# A Novel IBA-DE Hybrid Approach for Modeling Sovereign Credit Ratings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Differential Evolution

**initialization**in the first iteration when a set of m initial d-dimensional solutions (target vectors) are randomly or pseudo-randomly generated. The set size m, commonly referred to as population size, is usually predetermined and kept constant over time. After the initial generation is created, the iterative process begins so that each iteration consists of three steps: mutation, crossover, and selection.

**Mutation**is a process where every target vector from the population is altered in order to create the respective mutation vector. There are a vast number of mutation algorithms in the literature that differ by the crossover scheme, i.e., the number of difference vectors used to create a mutation vector, etc. One of the most used algorithms is DE/rand/1/bin, where the mutation vectors are generated as shown in the Equation (1):

**The crossover**phase takes place after the mutation and is used to increase the diversity between vectors in the generation. In this phase, a trial vector is generated for every target vector and its respective mutation vector. Binomial and exponential crossover are the two most commonly used strategies in practice. For every pair of target and mutation vector elements, a trial vector element is generated as shown in Equation (2).

**Selection**is the last phase in the iterative process. For every target vector in the current generation, a decision is made whether to keep it in the next generation or to replace it with the respective trial vector. The decision is based on comparing fitness function values for the vectors, as shown in the Equation (3):

#### 2.2. Interpolative Boolean Algebra

## 3. IBA-DE Approach

#### 3.1. The Main Steps of the IBA-DE Algorithm

- Input preparation;
- Model training;
- Model testing;
- Model interpretation.

**Input preparation:**The first step in the IBA-DE approach refers to the input preparation, and it consists of the normalization of training inputs and generating IBA atomic vectors. Since we are modeling and inferring in the IBA framework, all elements of input vectors must be valued on an $\left[0,1\right]$ interval. Any normalization function can be used for this purpose. Still, classical min-max normalization along with some methods for handling outliers is a common choice. Data preparation for optimization is performed by generating and valuing IBA atomic vectors. Let $V$ be the g-dimensional input vector and let A be the d-dimensional atomic vector. Each element of the input vector is an IBA attribute, while each element of the atomic vector is an IBA atom. Every element of the atomic vector represents one distinctive combination of attributes. It is represented as a product, $\prod _{i=1}^{g}{y}_{i}$, where ${y}_{i}\in \left\{{v}_{i},1-{v}_{i}\right\}$ and ${v}_{i}$ is a normalized value of an attribute. In Section 2.1, it was stated that the relationship between the dimensions of input and the atomic vector is $d={2}^{g}$, i.e., the number of atomic elements will exponentially increase with the increasing number of inputs. For instance, an atomic vector created from the two-dimensional input vector consists of four elements and it is shown in Equation (4).

**Model training:**After the inputs are prepared in a manner suitable for DE optimization, the model training step may begin. The main goal of this phase is to obtain the optimal structure vector which will minimize the objective function, e.g., mean squared error (MSE) of prediction. In our case, elements of a structured vector are in the unit interval. Therefore, the pseudo-logical aggregation function [51] is obtained as a final output. This function is easy to interpret since it is basically a weighted sum of several logical functions. The DE algorithm used for optimization is the basic one with the steps described in Section 2.1. In addition to a typical stopping condition defined as a maximal number of iterations, we have included an early stopping, a standard machine learning mechanism to avoid model overfitting. Three DE control parameters may be considered as hyper-parameters important for model training: $F$, $Cr$, and the population size. Their values should be assessed during the training phase in order to maximize the potential of the IBA-DE approach.

**Model testing:**The trained model is assessed and utilized in the model testing step of IBA-DE. A product of the atomic vector of an observed instance and the transposed optimal structure vector are the numerical values that are being forecasted, as explained in Section 3.2. All attributes should be scaled to a unit interval using the same normalization function as in the model training step. The final prediction may be validated by comparing the benchmarks, or re-evaluated if some input data changes in time.

**Model interpretation:**The final phase of the IBA-DE approach is the model interpretation step. The output of the DE algorithm is an optimal structure vector whose atoms are used to create the appropriate weight factors for each of the inputs. The weights show the impact of every input, compared to all other inputs, on the forecasted credit ratings. With this information, all inputs can be ranked and the ones with the lowest impact discarded or replaced with new ones. The model is therefore not only predicting the credit ratings but also giving the information necessary for further decision-making, and, therefore, cannot be considered a black box but rather an instrument that can be used not only for forecasting but also for the interpretation of the obtained results.

#### 3.2. Benefits and Limitations

## 4. Forecasting Sovereign Credit Ratings with the IBA-DE Approach

- A single-aspect approach where different groups of macroeconomic factors (stability, activity, and social) are used separately to forecast sovereign credit ratings;
- A multi-aspect approach that uses all groups of input variables to include a broader perspective in the prediction model;

#### 4.1. Inputs

#### 4.2. Output

#### 4.3. Forecasting Models

#### 4.3.1. Single-Aspect Model

- To investigate whether specific aspects of a country’s economic strength could produce satisfactory predictions and therefore reduce input space or substitute some group(s) of factors instead;
- To provide the aggregated (multi-aspect) model with optimal inputs.

#### 4.3.2. Multi-Aspect Model

## 5. Experiment and Results

#### 5.1. Data

#### 5.2. Hyper-Parameter Optimization and Training

#### 5.3. Test Results and Interpretation

_{1}), consumer price inflation (in

_{2}), country’s total reserves (in

_{3}), and gross savings (in

_{4}). As a result of DE optimization, we obtained a 16-dimensional optimal structure vector of weights. Table 3 presents structure vectors for the two most accurate models achieved with and without using historical credit ratings as inputs.

#### 5.4. Comparison with Neural Networks and Support Vector Machine Algorithms

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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Rating | Rep. Value | Rep. Interval | Rating | Rep. Value | Rep. Interval | Rating | Rep. Value | Rep. Interval |
---|---|---|---|---|---|---|---|---|

AAA | AA+ | 88 | (86.66, 90) | A+ | 78 | (76.66, 80) | ||

100 | (91, 100) | AA | 85 | (83.33, 86.66) | A | 75 | (73.33, 76.66) | |

AA− | 82 | (80, 83.33) | A− | 72 | (70, 73.33) | |||

BBB+ | 68 | (66.66, 70) | BB+ | 58 | (56.66, 60) | B+ | 48 | (46.66, 50) |

BBB | 65 | (63.33, 66.66) | BB | 55 | (53.33, 56.66) | B | 45 | (43.33, 46.66) |

BBB− | 62 | (60, 63.33) | BB− | 52 | (50, 53.33) | B− | 42 | (40, 43.33) |

CCC+ | 38 | (36.66, 40) | CC | 28 | (25, 30) | RD | 15 | (10, 20) |

CCC | 35 | (33.33, 36.66) | C | 22 | (20, 25) | |||

CCC− | 32 | (30, 33.33) | ||||||

D | 8 | (6.66, 10) | ||||||

DD | 5 | (3.33, 6.66) | ||||||

DDD | 2 | (0, 3.33) |

Series | Non-Historical | Historical | ||||||
---|---|---|---|---|---|---|---|---|

Inputs | Stability | Activity | Social | Multi-Aspect | Stability | Activity | Social | Multi-Aspect |

MSE | 333.03 | 320.63 | 339.04 | 311 | 12.01 | 11.99 | 11.92 | 11.82 |

Iterations | 189 | 189 | 197 | 6 | 167 | 175 | 174 | 5 |

Structure Vectors | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Inputs | s_{1} | s_{2} | s_{3} | s_{4} | s_{5} | s_{6} | s_{7} | s_{8} | s_{9} | s_{10} | s_{11} | s_{12} | s_{13} | s_{14} | s_{15} | s_{16} |

Macro | 0.71 | 0.88 | 0.62 | 0.94 | 1.00 | 0.68 | 0.81 | 0.65 | 0.64 | 0.67 | 0.51 | 0.79 | 0.91 | 0.58 | 0.59 | 0.63 |

Macro + Hist. ratings | 1.00 | 0.99 | 1.00 | 0.96 | 1.00 | 0.96 | 0.99 | 1.00 | 0.01 | 0.00 | 0.03 | 0.05 | 0.03 | 0.01 | 0.00 | 0.02 |

Weight | w_{1} (in_{1}) | w_{2} (in_{2}) | w_{3} (in_{3}) | w_{4} (in_{4}) |
---|---|---|---|---|

Equation | $\sum _{i=1}^{8}{s}_{i}}/{\displaystyle \sum _{i=1}^{16}{s}_{i}$ | $\sum _{i=1,9,}{\displaystyle \sum _{j=i}^{i+3}{s}_{1}}}/{\displaystyle \sum _{i=1}^{16}{s}_{i}$ | $\sum _{i=1,5,9,13}{\displaystyle \sum _{j=i}^{i+1}{s}_{1}}}/{\displaystyle \sum _{i=1}^{16}{s}_{i}$ | $\sum _{\begin{array}{c}i=1\\ i\mathrm{mod}2=1\end{array}}^{16}{s}_{i}}/{\displaystyle \sum _{i=1}^{16}{s}_{i}$ |

Macro | 0.541 | 0.495 | 0.523 | 0.499 |

Macro + Hist. ratings | 0.983 | 0.502 | 0.496 | 0.505 |

Input | Macro | Macro + Hist. Ratings | ||||||
---|---|---|---|---|---|---|---|---|

Model | Stability | Activity | Social | Multi-Aspect | Stability | Activity | Social | Multi-Aspect |

MSE | 303.15 | 317.79 | 319.21 | 324.37 | 6.52 | 6.53 | 7.34 | 7.22 |

Country | Year | Credit Rating | Country | Year | Credit Rating | Country | Year | Credit Rating | |||
---|---|---|---|---|---|---|---|---|---|---|---|

Real | Forecast | Real | Forecast | Real | Forecast | ||||||

Argentina | 2017 | B | B | Hungary | 2017 | BBB− | BBB− | Peru | 2017 | A− | A− |

2018 | B | B | 2018 | BBB− | BBB− | 2018 | A− | A− | |||

Australia | 2017 | AAA | AAA | Iceland | 2017 | A | A− | Philippines | 2017 | BBB | BBB− |

2018 | AAA | AAA | 2018 | A | A | 2018 | BBB | BBB | |||

Austria | 2017 | AA+ | AA+ | India | 2017 | BBB− | BBB− | Poland | 2017 | A− | A |

2018 | AA+ | AA+ | 2018 | BBB− | BBB− | 2018 | A− | A− | |||

Belgium | 2017 | AA− | AA | Indonesia | 2017 | BBB− | BBB− | Portugal | 2017 | BB+ | BB+ |

2018 | AA− | AA− | 2018 | BBB | BBB− | 2018 | BBB | BB+ | |||

Bolivia | 2017 | BB− | BB− | Ireland | 2017 | A | A | Romania | 2017 | BBB− | BBB− |

2018 | BB− | BB− | 2018 | A+ | A | 2018 | BBB− | BBB− | |||

Brazil | 2017 | BB− | BB− | Israel | 2017 | A+ | A+ | Russia | 2017 | BBB− | BBB− |

2018 | BB− | BB− | 2018 | A+ | A+ | 2018 | BBB− | BBB− | |||

Bulgaria | 2017 | BBB− | BBB− | Italy | 2017 | BBB | BBB+ | Rwanda | 2017 | B+ | B+ |

2018 | BBB | BBB− | 2018 | BBB | BBB | 2018 | B+ | B+ | |||

Cameroon | 2017 | B | B | Jamaica | 2017 | B | B | Saudi Arabia | 2017 | A+ | AA− |

2018 | B | B | 2018 | B | B | 2018 | A+ | A+ | |||

Canada | 2017 | AAA | AAA | Japan | 2017 | A | A | Serbia | 2017 | BB | BB− |

2018 | AAA | AAA | 2018 | A | A | 2018 | BB | BB | |||

Chile | 2017 | A+ | AA− | Kazakhstan | 2017 | BBB | BBB | Seychelles | 2017 | BB− | BB− |

2018 | A+ | A+ | 2018 | BBB | BBB | 2018 | BB | BB− | |||

China | 2017 | A+ | A+ | Kenya | 2017 | B+ | B+ | Singapore | 2017 | AAA | AAA |

2018 | A+ | A+ | 2018 | B+ | B+ | 2018 | AAA | AAA | |||

Colombia | 2017 | BBB | BBB+ | Korea, Rep. | 2017 | AA− | AA− | Slovakia | 2017 | A+ | A+ |

2018 | BBB | BBB | 2018 | AA− | AA− | 2018 | A+ | A+ | |||

Congo, D. p. | 2017 | CCC | CCC | Latvia | 2017 | A− | A− | Slovenia | 2017 | A− | BBB+ |

2018 | CCC | CCC | 2018 | A− | A− | 2018 | A− | A− | |||

Costa Rica | 2017 | BB | BB+ | Lithuania | 2017 | A− | A− | South Africa | 2017 | BB+ | BBB− |

2018 | BB | BB | 2018 | A− | A− | 2018 | BB+ | BB+ | |||

Croatia | 2017 | BB | BB | Luxemburg | 2017 | AAA | AAA | Spain | 2017 | BBB+ | BBB+ |

2018 | BB+ | BB | 2018 | AAA | AAA | 2018 | BBB+ | BBB+ | |||

Cyprus | 2017 | BB− | BB− | Malaysia | 2017 | A− | A− | Sri Lanka | 2017 | B+ | B+ |

2018 | BBB− | BB− | 2018 | A− | A− | 2018 | B+ | B+ | |||

Czech Rep. | 2017 | A+ | A+ | Malta | 2017 | A+ | A | Sweden | 2017 | AAA | AAA |

2018 | AA− | A+ | 2018 | A+ | A+ | 2018 | AAA | AAA | |||

Denmark | 2017 | AAA | AAA | Mexico | 2017 | BBB+ | BBB+ | Switzerland | 2017 | AAA | AAA |

2018 | AAA | AAA | 2018 | BBB+ | BBB+ | 2018 | AAA | AAA | |||

Dominican Rep. | 2017 | BB− | BB− | Mongolia | 2017 | B− | B | Thailand | 2017 | BBB+ | BBB+ |

2018 | BB− | BB− | 2018 | B | B− | 2018 | BBB+ | BBB+ | |||

Egypt | 2017 | B | B | Morocco | 2017 | BBB− | BBB− | Tunisia | 2017 | B+ | BB |

2018 | B | B | 2018 | BBB− | BBB− | 2018 | B+ | BB− | |||

El Salvador | 2017 | CCC | B+ | Namibia | 2017 | BBB− | BBB− | Turkey | 2017 | BBB− | BBB− |

2018 | B− | CCC | 2018 | BB+ | BBB− | 2018 | BB+ | BBB− | |||

Estonia | 2017 | A+ | A+ | Netherlands | 2017 | AAA | AAA | Uganda | 2017 | B+ | B+ |

2018 | A+ | A+ | 2018 | AAA | AAA | 2018 | B+ | B+ | |||

Finland | 2017 | AA+ | AA | New Zealand | 2017 | AA+ | AA+ | Ukraine | 2017 | B− | CCC+ |

2018 | AA+ | AA+ | 2018 | AA+ | AA+ | 2018 | B− | B | |||

France | 2017 | AA | AA | Nigeria | 2017 | B+ | B+ | United Kingdom | 2017 | AA | AA |

2018 | AA | AA | 2018 | B+ | B+ | 2018 | AA | AA | |||

Georgia | 2017 | BB− | BB− | N. Macedonia | 2017 | BB | BB+ | United States | 2017 | AAA | AAA |

2018 | BB− | BB− | 2018 | BB | BB | 2018 | AAA | AAA | |||

Germany | 2017 | AAA | AAA | Norway | 2017 | AAA | AAA | Uruguay | 2017 | BBB− | BBB− |

2018 | AAA | AAA | 2018 | AAA | AAA | 2018 | BBB− | BBB− | |||

Greece | 2017 | C | CC | Panama | 2017 | BBB | BBB | Zambia | 2017 | B | B |

2018 | B | CC | 2018 | BBB | BBB | 2018 | B | B | |||

Hong Kong | 2017 | AA+ | AA+ | Paraguay | 2017 | BB+ | BB+ | ||||

2018 | AA+ | AA+ | 2018 | BB+ | BB+ |

NN Learning Algorithm | Series | Non-Historical | Historical | ||||||
---|---|---|---|---|---|---|---|---|---|

Inputs | Stability | Activity | Social | Multi-Aspect | Stability | Activity | Social | Multi-Aspect | |

BFG | MSE | 328.49 | 329.92 | 359.07 | 316.42 | 16.61 | 18.61 | 17.99 | 11.82 |

Iterations | 27 | 26 | 17 | 37 | 59 | 45 | 42 | 38 | |

LM | MSE | 329.16 | 317.93 | 351.10 | 314.35 | 14.32 | 14.88 | 17.92 | 13.39 |

Iterations | 70 | 30 | 22 | 27 | 138 | 53 | 40 | 37 | |

RB | MSE | 343.75 | 331.57 | 347.42 | 327.26 | 11.97 | 15.37 | 15.21 | 14.87 |

Iterations | 24 | 80 | 13 | 8 | 39 | 91 | 42 | 34 |

SVM Kernel Function | Series | Non-Historical | Historical | ||||||
---|---|---|---|---|---|---|---|---|---|

Inputs | Stability | Activity | Social | Multi-Aspect | Stability | Activity | Social | Multi-Aspect | |

Linear | MSE | 344.99 | 362.83 | 363.82 | 322.08 | 14.27 | 14.27 | 14.19 | 11.32 |

Iterations | 647 | 701 | 701 | 582 | 311 | 363 | 363 | 295 | |

Gaussian RBF | MSE | 323.47 | 308.61 | 325.13 | 295.46 | 12.27 | 12.47 | 12.21 | 9.08 |

Iterations | 755 | 809 | 755 | 643 | 415 | 519 | 467 | 386 | |

Polynomial | MSE | 334.07 | 317.74 | 337.22 | 304.25 | 13.16 | 13.59 | 13.53 | 10.29 |

Iterations | 755 | 755 | 701 | 629 | 571 | 519 | 519 | 489 |

Model | Series | Non-Historical | Historical | ||||||
---|---|---|---|---|---|---|---|---|---|

Inputs | Stability | Activity | Social | Multi-Aspect | Stability | Activity | Social | Multi-Aspect | |

IBA-DE | MSE | 303.15 | 317.79 | 319.21 | 324.37 | 6.52 | 6.53 | 7.34 | 7.22 |

NN | 311.90 | 314.73 | 343.56 | 307.26 | 6.61 | 6.64 | 14.59 | 6.57 | |

SVM | 305.46 | 330.12 | 320.93 | 309.08 | 7.24 | 7.83 | 7.60 | 6.86 |

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## Share and Cite

**MDPI and ACS Style**

Jelinek, S.; Milošević, P.; Rakićević, A.; Poledica, A.; Petrović, B.
A Novel IBA-DE Hybrid Approach for Modeling Sovereign Credit Ratings. *Mathematics* **2022**, *10*, 2679.
https://doi.org/10.3390/math10152679

**AMA Style**

Jelinek S, Milošević P, Rakićević A, Poledica A, Petrović B.
A Novel IBA-DE Hybrid Approach for Modeling Sovereign Credit Ratings. *Mathematics*. 2022; 10(15):2679.
https://doi.org/10.3390/math10152679

**Chicago/Turabian Style**

Jelinek, Srđan, Pavle Milošević, Aleksandar Rakićević, Ana Poledica, and Bratislav Petrović.
2022. "A Novel IBA-DE Hybrid Approach for Modeling Sovereign Credit Ratings" *Mathematics* 10, no. 15: 2679.
https://doi.org/10.3390/math10152679