Representing the Countries Credit Ranking Problem with a Simple Weighted Non-Linear Formulation

: In this study a non - linear weighted sum model is proposed to rank countries based on economic factors. This ranking problem could be new and useful as most of previous researches rated countries not rank them. The countries are ranked from the best to the worst one by their score obtained by the model from credit point of view. As an advantage of the model, it is solvable by an analytical solution method manually instead of using optimization software. The analytical solution is useful for managers and decision makers to apply the model easily. The obtained ranking is compared with Moody’s rating to discuss the efficiency of the model.


INTRODUCTION
Understanding the contribution of economic and political factors on the evaluation of countries credit rating has crucial policy implications.Up to now there are many different methodologies were suggested by researchers to calculate country credit rating (sovereign rating), however some of them were valuable.Most of these approaches and techniques are based on mathematical models ( [1][2][3]) and some the others based on probabilistic and stochastic methods ( [4][5][6]).Famous rating agencies such as Standard & Poor's, Moody's, Fitch Rating, etc. announce a rating list monthly, quarterly, semiannually and annually which rate counties by considering several factors (economic and political).They employ different methodologies and techniques in addition to idea of expertise to rate the countries.In fact, it is impossible to create a unique model to rate countries, since there are many different quantitative and qualitative factors directly or indirectly affecting the outcome.
Credit rating announcement that is provided by the rating agencies affect the financial market in different ways.Recently, information provided by rating agencies is important for market contributors and regulators much more than past as they are under pressure after they fail to predict world financial crises 2007-2008 [7].The latest crisis demonstrated that the country credit rating is an important issue in the international market for both develop and developing countries [8].For that reason nowadays credit rating agencies are not the only provider for rating information as econometrician, operation researcher, financial investigator and statisticians, etc. propose valuable approaches to rate countries [9].
Credit rating methods mostly consider some predetermined levels for countries and each country is assigned to an appropriate level.The assignment of each country is done just by evaluation of the performance of that country in the economic factors considered for the rating.On the other hand credit ranking prepare a complete ranking of countries from the best to the worst one according to the economic factors.So, the performances of countries in the economic factors are compared to rank the countries.
In this study a mathematical model is proposed to rank the countries and it is subjected to selected economic factors.The suggested model is used to rank the countries (from best to worst) rather than rating them from credit point of view as the most of researches have focused on countries credit rating.The model calculate counties' score and the countries are ranked based on their obtained score.In other word, the result is subjective to the factors and number of countries which are selected.Therefore, if new counters or factors are selected and added to the dataset, then the ranking result may change accordingly.
As the rest of the paper, Section 2 reviews the existing literature of methodologies and techniques which were utilized in country credit rating.Section 3 proposes a nonlinear data envelopment analysis model for countries credit ranking.Data collection and factor selection are discussed in Section 4. Empirical result and comparison of result with Moody's rating is covered by Section 5. Concluding remarks are discussed in the last section.

LITERATURE REVIEW
Sovereign rating also known as country rating, is one of the import topics in the global financial market and is affected mostly by economic factors.Up to now, researchers proposed different methodologies which utilize mathematical, statistical, probabilistic and/or stochastic models to rate or classify countries.For instances, a model that is proposed in 2001 was based on the multi-criteria decision aid (MCDA) and multi-group hierarchical discrimination (M.H.DIS), which use different factors to classify a number of countries in to specified classes.The model was revised many times during next 2 years starting from 2000 to 2002 ( [1,2]).Later on, the proposed model have modified and improved by [9] and used to reconstruct the World Bank classification.Ref. [4] constructed rating transition matrices for countries as an input of rating-based credit portfolio model.Ref. [10] applied reverse engineering by utilizing Logical Analysis of Data in the case of financial risk rating and the results were compared with Standard & Poor's rating result to prove the model accuracy.
One of the important issues in country rating is factors' selection.Although, economic and political factors both have essential effect on the country credit rating result, many economic theories emphasis in importance of macroeconomic factors for credit and default rating of countries [11].It seems instinctively obvious that macroeconomic conditions of a country should effect on the credit rating of that country [12].The relations between macroeconomic factors and risk default were examined by several researchers such as [13][14][15][16][17], and so on.The macroeconomic factors which appear to be significant for country credit rating are GDP, GDP per capita and GDP growth ( [18][19][20][21]), current account balance and public [22] and debt budget balance [23].Later, [24] mentioned that six factors appear to be important in deciding about country credit rating.Those six factors are GDP per capita, external debt, level of economic development, default history, real growth rate and inflation.Ref. [25] investigated the impact of worldwide financial conditions, domestic fundamentals and U.S. macroeconomic factor (news) on the emerging market bond index spread based on daily data.They found out that in the long run evolution of emerging market bond index spread is affected by global financial conditions, crises and domestic fundamentals which depend on sovereign rating.Ref. [26] investigated on the interactions between sovereign ratings and macroeconomic factors using a Panel Vector Autoregressive (PVAR) approach.He used annual data for European countries from 1996-2013.The results of the study proved that there is a significant tow-way interaction between macroeconomic factors and sovereign rating.Ref. [27] studied the impact of tax convexity on the decisions taken for investing purposes.
Based on the information in the literature, several economic factors are chosen for this study which are discussed in Section 4.1.

THE PROPOSED MODEL
In this section a mathematical model is introduced to rank some given countries according to some criteria (economic factors) in a way that the countries with higher ranking provide better environment for making investment by investors.The model applies the following notations and their definitions: I -number of countries to be ranked (parameter), J -number of criteria (economic factors) that effect ranking obtained for the countries (parameter), i -index used to show each country (index), j -index used to show each economic factor (index), wj -relative importance obtained for the j th economic factor (variable), S i -maximal possible score obtained for country i (variable), x ij -performance of country i in economic factor j (parameter), r ij -normalized performance of country i in economic factor j (parameter).
As the economic factors are divided to two types of positive (factors that higher performance of them is favored e.g.income) and negative (factors that lower performance of them is preferred e.g.inflation rate) factors, the normalization of x ij is obtained by, where POS and NEG are the sets of positive and negative economic factors, respectively.
Assuming that the factors are allowed to have positive relative importance such that w 1 , w 2 , …, w J ≥ 0, the aim is to aggregate multiple performance scores of each country obtained from different factors into a single score for the credit ranking problem.Therefore, a country with the highest score is ranked at the first place and so on.
To achieve this purpose, the following weighted sum model is proposed, 1 max , The model ( 2)-( 4) is applied to each country separately.The objective function calculated by Eq. ( 2) is a simple weighted sum of the normalized performances of each country in all the factors which maximizes the possible score obtained for the country.In constraint (3), the Euclidean norm of the relative importance of the factors is equal to 1. Applying this constraint, the model is an endogenous type model.Meaning that the relative importance of each factor is determined by the data of the model (normalized performances) endogenously instead of being determined by a decision maker.Constraint set (4) guarantee a positive weight for each factor.
Using Euclidean norm in the constraint ( 3) is an advantage for the model which gives a possibility to the model to be solved analytically.Thus, any financial manager and decision maker can apply and solve the model manually without any optimization solver.The analytical optimal solution of the model can be obtained by the method of Lagrange multipliers.
A simple explanation of Lagrange multipliers method can be found in the following steps: Step 1: A model to be solved: max f(x) s.
Finally, applying ( 16) in ( 6), the optimal (maximum) score for each country is obtained by the following equation.

COMPUTATIONAL EXPERIMENTS ON A CASE STUDY
In this section the efficiency of the proposed model ( 2)-( 4) and its analytical optimal solution obtained by (11) and ( 12) is measured.Some economic factors and countries are considered to be raked accordingly.Finally some economical comments on the obtained ranking are discussed.

Data Collection and Modification
All data in this study were collected from International Monetary Fund (IMF) database.Totally 25 factors (economical) and 53 countries are selected to elaborate the approximate ranking method.The number of selected countries is depended to availability of information in IMF database 2014.In addition, information related to the local currency rating of Moody's investor service (government bond rating for August 2014) is used to compare with the result of the proposed model.The 25 economic factors are listed in Tab. 1.The performance of each country in each economic factor (x ij ) also is obtained from IMF database.The x ij values of all countries in all 25 selected factors are normalized (r ij ) by the method mentioned in Eq. ( 1).The obtained r values of two factors (for instance) of Tab. 1 for all countries are depicted in Tabs. 2 and 3.

Credit Ranking of the Countries
After normalizing the performances of each country in all factors, the model ( 2)-( 4) is solved separately for each country using its analytical solution.The countries are sorted by decreasing order of their optimal scores.The ranking obtained for the countries is illustrated by Tab. 4. It appears that country rating remain an important determinant of agencies credit rating.Although this study used a non-linear ranking model to rank the countries, it can be even used as an approach to rate countries.The model try to compare all countries by each other base on all economic factor levels, and rank the countries from the highest score to lowest score.Such a measure is suited for any country that have available information regarding to those economic factors.
The results that are depicted in Tab. 4 compare 54 countries, and rank them from the highest score to the lowest one.As it is expected, developed countries in Europe, China and US are listed among the top 10 countries.This is a valuable result and reflect the accuracy of the model in ranking the countries.If we compare our results with Moody's rating which is illustrated in Tab. 5, it can be concluded that the ranking model was successful in application.Also it is important to remind that the model which is utilized in this study is used to rank the countries not rating them.Since all of the rating agencies rate the countries and none of them rank them, there is no other source to compare the result of this study with them.Although, there are some miss ranking or error in ranking the model, overall result satisfy and show validity of the model.The reason for miss ranking may occur due to some political factors which were not the interest of this research.Since all of the factors which employed in this study are economic factors and are quantitative, another reason that may be cause of the miss ranking is lake of availability of some economic or other quantitative factors.In addition, as another reason of the miss ranking, some rating agencies are not willing to downgrade some countries since some of them are their clients and many other reasons that are not interest of this research.
One important note that was also mentioned earlier is that the model rank the countries not rate them.So, some time there is a small difference between two or more different ranks.For instance, Sweden and Singapore are ranked in 6 th (with score 3.239604525) and 7 th (with score 3.234437298) respectively in Tab. 4. Clearly, there is a minor difference between the scores of two countries, but with a small difference Sweden is preferred to Singapore.

CONCLUSION
Country credit rating changes have influences on investment and every sector of the related countries.Most of researches in the literature focus on rating changes of countries and those rating affected mostly by macroeconomic outcomes.Numerous downgraded of European countries in past years have shown how vital it is to examine the issue.
This study starts for the first time to rank a set of countries based on several important macroeconomic factors instead of rating them.A weighted sum model was proposed and solved analytically.The analytical optimal solution was obtained by the Lagrange Multipliers method easily.This easy solution method is an advantage of the model that helps the managers to apply the model easily without any optimization software.The results was compared to Moody's rating (in 2013) to show accuracy of the model.This study is a worthy empirical analysis for comparing several specified countries for investment.In particular, if there is a set of specified countries which are interested for investment, it is possible to compare them and find the best candidate.The presented model can be an alternative to country risk rating

Table 1
Economic factors selected from IMF database

Table 2
The selected countries and original and normalized values of their performances in a positive factor for instance.

Table 3
The selected countries and original and normalized values of their performances in a negative factor for instance.

Table 4
Complete credit ranking of the countries obtained by the model (2)-(4).