# Proposal for Mathematical and Parallel Computing Modeling as a Decision Support System for Actuarial Sciences

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## Abstract

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## 1. Introduction

## 2. Background

#### 2.1. Actuarial Tables

#### 2.2. Goodness-of-Fit (GoF) Tests

_{0}that X, a random variable, follows a declared probability law F(x). The techniques of these tests consist of mathematical models to measure the conformity of the data of a sample, that is, set of values of x with the hypothetical distribution; or, equivalently, with its discrepancy about it [59]. In other words, the basic concept is that, given a random sample of size n, observed from a random variable X, it is desired to test the null hypothesis H

_{0}that the sample follows a certain distribution function F(x), confronting it with the alternative hypothesis H

_{1}, that the sample does not follow the distribution function F(x):

**H**

_{0}.**H**X has no distribution F(x).

_{1}._{0}can be a simple hypothesis when F(x) is specified completely or H

_{0}can provide an incomplete specification and then it will be a compound hypothesis.

#### 2.3. Chi-Square Adherence Test

- ${X}^{2}$ = chi-square test statistics;
- n = maximum age in the actuarial tables;
- ${O}_{i}$ = observed frequency of deaths/disability with age i;
- $E$ = average death/disability frequency with age i.

## 3. Methodology

- ▪
- Step 1 creating a trusted database;
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- Step 2 define/choose a heuristic to be applied to the data;
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- Step 3 database heuristic adherence test;
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- Step 4 selection of the best model for the study;
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- Step 5 selection of the computational model; and
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- Step 6 analysis of results.

#### 3.1. Creation of SMIB

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- Greater consistency of information based on the maintenance of a unique historical basis;
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- Higher reliability of the database, these being correct and applicable in any situation;
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- Ability to consolidate information, with the possibility of aggregating remunerative installments of various forms;
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- Ability to detail the remuneration structure of the military, with the possibility of explaining the composition of the military remuneration; and
- ▪
- Ability to detail payroll by nature of expenses (NE), which provides greater precision in comparisons with the federal government’s integrated financial administration system (FGIFAS).

#### 3.2. Heuristics

#### 3.2.1. The Probabilistic Multidecrements

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- An active military (state 1) can be declared invalid (state 2), marry (state 4), go to reserve (state 9), resign (state 3), or die (state 11);
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- An invalid military (state 2) can return to active (state 1), marry (state 2), or pass away (state 11);
- ▪
- If the military member resigns (state 3), he leaves the system and does not generate a pension;
- ▪
- A married soldier (state 4) can divorce (state 5), become a widower (state 6), or become a parent (state 8). Once married, the dotted line shows that the military can generate a pension (state 12);
- ▪
- A divorced military (state 5) may enter into a new marriage (state 7);
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- A widowed soldier (state 6) may contract a new marriage (state 7);
- ▪
- A military who contracted a new marriage (state 7) can become a parent state 8);
- ▪
- A military man who became a father (state 8) can generate a pension (state 12) as indicated by the dotted line;
- ▪
- A military in the Remunerated Reserve (state 9) can contract marriage (state 4), go to the reserve by age (state 10), or die (state 11);
- ▪
- A military in the reserve by age (state 10) may marry (state 4) or die (state 11); and
- ▪
- Generation of pension (state 12) can be generated by marriage (state 4), by a new marriage (state 7), and by paternity (state 8) once the military passed away (state 12).

#### 3.2.2. The Actuarial Tables

#### 3.2.3. The Mortality Tables

#### 3.2.4. Invalidity Entry Tables

#### 3.2.5. Tables of Invalidity Mortality

#### 3.2.6. Selection of the Model

#### 3.3. Computational Model

#### 3.4. Parallel Computing

#### 3.4.1. Parallel Programming with C #

#### 3.4.2. Asynchro Programming with Async and Await

#### 3.5. The Software Developed

## 4. Results

#### 4.1. Result of Mortality of Assets, Inactive and Pensioners of the Armed Forces

#### 4.2. Results of Invalidity of Mortality of Armed Forces

#### 4.3. Result of Entry into Disability of the Armed Forces

- ▪
- ALLG-72 for all redemptions between 28% and 39%;
- ▪
- USTP-61 for all redemptions between 38% and 48%; and
- ▪
- X17 for all redemptions between 50% and 55%.

#### 4.4. Processing Time Reduction Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Methodology for the actuarial projection of the costs with pensioners of the armed forces.

CS0-41 | CSO-58 | CSO-80 | AT-49 | AT-50 | AT-55 |
---|---|---|---|---|---|

AT-71 | American Experience | GAM-1971 | SGB-51 | SGB-71 | SGB-75 |

IAPC | Hunter Semitropical | Rentiers Français | Grupal Americana | USTP-61 | GKM-70 |

GKM-80 | ALLG-72 | X-17 | CSG-60 | Prudential 1950 | GAM 1994 Male |

RP-2000-1992 Base-Male Aggregate | AT-2000 | AT-2000 F | AT-83 | AT-83 male | UP-84 |

UP94Men | UP94Woman | UP-94 MT-M-ANB | GRM-80 | GRF-80 | GRM-95 |

GRF-95 | BR-EMSsb-v.2010-m | BR-EMSsb-v.2010-f | BR-EMSmt-v.2010-m | BR-EMSmt-v.2010-f | BR-EMSsb-v.2015-m |

BR-EMSsb-2015-f | BR-EMSmt-2015-m | BR-EMSmt-2015-f | CSO2001MALE | CSO2001FEMALE | IBGE-2011-M |

IGBE-2011-F | IBGE-2011 | IBGE-2012-M | IBGE-2012-F | IBGE-2012 | - |

Invalidity and Mortality Tables for Invalids Used | |||||
---|---|---|---|---|---|

IAPB-57 Weak | IAPB-57 Strong | Zimmermann | Zimmermann (Ferr. Germans) | Zimmermann (Empre. Write.) | Grupal Americana |

Álvaro Comings | TASA-1927 | Prudential (Ferr. Retired.) | IBA (Railways) | Muller | Hunter’s |

IAPB-57 (AJUST/ITAU) | Winklevoss | Bentzien | IAPC | IAPB-57 | ALLG72 |

USTP61 | Rentiers Français | X17 | - | - | - |

Invalidity and Mortality Tables for Invalids Used | |||||
---|---|---|---|---|---|

IAPB-57 Weak | IAPB-57 Strong | Zimmermann | Zimmermann (Ferr. Germans) | Zimmermann (Empre. Write.) | Grupal Americana |

Álvaro Comings | TASA-1927 | Prudential (Ferr. Retired.) | IBA (Railways) | Muller | Hunter’s |

IAPB-57 (AJUST/ITAU) | Winklevoss | Bentzien | IAPC | IAPB-57 | ALLG72 |

USTP61 | Rentiers Français | X17 | - | - | - |

Process | Previous Version [Approximate] HH:MM:SS | Refactored Version [Approximate] HH:MM:SS | Time Reduction (%) |
---|---|---|---|

Importing databases. | 04:00:00 | 00:08:00 | 96.7% |

Actuarial calculation of present value. | 23:00:00 | 00:07:00 | 99.5% |

Actuarial projection with a term of 75 years. | 16:00:00 | 00:05:00 | 99.5% |

Total Time | 43:00:00 | 00:20:00 | 99.2% |

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## Share and Cite

**MDPI and ACS Style**

Santos, M.d.; Gomes, C.F.S.; Pereira Júnior, E.L.; Moreira, M.Â.L.; Costa, I.P.d.A.; Fávero, L.P.
Proposal for Mathematical and Parallel Computing Modeling as a Decision Support System for Actuarial Sciences. *Axioms* **2023**, *12*, 251.
https://doi.org/10.3390/axioms12030251

**AMA Style**

Santos Md, Gomes CFS, Pereira Júnior EL, Moreira MÂL, Costa IPdA, Fávero LP.
Proposal for Mathematical and Parallel Computing Modeling as a Decision Support System for Actuarial Sciences. *Axioms*. 2023; 12(3):251.
https://doi.org/10.3390/axioms12030251

**Chicago/Turabian Style**

Santos, Marcos dos, Carlos Francisco Simões Gomes, Enderson Luiz Pereira Júnior, Miguel Ângelo Lellis Moreira, Igor Pinheiro de Araújo Costa, and Luiz Paulo Fávero.
2023. "Proposal for Mathematical and Parallel Computing Modeling as a Decision Support System for Actuarial Sciences" *Axioms* 12, no. 3: 251.
https://doi.org/10.3390/axioms12030251