 53.395, R ≥ 6285000000, UN ≤ 7.67), OR (IM ≤ 48.27, U ≥ 54.29, EX ≥ 30.805)BP(SD ≥ 32.355, IV ≥ 16.485, SD ≤ 59.535), OR (G ≥ 1873.54, SD ≥ 32.355, IV ≥ 21.35), OR (IM ≥ 31.85, IV ≥ 28.755, U ≥ 41.26), OR (GG ≤ 2.93, I ≤ 5.455, UN ≤ 16.45), OR (PG ≤ 1.485, GS ≥ 15.50, PG ≥ −0.325)B(CG ≤ 141.95, E ≥ 31.395), OR (IV ≥ 11.60, SD ≥ 0.16, TD ≤ 24.205)BM(R ≤ 455530000000, G ≤ 11380.10), OR (SD ≥ 0.16, TD ≤ 35.125)Table 5
Decision tree of the year 2015.
RatingYear 2015ClassDecision treeAAA(PA ≤ 19.675, E ≤ 42.43, G ≥ 47533.40)AAP(C ≤ −1.3747, G ≥ 45329.50, M ≥ 116.68), OR (UN ≤ 8.35, PPP ≤ 10.825, G ≥ 45329.50)AA(GS ≥ 24.55, PG ≥ 0.42, G ≥ 24204.20), OR (GG ≤ 4.24, PPP ≤ 10.825, G ≥ 38710.90)AAM(U ≤ 87.21, UN ≤ 6.55, G ≥ 24204.20), OR (GG ≤ 4.24, PPP ≤ 10.825, G ≥ 38710.90)AP(U ≤ 87.585, G ≥ 17495.60, C ≤ −1.479), OR (U ≥ 53.66, GS ≥ 25.535, R ≥ 52161000000), OR (PPP ≤ 10.825, U ≥ 65.38, G ≥ 38710.90), OR (GG ≤ 1.655, GG ≥ 0.74, U ≥ 79.13)A(PPP ≤ 117.435, U ≥ 65.38, G ≥ 35666.20), OR (C ≤ 0.305, UN ≤ 6.20, PA ≤ 20.835), OR (M ≥ 117.495, GS ≥ 23.705, G ≥ 14451.70), OR (M ≤ 131.47, M ≥ 114.29, G ≥ 15065)AM(GS ≥ 20.755, G ≥ 14179.60, C ≤ −1.63), OR (UN ≤ 10.90, R ≥ 22763500000, G ≥ 14179.60), OR (I ≤ 2.055, M5≥111.635, U ≥ 73.88)BBBP(EX ≥ 23.73, C ≤ −1.245, R ≥ 82865000000), OR (GS≥16.94, R ≥ 8469500000, G ≥ 14179.60), OR (UN≤5.15, R ≥ 16708000000, G ≥ 10561.40), OR (M ≤ 147.075, RE ≤ 29.38, SD ≥ 42.33)BBB(GS ≥ 19.735, PPP ≤ 14.99, G ≥ 10327.40), OR (R ≥ 11365000000, UN ≤ 5.25, G ≥ 10383.10), OR (I ≤ 6.34, PG ≤ 1.345, R ≥ 44997500000), OR (M ≤ 134.165, U ≥ 73.325, M ≥ 111.15))BBBM(PPP ≤ 1.77, I ≤ 299, GS ≥ 14.77), OR (GS ≥ 23.47, G ≥ 14179.60, PA ≤ 28.295), OR (IM ≤ 33.47, M ≥ 103.755, PA ≤ 29.275), OR (PA ≤ 37.02, PPP ≤ 117.435, R ≥ 19915000000), OR (U ≥ 72.05, PG ≤ 1.06, GS ≥ 14.77)BBP(GG ≥ 0.09, TD ≥ 10.915, R ≥ 8469500000), OR (TD ≥ 5.145, R ≥ 15619500000, SD ≥ 11.155), OR (G ≥ 9845.96, PG ≤ 1.63, M ≥ 111.15), OR (PA ≤ 22.905, GS ≥ 20.74, M ≥ 90.31)BB(GS≥20.575, M ≥ 104.745, G ≥ 11039.60), OR (M ≥ 111.15, PG ≤ 1.52, G ≥ 10593.80), OR (R ≥ 66.115, IM ≤ 47.05, U ≥ 48.055), OR (PPP ≤ 5808.54, PG ≤ 2.295, R ≥ 15028000000), OR (PA ≤ 22.905, GS ≥ 20.575, M ≥ 104.745)BBM(I ≤ 6.80, SD ≥ 20.235, GG ≥ 1.335), OR (IV ≥ 28.14, SD ≥ 27.36, M ≥ 105.065), OR (PG ≤ 1.35, IV ≥ 28.14, PA ≥ 16.725), OR (PPP ≤ 134.13, UN ≤ 7.61, PPP ≥ 1.095), OR (PA ≤ 28.07.5, PPP ≤ 134.135, GS ≥ 14.77)BP(IM ≤ 47.345, G ≥ 1889.15, PPP ≥ 2.585), OR (SD ≥ 18.905, I ≤ 7.545, U ≤ 65.38), OR (PPP ≤ 45.59, UN ≤ 19.60, IV ≤ 27.45), OR (IV ≥ 17.98, C ≤ 1.005, PG ≤ 1.35), OR (TD ≥ 4.865, I ≤ 9.515, M ≥ 111.15)B(GG ≥ −4.645, PG ≥ 1.12), OR (I ≤ 8.175, R ≤ 4104500000), OR (C ≤ 1.005, I ≤ 38.83, RE ≤ 44.995)BM(RE ≤ 44.995, I ≤ 38.83, C ≤ 1.005), OR (PG ≥ −0.62, IV ≤ 28.125), IV ≥ 29.46)Table 6
Applied variables for exploring the hidden pattern of the Fitch rating system.
NotationAttributeCCash surplus/deficit (% age of GDP)EXExports of goods and services (% of GDP)GGDP per capita (current US$)IMImports of goods and services (% age of GDP)RERevenue, excluding grants (% age of GDP)SDShort-term debt (% age of total reserves)TDTotal debt service (% age of exports of goods, services, and primary income)CGCentral government debt, total (% age of GDP)EExpense (% age of GDP)GGGDP per capita growth (annual % age)GSGross savings (% age of GDP)IVIndustry, value added (% age of GDP)IInflation, consumer prices (annual % age)PPPPPP conversion factor, GDP (LCU per international $)RTotal reserves (includes gold, current US$)UUrban population (% age of total)PGPopulation growth (annual % age)PAPopulation ages 0–14 (% age of total)UNUnemployment, male (% age of male labor force) (modelled ILO estimate)MMobile cellular subscriptions (per 100 people)The dataset of 116 countries contains populations of more than half a million in the form of two sets, namely, training and test: 68 ± 2 and 48 ± 2 countries, respectively. These are gathered from World Bank data covering the period 2012–2015. Countries with very small populations were excluded, as they have a special risk. This fact has been proven by Iceland, which experienced an extreme financial crisis in 2008. The Fitch ratings are listed in Table1. There are further scales for risky countries. However, the occurrence of these scales is so rare that they are omitted from the calculation.The following decision trees indicate the prediction of the rating for each country in the set.


### 3.3. Decision Trees

The decision trees obtained from the multiple applications of LAD are summarized in Table2–5. These tables contain the basic logical rules in Boolean form.The use of decision tree is simple. Only the values of the attributes of the country in question must be available. The user must only compare these values with the values of the cut-points given in the table.


### 3.4. Key Variables of the Decision Trees

The four decision trees share certain properties. The most important attributes are listed in Table7.Table 7
The most common properties of the classified ranking classes.
Rating classCommon variables in all treesAAAGAAP-AA-AAMGAP-A-AMG-PA-IBBBP-BBB-BBBMR-GS-G-UN-PPP-I–UBBP-BB-BBMR-G-GG-SD-IV-I-GS-PGBP-B-BMUN-IVThe GDP per capita was an essential factor in all rating classes except the lowest. For ratings from AAA to AAM, there was no pattern without it. GDP occurs from AP to AM in more than 50% of the patterns. In addition to GDP, total reserves, gross savings, and inflation are the dominant variables in the classification of countries with BBB and BB rating levels. In the lowest rating class, that is, B,male unemployment rate and industry value added are the major variables.


## 3.1. How to Apply LAD to Sovereign Credit Rating?

LAD was designed to separate two classes from one another. Sovereign rating has many classes. These classes are ordered according to the risk represented by the countries of the classes. Thus, it is possible to decompose the multiclass rating into a sequence of binary classifications. Countries representing a certain level of risk consist of one class and all other countries with a higher risk are in the other. For example, countries with a rating from AAA to AAM are in the first class and countries having AP or lower are in the second. Every such separation creates a problem for LAD. The DNFs provided by LAD form a decision tree. If a country satisfies the DNF separating AAA countries from other countries, then it is an AAA country. Otherwise, if it satisfies the DNF separating AAA and AAP countries from the others, then it is an AAP country.


## 3.2. Parameters Used in LAD Calculation

To apply LAD, we made a number of assumptions. The extent of a pattern is the number of Boolean variables in a conjunction. The prevalence of a positive (negative) pattern is the ratio of positive (negative) observations covered by the pattern to the number of all positive (negative) observations in the dataset. The homogeneity of a positive (negative) pattern is the percentage of the positive (negative) observations covered by the pattern to the number of all observations covered by it. Every database may require a different value for the LAD parameters. The highest permitted degree for a pattern was 3. It means that a pattern may have three conditions only and not more. The prevalence was at least 70%. The homogeneity was claimed to be 100%. The decision trees for four years from 2012 to 2015 are shown in Tables2–5. The abbreviations of the selected variables that are used in the tables are listed in Table 6.Table 2
Decision tree of the year 2012.
Rating classYear 2012Decision treeAAA(U ≥ 73.715, G ≥ 52456.10), OR (U ≤ 90.02, C ≥ −9.54, G ≥ 40460.80)AAP(UN ≤ 3.60, GS ≥ 26.17.5, G ≥ 34424.90), OR (C ≥ −9.54, U ≤ 90.02, G ≥ 40460.80)AA(G ≥ 34424.90, CG ≤ 107.355)AAM(G ≥ 34424.90, CG ≤ 107.355), OR (GS ≥ 25.435, U ≥ 73.715, G ≥ 22801.90)AP(GS ≥ 23.5438, U ≥ 65.98, G ≥ 22293.20), OR (GS ≥ 12.64, U ≥ 73.03, G ≥ 34424.90), OR (GG ≥ 1.285, PA ≤ 16.985, G ≥ 16463.90), OR (SD ≤ 26.6415, PA ≥ 15.88, PA ≤ 18.755)A(GS ≥ 12.83, U ≥ 84.915, G ≥ 22293.20), OR (CG ≤ 99.96, I ≥ −0.32, G ≥ 16463.90), OR (PG ≥ 0.35, GS ≥ 22.085, PA ≤ 19.7.5)AM(GG ≥ −1.37, CG ≤ 109.17, G ≥ 16463.90), OR (CG ≤ 99.96, E ≥ 29.4973, U ≥ 73.03), OR (CG ≥ 42.545, UN ≤ 0.91, PA ≤ 17.785)BBBPGS ≥ 12.83, GG ≥ −1.37, G ≥ 15008.10), OR (I ≤ 5.60, R ≥ 26351500000, PPP ≤ 6.075), OR (GS ≥ 21.18, G ≥ 25449.20, U ≥ 74.11)BBB(EX ≥ 39.785, CG ≤ 54.94, G ≥ 8405), OR (GG ≤ 7.92, GS ≥ 12.83, G ≥ 15008.10), OR (GS ≥ 14.58, E ≥ 16.33, R ≥ 49114000000), OR (M ≥ 91.745, PPP6≤116.135, R ≥ 48396000000)BBBM(G ≥ 5435.98, EX ≥ 38.185, PPP ≤ 6.075), OR (C ≥ −5.88, R ≥ 17824000000, I ≤ 9.165), OR (E ≤ 53.195, UN ≤ 8.5, G ≥ 14189.30)BBP(R ≥ 47951500000, G ≥ 1357.30), OR (EX ≥ 30.42, GG ≤ 4.36, SD ≥ 8.625), OR (IV ≥ 19.845, R ≥ 3462500000, I ≤ 6.45), OR (PA ≤ 30.175, I ≤ 6.68, GS ≥ 14.04), OR (IV ≥ 21, M ≥ 106.835, G ≥ 13587.10)BBC ≥ −5.75, R ≥ 3462500000, I ≤ 9.165), OR (GS ≥ 14.715, M ≥ 96, G ≥ 13587.10), OR (GS ≥ 16.51, IV ≤ 31.075, M ≥ 96), OR (I ≤ 6.68, IV ≥ 19.555, PA ≤ 30.175)BBM(SD ≤ 23.1219, C ≥ 37.075), OR (GS ≥ 16.925, GG ≤ 4.125, GS ≤ 30.36), OR (IV ≥ 21.655, GG ≤ 5.035, IM ≥ 26.645), OR (I ≤ 9.165, IV ≥ 21.095, R ≥ 3462500000), OR (IV ≥ 21.655, PG ≤ 1.03, PA ≥ 14.14), OR (PA ≤ 30.175, I ≤ 7.905, GS ≥ 13.505)BP(EX ≥ 31.19, PG ≥ 0.175, RE ≥ 26.4706), OR (GG ≤ 5.675, GS ≥ 9.27, SD ≥ 10.075), OR (EX ≥ 23.73, GS ≥ 15.78, UN ≥ 0.325), OR (IV ≥ 21.655, PA ≤ 30.175, M ≤ 121.28))B(E ≤ 53.195, SD ≤ 23.1219), OR (GS ≥ 9.27, PPP ≥ 0.705), OR (GS ≥ 9.27, UN ≥ 0.325)BM(E ≤ 53.195)Table 3
Decision tree of the year 2013.
RatingYear 2013ClassDecision treeAAA(G ≥ 47058.30, PA ≤ 19.40), OR (G ≥ 42861.50, EX ≤ 68.985, GS ≥ 24.62)AAP(G ≥ 35760.90, CG ≤ 102.88, GG ≤ 1.515), OR (U ≥ 78.615, M ≥ 122.63, R ≥ 1443740000000)AA(CG ≤ 118.525, PPP ≤ 10.545, G ≥ 35760.90), OR (G ≥ 45717.70, U ≤ 90.575, CG ≤ 118.52)AAM(PA ≤ 20.81, CG ≤ 118.52, G ≥ 23758.60), OR (G ≥ 23029.80, CG ≤ 63.955, PA≥14.835)AP(G ≥ 15203.90, GG ≥ −0.96, PA ≤ 19.40), OR (IV ≥ 32.155, I ≥ 2.805, G ≥ 15172.90), OR (G ≥ 15172.90, PPP ≥ 0.81, R ≥ 3240000000), OR (UN ≥ 3.05, PA ≤ 21.59, IV ≥ 35.055)A(U ≥ 78.615, PPP ≥ 0.505, G ≥ 15203.90), OR (PA ≤ 19.40, GG ≥ −0.96, G ≥ 15203.90)AM(U ≥ 72.18, G ≥ 15203.90, PPP ≥ 0/685), OR (PA ≤ 19.40, GG ≥ 0.305, CG ≥ 51.565), OR (CG ≤ 58.835, IV ≥ 31.855, EX ≥ 54.065), OR (UN ≤ 13.55, I ≤ 4.02, EX ≥ 74.515), OR (SD ≤ 57.89, IV ≥ 31.85.5, PA ≤ 19.68)BBBP(I ≤ 4.24235, R ≥ 43160000000, G ≥ 5808.70), OR (CG ≤ 50.22, I ≤ 3.13, G ≥ 9139.48), OR (I ≤ 2.685, C ≥ −3.66, E ≥ 28.825)BBB(G ≥ 10738.50, I ≤ 5.125, UN ≤ 15.95), OR (IV ≤ 23.88, TD ≥ 11.425, CG ≤ 116.03), OR (IM ≤ 31.14, GG ≤ 1.46, EX ≤ 41.555)BBBM(PA ≤ 34.91, CG ≤ 67.12.71, SD ≤ 68.87), OR (PPP ≤ 6.405, EX ≥ 38.815, G ≥ 5669.61), OR (GS ≥ 21.57, GS ≤ 23.945, U ≤ 94.895)BBP(TD ≥ 6.66, GS ≥ 18.72, IV ≤ 32.025), OR (G ≥ 5669.61, IV ≤ 39.53, IV ≥ 20.625), OR (GS ≥ 17.485, PA ≤ 29.015, I ≤ 6.735), OR (GG ≤ 1.46, R ≥ 4830000000, C ≥ −3.37)BBNo caseBBM(IV ≥ 22.675, EX ≥ 26.235, GG ≤ 3.91), OR (IV ≥ 25.265, I ≤ 9.835, TD ≥ 10.365), OR (SD ≥ 7.53, I ≤ 5.09, PG ≥ −0.53), OR (PPP ≤ 266.50, PG ≤ 1.425, IV ≥ 25.265)BP(UN ≥ 4.35, GS ≥ 13.82, C ≤ 3.65), OR (R ≥ 2505000000, M ≥ 60.91, C ≥ −8.36)B(G ≥ 5808.70, GS ≥ 12.985), OR (IV ≤ 39.53, C ≥ −8.905)BMNo caseTable 4
Decision tree of the year 2014.
RatingYear 2014ClassDecision treeAAA(CG ≤ 97.51, G ≥ 44264.40)AAP(CG ≤ 97.14, G ≥ 44264.40), OR (IM ≥ 31.69, R ≥ 76865000000, G ≥ 37322.70)AA(GG ≤ 3.325, CG ≤ 102.37, G ≥ 37322.70), OR (G ≥ 23895.10, UN≤7.20, CG ≤ 76.265)AAM(PA ≤ 20.59, CG ≤ 117.265, G ≥ 24571)AP(C ≤ 0.365, G ≥ 15730.30, E ≤ 30.185), OR (C ≥ −4.20, CG ≤ 102.37, G ≥ 24512.50), OR (E ≤ 42.62, PG ≥ 0.465, PA ≤ 19.10)A(GG ≥ −0.725, G ≥ 15730.30, PA ≤ 18.245), OR (GS ≥ 22.90, U ≥ 66.17, GG ≤ 1.985), OR (M ≤ 138.415, I ≤ 2.72, R ≥ 65565000000)AM(GG ≥ −0.225, GG ≤ 1.545, I ≤ 3.66), OR (M ≤ 152.09, R ≥ 6285000000, G ≥ 13680.90), OR (GG ≥ −0.99, PA ≤ 19.29, G ≥ 15618.90), OR (GG ≤ 3.295, GS ≥ 22.90, U ≥ 72.715), OR (M ≤ 138.415, R ≥ 54060000000, I ≤ 2.805)BBBP(I ≤ 2.91, I ≥ 0.315, G ≥ 13680.90), OR (R ≥ 60320000000, UN ≤ 7.82, PPP ≤ 14.815), OR (M ≥ 83.455, U ≥ 77.645, GG ≤ 2.285)BBBR ≥ 565000000, G ≥ 13680.90, GS ≥ 17.065), OR (PG ≥ −0.365, R≥93800000000, I ≤ 5.525), OR (UN ≤ 9.90, U ≥ 66.505, I ≤ 2.91)BBBM(R≥565000000, G ≥ 13680.90, UN ≤ 15.10), OR (PG ≥ 0.15, R ≥ 32595000000, SD ≥ 9.635), OR (U ≥ 54.17, M ≥ 121.215, PA ≤ 27.475), OR (UN ≤ 7.82, PA ≤ 22.905, M ≥ 103.16)BBP(GG ≤ 3.265, TD ≥ 13.52, GS ≥ 20.40), OR (TD ≥ 7.595, GG ≥ 0.195, R ≥ 16430000000), OR (I ≤ 4.345, GG ≥ −0.03, R ≥ 6935000000), OR (G ≥ 22493.40, I ≤ 2.76, M ≥ 83.455)BB(TD ≤ 32.585, IM ≤ 81.065, R ≥ 12170000000, SD ≥ 18.68), OR (I ≥ 0.93, M ≤ 146.655, M ≥ 105.905, I5≤5.66)BBM(PG≤1.26, GS ≥ 13.755, GS ≤ 3.63), OR (SD ≥ 32.355, IV ≥ 23.23, GG ≤ 4.30), OR (IM ≤ 53.395, R ≥ 6285000000, UN ≤ 7.67), OR (IM ≤ 48.27, U ≥ 54.29, EX ≥ 30.805)BP(SD ≥ 32.355, IV ≥ 16.485, SD ≤ 59.535), OR (G ≥ 1873.54, SD ≥ 32.355, IV ≥ 21.35), OR (IM ≥ 31.85, IV ≥ 28.755, U ≥ 41.26), OR (GG ≤ 2.93, I ≤ 5.455, UN ≤ 16.45), OR (PG ≤ 1.485, GS ≥ 15.50, PG ≥ −0.325)B(CG ≤ 141.95, E ≥ 31.395), OR (IV ≥ 11.60, SD ≥ 0.16, TD ≤ 24.205)BM(R ≤ 455530000000, G ≤ 11380.10), OR (SD ≥ 0.16, TD ≤ 35.125)Table 5
Decision tree of the year 2015.
RatingYear 2015ClassDecision treeAAA(PA ≤ 19.675, E ≤ 42.43, G ≥ 47533.40)AAP(C ≤ −1.3747, G ≥ 45329.50, M ≥ 116.68), OR (UN ≤ 8.35, PPP ≤ 10.825, G ≥ 45329.50)AA(GS ≥ 24.55, PG ≥ 0.42, G ≥ 24204.20), OR (GG ≤ 4.24, PPP ≤ 10.825, G ≥ 38710.90)AAM(U ≤ 87.21, UN ≤ 6.55, G ≥ 24204.20), OR (GG ≤ 4.24, PPP ≤ 10.825, G ≥ 38710.90)AP(U ≤ 87.585, G ≥ 17495.60, C ≤ −1.479), OR (U ≥ 53.66, GS ≥ 25.535, R ≥ 52161000000), OR (PPP ≤ 10.825, U ≥ 65.38, G ≥ 38710.90), OR (GG ≤ 1.655, GG ≥ 0.74, U ≥ 79.13)A(PPP ≤ 117.435, U ≥ 65.38, G ≥ 35666.20), OR (C ≤ 0.305, UN ≤ 6.20, PA ≤ 20.835), OR (M ≥ 117.495, GS ≥ 23.705, G ≥ 14451.70), OR (M ≤ 131.47, M ≥ 114.29, G ≥ 15065)AM(GS ≥ 20.755, G ≥ 14179.60, C ≤ −1.63), OR (UN ≤ 10.90, R ≥ 22763500000, G ≥ 14179.60), OR (I ≤ 2.055, M5≥111.635, U ≥ 73.88)BBBP(EX ≥ 23.73, C ≤ −1.245, R ≥ 82865000000), OR (GS≥16.94, R ≥ 8469500000, G ≥ 14179.60), OR (UN≤5.15, R ≥ 16708000000, G ≥ 10561.40), OR (M ≤ 147.075, RE ≤ 29.38, SD ≥ 42.33)BBB(GS ≥ 19.735, PPP ≤ 14.99, G ≥ 10327.40), OR (R ≥ 11365000000, UN ≤ 5.25, G ≥ 10383.10), OR (I ≤ 6.34, PG ≤ 1.345, R ≥ 44997500000), OR (M ≤ 134.165, U ≥ 73.325, M ≥ 111.15))BBBM(PPP ≤ 1.77, I ≤ 299, GS ≥ 14.77), OR (GS ≥ 23.47, G ≥ 14179.60, PA ≤ 28.295), OR (IM ≤ 33.47, M ≥ 103.755, PA ≤ 29.275), OR (PA ≤ 37.02, PPP ≤ 117.435, R ≥ 19915000000), OR (U ≥ 72.05, PG ≤ 1.06, GS ≥ 14.77)BBP(GG ≥ 0.09, TD ≥ 10.915, R ≥ 8469500000), OR (TD ≥ 5.145, R ≥ 15619500000, SD ≥ 11.155), OR (G ≥ 9845.96, PG ≤ 1.63, M ≥ 111.15), OR (PA ≤ 22.905, GS ≥ 20.74, M ≥ 90.31)BB(GS≥20.575, M ≥ 104.745, G ≥ 11039.60), OR (M ≥ 111.15, PG ≤ 1.52, G ≥ 10593.80), OR (R ≥ 66.115, IM ≤ 47.05, U ≥ 48.055), OR (PPP ≤ 5808.54, PG ≤ 2.295, R ≥ 15028000000), OR (PA ≤ 22.905, GS ≥ 20.575, M ≥ 104.745)BBM(I ≤ 6.80, SD ≥ 20.235, GG ≥ 1.335), OR (IV ≥ 28.14, SD ≥ 27.36, M ≥ 105.065), OR (PG ≤ 1.35, IV ≥ 28.14, PA ≥ 16.725), OR (PPP ≤ 134.13, UN ≤ 7.61, PPP ≥ 1.095), OR (PA ≤ 28.07.5, PPP ≤ 134.135, GS ≥ 14.77)BP(IM ≤ 47.345, G ≥ 1889.15, PPP ≥ 2.585), OR (SD ≥ 18.905, I ≤ 7.545, U ≤ 65.38), OR (PPP ≤ 45.59, UN ≤ 19.60, IV ≤ 27.45), OR (IV ≥ 17.98, C ≤ 1.005, PG ≤ 1.35), OR (TD ≥ 4.865, I ≤ 9.515, M ≥ 111.15)B(GG ≥ −4.645, PG ≥ 1.12), OR (I ≤ 8.175, R ≤ 4104500000), OR (C ≤ 1.005, I ≤ 38.83, RE ≤ 44.995)BM(RE ≤ 44.995, I ≤ 38.83, C ≤ 1.005), OR (PG ≥ −0.62, IV ≤ 28.125), IV ≥ 29.46)Table 6
Applied variables for exploring the hidden pattern of the Fitch rating system.
NotationAttributeCCash surplus/deficit (% age of GDP)EXExports of goods and services (% of GDP)GGDP per capita (current US$)IMImports of goods and services (% age of GDP)RERevenue, excluding grants (% age of GDP)SDShort-term debt (% age of total reserves)TDTotal debt service (% age of exports of goods, services, and primary income)CGCentral government debt, total (% age of GDP)EExpense (% age of GDP)GGGDP per capita growth (annual % age)GSGross savings (% age of GDP)IVIndustry, value added (% age of GDP)IInflation, consumer prices (annual % age)PPPPPP conversion factor, GDP (LCU per international $)RTotal reserves (includes gold, current US$)UUrban population (% age of total)PGPopulation growth (annual % age)PAPopulation ages 0–14 (% age of total)UNUnemployment, male (% age of male labor force) (modelled ILO estimate)MMobile cellular subscriptions (per 100 people)The dataset of 116 countries contains populations of more than half a million in the form of two sets, namely, training and test: 68 ± 2 and 48 ± 2 countries, respectively. These are gathered from World Bank data covering the period 2012–2015. Countries with very small populations were excluded, as they have a special risk. This fact has been proven by Iceland, which experienced an extreme financial crisis in 2008. The Fitch ratings are listed in Table1. There are further scales for risky countries. However, the occurrence of these scales is so rare that they are omitted from the calculation.The following decision trees indicate the prediction of the rating for each country in the set.


## 3.3. Decision Trees

The decision trees obtained from the multiple applications of LAD are summarized in Table2–5. These tables contain the basic logical rules in Boolean form.The use of decision tree is simple. Only the values of the attributes of the country in question must be available. The user must only compare these values with the values of the cut-points given in the table.


## 3.4. Key Variables of the Decision Trees

The four decision trees share certain properties. The most important attributes are listed in Table7.Table 7
The most common properties of the classified ranking classes.
Rating classCommon variables in all treesAAAGAAP-AA-AAMGAP-A-AMG-PA-IBBBP-BBB-BBBMR-GS-G-UN-PPP-I–UBBP-BB-BBMR-G-GG-SD-IV-I-GS-PGBP-B-BMUN-IVThe GDP per capita was an essential factor in all rating classes except the lowest. For ratings from AAA to AAM, there was no pattern without it. GDP occurs from AP to AM in more than 50% of the patterns. In addition to GDP, total reserves, gross savings, and inflation are the dominant variables in the classification of countries with BBB and BB rating levels. In the lowest rating class, that is, B,male unemployment rate and industry value added are the major variables.


## 4. Discussion

It can be seen from Tables2–5 that LAD gives simple formulas for the classification that can be interpreted easily. The decision trees make it possible to predict an approximate rate to rank countries that are not rated by Fitch; then any user can do this prediction.


### 4.1. Fitch Rating Predictions

The dataset of the study was divided into two subsets, namely, training and test sets. These included 67 and 45 countries, respectively. The decision trees are estimated for the whole years as well as separately for the two subsets. In our result, the explored patterns in the form of the decision trees could predict the same credit ratings of the countries by high percentage as Fitch has reported. We obtained 93%, 89%, 91.5%, and 90.5% identical ratings with Fitch agency’s country ratings for each of the four years. The high accuracy confirms the value of prognosis and robustness of the LAD rating system. On average, we matched the Fitch rating results in the training set by 98% and in the test set by 84%. The mismatched ratings according to country and the corresponding ratings are shown in Tables8−11.Table 8
Mismatched cases of the test set for 2012.
CountryRating by LADFitch ratingUkraineBBBMBSurinameBBBMBBMRwandaBPBGhanaBBBPDominican RepublicBBBTable 9
The mismatched cases and mismatched result of the test set for 2013.
CountryRating by LADFitch ratingMismatched casesMaltaAPAVietnamBBMBPUkraineBBMBKazakhstanBBBBBBPZambiaBPBCongo, Dem RepublicBBMBPParaguayBBPBBMPortugalBBBBBPIcelandAAMBBBSpainAAMBBBCyprusBBBBMismatched resultIsraelAAMATable 10
The mismatched cases and mismatched result of the test set for 2014.
CountryRating by LADFitch ratingMismatched casesSouth AfricaBBBMBBBPortugalABBPPanamaBBBBIcelandAAMBBBCyprusAPBMCosta RicaBBBMBBPMismatched resultNamibiaBBPBBBMTable 11
Mismatched ratings of the test set and training set for 2015.
CountryRating by LADFitch ratingTest setCongo, Dem RepublicBBPRwandaBBPLatviaAAMIcelandBBMBBBArmeniaBBMBPTraining setIrelandBBBAMMalaysiaAPAMPeruBBBBBBPNamibiaBBMBBBMThe year 2012 showed a ratio of matched consequences of 100% for the training set and 85.7% for the test set.The year 2013 showed a ratio of matched consequences of 98.5% for the training set and 78% for the test set. The different output of the two sets is shown in Table9.For the year 2014, the ratio of matched outcomes for training and test sets is 98.5% and 85%. The different output of the two sets is shown in Table10.For the year 2015, the ratio of the matched result of training and test sets is 94% and 87%. The differences in rating of the two sets are shown in Tables10 and 11.First, the high proportions of matched ratings with the Fitch rating system show that our LAD methodology was successful. Second, in the majority of the misclassified cases through the years, Fitch has showed a bias towards downgrading rather than upgrading the countries, as it has already been proven by previous studies on CRAs. The incentive for being conservative is that it may deflect criticism, especially in periods of economic and financial crisis. Using our methodology, downgrading outnumbered upgrading considerably −75.8%, compared with 24.2% in mismatched cases. Fitch had misrated several countries twice or more than twice in the four years (Table12).Table 12
Significant misclassified countries by Fitch between 2012 and 2015.
No. of misclassificationCountryTwiceMore than twiceUkraine√Rwanda√Portugal√Namibia√Iceland√Cyprus√Congo, Dem Republic√As can be seen, seven countries were misrated at least twice. The Democratic Republic of Congo, which is rich in minerals, is the main producer of cobalt in the world. There were several military conflicts in the area, and thus, the situation was uncertain. Cyprus became a divided country after the events of 1974. No solution has been found yet. The country was also affected by the Greek financial crisis. Iceland was the first victim of the 2008–2009 economic crisis. Its population is less than 400,000. Therefore, its currency has also a small total value, which was considered by investors to be a source of instability. The Namibian economy is tied to the strong economy of South Africa. Fitch considers Namibia stronger than is merited by its attributes. Portugal is a European Union country in the Mediterranean region. As with other similar countries, it suffered a great deal after the 2008–2009 crisis and accumulated high debts. Rwanda is a neighbor of the Democratic Republic of Congo. Genocide was committed to the country’s Tutsi minority in 1994, and the region as a whole remains unstable. Ukraine was another of the countries that were hit by the financial crisis in 2008–2009. It is also unstable politically. It has recently gone through a civil war, and it has been in serious conflict with Russia. The misratings by the LAD model can have different reasons including both technical and political ones.


## 4.1. Fitch Rating Predictions

The dataset of the study was divided into two subsets, namely, training and test sets. These included 67 and 45 countries, respectively. The decision trees are estimated for the whole years as well as separately for the two subsets. In our result, the explored patterns in the form of the decision trees could predict the same credit ratings of the countries by high percentage as Fitch has reported. We obtained 93%, 89%, 91.5%, and 90.5% identical ratings with Fitch agency’s country ratings for each of the four years. The high accuracy confirms the value of prognosis and robustness of the LAD rating system. On average, we matched the Fitch rating results in the training set by 98% and in the test set by 84%. The mismatched ratings according to country and the corresponding ratings are shown in Tables8−11.Table 8
Mismatched cases of the test set for 2012.
CountryRating by LADFitch ratingUkraineBBBMBSurinameBBBMBBMRwandaBPBGhanaBBBPDominican RepublicBBBTable 9
The mismatched cases and mismatched result of the test set for 2013.
CountryRating by LADFitch ratingMismatched casesMaltaAPAVietnamBBMBPUkraineBBMBKazakhstanBBBBBBPZambiaBPBCongo, Dem RepublicBBMBPParaguayBBPBBMPortugalBBBBBPIcelandAAMBBBSpainAAMBBBCyprusBBBBMismatched resultIsraelAAMATable 10
The mismatched cases and mismatched result of the test set for 2014.
CountryRating by LADFitch ratingMismatched casesSouth AfricaBBBMBBBPortugalABBPPanamaBBBBIcelandAAMBBBCyprusAPBMCosta RicaBBBMBBPMismatched resultNamibiaBBPBBBMTable 11
Mismatched ratings of the test set and training set for 2015.
CountryRating by LADFitch ratingTest setCongo, Dem RepublicBBPRwandaBBPLatviaAAMIcelandBBMBBBArmeniaBBMBPTraining setIrelandBBBAMMalaysiaAPAMPeruBBBBBBPNamibiaBBMBBBMThe year 2012 showed a ratio of matched consequences of 100% for the training set and 85.7% for the test set.The year 2013 showed a ratio of matched consequences of 98.5% for the training set and 78% for the test set. The different output of the two sets is shown in Table9.For the year 2014, the ratio of matched outcomes for training and test sets is 98.5% and 85%. The different output of the two sets is shown in Table10.For the year 2015, the ratio of the matched result of training and test sets is 94% and 87%. The differences in rating of the two sets are shown in Tables10 and 11.First, the high proportions of matched ratings with the Fitch rating system show that our LAD methodology was successful. Second, in the majority of the misclassified cases through the years, Fitch has showed a bias towards downgrading rather than upgrading the countries, as it has already been proven by previous studies on CRAs. The incentive for being conservative is that it may deflect criticism, especially in periods of economic and financial crisis. Using our methodology, downgrading outnumbered upgrading considerably −75.8%, compared with 24.2% in mismatched cases. Fitch had misrated several countries twice or more than twice in the four years (Table12).Table 12
Significant misclassified countries by Fitch between 2012 and 2015.
No. of misclassificationCountryTwiceMore than twiceUkraine√Rwanda√Portugal√Namibia√Iceland√Cyprus√Congo, Dem Republic√As can be seen, seven countries were misrated at least twice. The Democratic Republic of Congo, which is rich in minerals, is the main producer of cobalt in the world. There were several military conflicts in the area, and thus, the situation was uncertain. Cyprus became a divided country after the events of 1974. No solution has been found yet. The country was also affected by the Greek financial crisis. Iceland was the first victim of the 2008–2009 economic crisis. Its population is less than 400,000. Therefore, its currency has also a small total value, which was considered by investors to be a source of instability. The Namibian economy is tied to the strong economy of South Africa. Fitch considers Namibia stronger than is merited by its attributes. Portugal is a European Union country in the Mediterranean region. As with other similar countries, it suffered a great deal after the 2008–2009 crisis and accumulated high debts. Rwanda is a neighbor of the Democratic Republic of Congo. Genocide was committed to the country’s Tutsi minority in 1994, and the region as a whole remains unstable. Ukraine was another of the countries that were hit by the financial crisis in 2008–2009. It is also unstable politically. It has recently gone through a civil war, and it has been in serious conflict with Russia. The misratings by the LAD model can have different reasons including both technical and political ones.


## 5. Conclusion

Sovereign debt rating plays an important role in the world economy. The method of the most important rating companies is not public. This can be explained by two facts: the algorithms are considered as individual properties of these enterprises, and the rating is based on some tacit knowledge of experts, built of their experiences, personal interviews, gray literature, open-source intelligence, etc. That is why a perfect reproduction of the classifications of these agencies does not seems possible. It would be important for both countries and investors to know the method or a good approximation of the method to all countries.The present study reconstructed the Fitch rating of sovereign debts in years 2012 to 2015 by applying the method of LAD. The analysis is based on 20 important attributes of countries. The data were obtained from the database of World Bank. The outcomes of the calculation are summarized in the form of decision trees. An approximate rating of an unrated country can be obtained by using the decision tree. This application does not need any expertise or professional understanding of rating methods. Previous papers were not providing such tool for investors. The sovereign debt rating given by the decision trees is largely the same as Fitch’s rating. The discrepancies may be due in part to the fact that an unknown function was approximated by regression. Another important reason is that for some countries the rating is biased from what is expected in some direction. A deeper analysis of the discrepancies is one of the research topics of the future.The LAD technology was originally developed to separate only two classes. The paper also showed that LAD may be able to distinguish several classes simultaneously in the case of a systematic application. In the future, it is worth exploring the conditions for such applications and creating additional applications accordingly.



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*Source: 2882930-2021-08-02.xml*