# Where can I get actuarial tables

## (IN) contributory equity in the general social security system: RGPS from the perspective of actuarial academic income and the actuarial social security factor

**ORIGINAL ITEM**

ROCHA, Matheus Carneiro ^{[1]}, ARAÚJO, Jamille Carla Oliveira ^{[2]}, SANTOS, Neuma Teixeira Dos ^{[3]}

ROCHA, Matheus Carneiro. ARAÚJO, Jamille Carla Oliveira. SANTOS, Neuma Teixeira Dos. **(IN) contributory justice in the general social security system: RGPS from the point of view of actuarial science from the point of view of income and the factor of actuarial social security. **Revista Científica Multidisciplinar Núcleo do Conhecimento. 04 year, Ed. 06, Volume 03, pp. 135-161. June 2019. ISSN: 2448-0959.

Contents

### SUMMARY

Public social security issues have sparked wide debate between society and government. The average person cares about the fair measure between what he pays in the present and what he will receive from his pension in the future. With that in mind, this article aims to define, following the principles of actuarial science, the fair measure between the value of contributions (payments) and the value of benefits (retirement) administered by the General System of Social Security - RGPS as an instrument of social (in-) justice. The method used to obtain all the parameters relevant to the RGPS, as well as the social security actuarial system, was to create situations in which men and women, under certain conditions, entered the social security system with the starting age, the retirement age and the contribution salary were involved, and from this input data the contribution values and benefits within the framework of Matlab-P*rogram*s calculated. , with calculations being operationalized as a calculation routine. The results were segmented by gender (male or female) and income range, it was observed that the social security contribution rate, the main parameter defining the value of contributions to the RGPS, ranged from 28% to 31% compared to that of the actuarial Science calculated rate of 16% to 17% for men and 13% to 14% for women is very high. It is concluded that for the ordinary citizen, subject to the applicable rules of the RGPS, the amount of contributions paid, reflecting the retirement benefits received, deviates from the fair measure calculated by actuarial science, therefore no factors have been made observed who testify that the RGPS is fair to the taxpayer and therefore to society.

Keywords: social security, RGPS, actuarial science.

### 1. INTRODUCTION

The public pension system is administered by the National Social Security Institute (INSS) implemented by Law No. 8.029 of April 12, 1990 and regulated by Decree No. 99.350 of June 27, 1990. It regulates the question of social security in the country which is responsible for the INSS for the administration and maintenance of the social security services, which result from the general social security system - RGPS, in the question of 8.213 of April 24, 1999 - essentially around the workers the private sector and the civil servants’s own social security system (RPPS).

The fundamental point of any social security system is how it is funded; H. ensure a continuous flow of resources to cover expenses with administrative services and expenses. Also, Iyer (2002), it is necessary to equal the flows of resources proportional to the contribution period and the final salary of the taxpayer.

The terms contributions and contribution rate naturally arise in the central discussion area when it comes to social security, since it is society as a whole that finances this system.

The complex dynamics of social security systems are discussed in detail by Iyer (2002), who informs that the parameters applied influence the social security system in demographic and economic aspects, the dynamics of the number of workers who contribute to the system and the number of Retirees or those who benefit from this system.

With the increase in the number of pensions and social assistance benefits and the decrease in the labor force, i. H. The number of social security taxpayers is thus confirmed to bring about a significant change in demographic profile, which can lead to an increase in social security spending to 10 times the value of gross domestic product (GDP) by 2050. This fact increases paying for a society so as not to affect other activities, with important public policies, according to Cruz, (2015).

According to Campani and Dias da Costa (2016), spending on public pensions is stratoferically high, according to government officials it is in the home of hundreds of billions of reais and in just over 50 years the spending will be more than 40% of the Union's budget revenues . Despite the size of the social security, no one has discussed whether the value of the contributions paid and received, as well as the discount rates applied, are really fair.

For the analysis of social security systems. Understanding how complex and stochastic parameters relate over time is a key concept for other aspects of the actuarial math of social security systems (Afonso and Lima, 2011).

The author's interest in the actuarial equity study was made even clearer after reading the work of Giambiagi and Afonso (2009), which positions the ideal social security system as an actuarial balance between the expected present value versus an interest rate equal to the flow of benefits.

The issue has social relevance to the attempt in the judiciary to check whether a pension reform is necessary, given the recent changes related to the non-incidence of the social security factor in the retirement age for the contribution period introduced by the new Law No. 13,183 / Set in 2015, reflect the economic and financial sustainability of social security.

The core question is: is there equity in the system of contribution to the RGPS from the perspective of actuarial science? That is, the income and insurance factor applied can determine the equity of contributory memory?

### 2. THEORETICAL FRAMEWORK

### 2.1 RGPS CONTRIBUTION FLOW - GENERAL SOCIAL SECURITY SCHEME

The INSS (National Social Security Institute) was implemented by Law No. 8,029 of April 12, 1990 and regulated by Decree No. 99,350 of June 27, 1990. It is The INSS is to administer and maintain the Social Insurance Services derived from the General Social Insurance System - RGPS, at the center of Law No. 8,213 of April 24, 1999.

The RGPS has an important benefit system for the retired citizen. This pension system has three rules, namely the progressive 85/95, the 30/35 contribution years and the proportional retirement rule. As the current rule is progressive 85/95, work will be carried out on this.

The contribution to the RGPS is characterized as a simple distribution, where companies contribute around 20% and the remaining rate is imposed by the employee. The contribution to the employee is calculated according to the income of the taxpayer (), whereby it is divided by the contribution rate (c) according to the income range e*r*Contributed significantly as shown in Table 1, which is available on the INSS website.

Table 1 - Contribution rates according to the wage range of the RGPS contribution rates 2017.

Contribution salary (R) | rate |

Up to 1,659.38 | 8% |

From 1,659.39 to 2,765.66 | 9% |

From 2,765.67 to 5,531.31 | 11% |

*Source - INSS (2019).*

Then the contribution flow or the present value of the contribution*rag*e (VPC) simply calculated according to equation 1. The accumulation of resources promoted by Equation 1 is increasing and progressive.

### 2.2 SOCIAL SECURITY FACTOR AND FLOW OF BENEFITS IN RGPS

The social security factor was introduced in 1998 in the FHC government through constitutional amendment 20. The following year, Law 9.876 created the social security factor with the mandate to discourage early retirement schemes, because the longer the contribution period, the higher the amount of retirement received and vice versa. Equation 2 shows how the social security factor is calculated (Penafieri and Afonso, 2013).

Where:

*T*_{c}:: Contribution time.

*α*: Contribution rate (0.31).

*E.*_{s}: Life expectancy reported in the IBGE mortality table for both sexes.

*I.*_{d}: Age at the moment of retirement.

Hence, Equation 2 shows the relationship of three important variables, time of contribution, age at retirement, and life expectancy. In mathematical terms, the longer the contribution period, the higher the social security factor, the longer the time and age at the moment of retirement, the higher the social security factor and, in the opposite direction, the higher the life expectancy, the lower the factor social security.

Law 9.876 / 99 was supplemented by decree 3265/99, with which a change to the RGPS in the calculation of the pension or benefit payment (p_{b}) was introduced*e _{,}* so that the salary is multiplied by the social security factor (

*f*) with d

*e*m real arithmetic mean

*M.*the 80% higher contribution salary is calculated according to equation 3.

Equation 4 is an integral that represents the basis for calculating retirement benefits according to the contribution period. The equation means the average of the 80% higher contribution*ge*container, *T _{c}*: represents the contribution time

*r*0,2

_{,}*T*represents 20% of the contribution time as a result of the interpretation of the constitutional rule of Amendment N

_{c}^{O }20 represents.

Where is the value of the Individual Salary in Time*t*, *w _{0}* the taxpayer's starting salary,

*H*b

*e*indicates the growth rate of the economy per year and

*G*di

*e*Rate of wage growth per year, the parameters of Equation 5 together explain the progressive and exponential growth in taxpayer income over time.

The parameters important for the RGPS are inserted into the economy as exogenous variables for the social security system of the RGPS. One of the variables is the taxpayer's salary, this follows the mathematical model of Gremaud and Patrick (2004); Penafieri and Afonso (2013) according to equation 5 and previously shown in equation 4.

### 2.3 NOTIONS OF ACTUARIAL MATHEMATICS

Filho (2010) presents the important terms of the actuarial assumptions, an important parameter to be considered is the survival board. The literature has a number of tables, but the one the author used in drafting this work was the survival plank devised by IBGE. The main meaning of the survival board is to get the life expectancy and the probability of survival in a given section of the table.

The ideas put forward by Afonso and Lima (2011) correspond to Filho (2010) with regard to the idea of the likelihood that a person ages *x *until the age of *x + t* Years at Le*ben* remains. This probability is given by P_{x} according to*Ä _{ß}* Equation 6 shown.

*I.*represents on the survival board the number of individuals on average that corresponds to their age

_{x + t}*x + t I*, which corresponds to what the hypothetical amount of individuals in the cut of the survival table in old age

_{x}*x*represents.

Similarly, Equation 6 can calculate the probability of an individual dying in t years, hence the difference in each cohort of individuals between the *x* and x + t intervals divided by the number of individuals in the cohort *x* died, given by Equation 7.

Reading the article written by Afonso and Lima (2011) was important for this work, as the authors treated the topic of the social security system from an actuarial point of view compared to the RGPS on the agenda of the relevant contribution system. The theoretical basis of the contribution and accumulation model of social benefits was presented in detail in the article.

Figure 1 shows two different pages, the left side shows the contribution period, i. H. it is the monetary values that the taxpayer regularly pays to the social security system. The right hand side represents the duration of retirement; H. the time in which the beneficiary received his performance salary up to his death in the time*t*.

Figure 1: Contribution flows and benefit flows as a function of time.

From financial mathematics, the contribution flow over time should be based on the present value for age x multiplication with the discount factor connection vt according to equation^{G }8, where t d*e*r is the period and i is the *D.*The discount rate for the computational effect varies between 1% and 3%, as proposed in Afonso and Lima (2011).

### 2.4 ACTUARIAL SOCIAL SECURITY FACTOR IN RAP - ACTUARIAL SOCIAL SECURITY REGIME

The actuarial justice claim piqued the author's interest, this interest became even more apparent upon reading the work of Giambiagi and Afonso (2009) which positioned the ideal social security system as an actuarial balance between the expected present value by placing an interest rate in The amount of the inflow of the services received is discounted. Therefore, for a fair actuarial balance, it is important to determine the insurance premium rate (C._{at}) based on Equation 9, where *f *the social factor*l*represents security, *w* the rate of wages*above*ression means *i* the discount rate, *T* is the contribution period*d*e and *N* the age limit of the survival plank used, in the case of this work it is the IBGE 2017 plank.

According to previous studies, the main difference between the two approaches to contribution flow calculation based on RGPS and the other based on actuarial science became apparent. The RGPS already submitted accumulates the contributions in a percentage without actuarial criteria that is focused on income. The other actuarial fairness, as has also been shown, is the multiplication factor or percentage determined by strictly scientific criteria based on actuarial mathematics.

### 2.5 CONTRIBUTION FLOW IN RAP

The contributions according to Afonso and Lima (2011) can be described mathematically by the actuarial formulation shown in equation 10. So mean*t de*r VPC that the present value of the contributions is obtained at present value updated by the compound discount factor determined by the v^{t} Product of P_{x} is given and multiplied by that of C_{at}W._{t}, specified term of income, where C_{at} by equation 10 and W._{t} is given by equation 5. The contribution flow is described as a finite series of instantaneous temporary variable prepayments.

Therefore, we thought of a hypothetical situation related to the pension situation through contribution time from four input data (gender, entry age, retirement age and starting income), in which the results could be easily replicated, so that it was possible to extend the discussion on the coherence of the RGPS- System and thus direct the judiciary by paying a certain amount to the INSS each month in order to retire with a certain amount.

### 2.6 PRESENT VALUE OF THE FLOW OF PENSION PENSION IN RAP

Calculating the power flow results in the power salary, the same amount calculated by the RGPS in equation 3. The social security actuarial system seeks to calculate the present value of the benefit flow, as in the Afonso and Lima (2011) article, as it will help interpret the gap between contributions to the RGPS in relation to the social security actuarial system . The calculation of the present value of the power flow in RAP is given by Equation 11, where S_{b} the performance salary and FA of the insurance*gs*mathematical factor is.

The current factor of equation 12 is given by the product between the compound discount rate V^{t} of equation 8 with the probability of survival P._{x} the equation_{i}chung 6 determined. Therefore, the current factor becomes F*A. *given by equation 11.

Based on this information from the calculation of the present value of the utility flow in RAP and the current factor of equation 10, some studies follow on the presented topic.

### 3. PREVIOUS STUDIES AND RESEARCH HYPOTHESES

In this section of the course thesis, according to Table 01, the general design of the works that will contribute to the vision and theoretical support of the author is presented.

Table 1 - Main studies on the subject

Quotes | AIMS | Results | APPLICATIONS IN THE STUDY |

Iyer, (2002) | Analyze and develop the mathematics of social security systems in terms of integral and differentiated formulation. | Actuarial mathematics of social security systems shows how social security systems are developed in their aspects of actuarial balance. | The social security systems presented in this thesis include the most widespread worldwide, such as the simple distribution system (in Brazil) and the system of direct capitalization. |

Gilat, (2006) | Matlab with Engineering Applications is one of the standard textbooks used in mechanical engineering, electrical and etc. courses. | The author used this book to help create the routine of social security calculations for the general social security system and for the actuarial social security system, the calculation routine is compiled in the appendix chapter. | Matlab applications as a solution and programming tool. |

Chapman, (2006) | Textbook for programming structural concept applications in Matlab. | Programming in Matlab for Engineers was another textbook that was widely used by the author. Along with Gilat's book (2006), there was major training for the author to develop the routines of social security calculations. | This author's contribution to this work was to elucidate concepts of the decision-making structure when - otherwise - they assess wage values and correctly apply contribution rates to the RGPS and the actuarial social security system in order to direct the flow of routine between the one paths. In addition to working together for the entity of the style for -end repeating structures, to finally calculate the value of the compound discount rates in order to bring the contribution values and benefits to the present value. |

Giambiagi & Afonso, (2009) | Calculate what would be a fair rate from an actuarial point of view. | The main conclusion of the authors was that the social security rate of 31% applicable to the RGPS is inordinately high for some income groups. Through this work, the author has elaborated the issue of justice in the RGPS from the perspective of the contributions and achievements of the users. | Calculating the Actuarial Balanced Social Security Contribution Rate: An Application to the Brazilian Case is a publication article by Professor Giambiagi, one of the great social security researchers in Brazil. |

Gonçalves & Letieri, (2010) | Social Security Contributions: An analysis in the annual accounts of the municipalities of Pernambuco with a view to the rakeliability. | This was a course thesis consulted by the author, the importance of which was the description of closed social security systems. | The theoretical foundations and social security laws mentioned in this study have been widely consulted and the author has therefore used this publication as a reference. |

Plamondon, et al., (2011) | Describe and analyze macroeconomic problems that affect the dynamics of the flow of contributions and benefits. | The main conclusion of the authors was the positive relationship between the rate of wage progression, level of economic development and the compound discount rate. | Actuarial practice in social security, this work, has had an impact on the study of social security from an actuarial point of view. |

Afonso & Lima, (2011) | Include aspects of actuarial math as it involves biometric risk and compound rebate. | It concluded that, while maintaining biometric and macroeconomic factors, income greatly affects the contribution profile of the individual associated with the RGPS. | Analysis of the distribution aspects of the pension according to the contribution period of the INSS with the occupation of actuarial mathematics, |

Lima, Wilbert, Pereira, & Paulo, (2012) | Examine the social security factor, one of the most important parameters to calculate how much benefit will be received for the contribution period. | The most important conclusion of this work is the positive correlation between the social security factor and the amount of funds that the federal government has already saved in the public coffers for the pensions granted since the social security factor came into force. | The impact of social security on the large number of social security insurers, this article quoted Afonso and Lima (2011). |

Penafieri & Afonso, (2013) | Create an actuarially fair Social Security Factor (FPAJ) that is less “punishable” for early retirement. | The authors examine the actuarial fairness of the social security factor to the contributions of the RGPS. The authors concluded that the social security factor diminishes the value of taxpayers' benefits well beyond the actuarial equilibrium values. The authors create a substitute for the current social security factor. | The discussion of social security equity is an important and much debated topic as it is a factor in social balance and therefore its importance. |

*Source: Created by the authors (2019).*

The complex dynamics of social security systems are discussed extensively by (Iyer, 2002); Gremaud and Patrick (2004). According to these authors, the parameters that influence the social security system can be modeled by assuming the following simplifying hypotheses:

H1: The taxpayer is not unemployed during the contribution period, i.e. his income is referred to as a continuous function.

H2: The values of the economic parameters are considered to be constant throughout the contribution period.

H3: The pensions in question are only valid for the contribution period and do not count as pensions due to illness, disability or death.

The aim of the work does not include the precise assessment of the effects of the economic growth rate, real inflation and wage progression that distinguish professions, education, etc. The author assumed values around the mean values used in the work of Gremaud and Patrick (2004), and it becomes impossible to predict how economic parameters will behave over decades.

In this way, the analyzes, discussions and the methodology of the present in the work could be created from these studies. Therefore, the next section to be presented is the methodology which starts with a brief history of the RGPS, then the actuarial theory for developing important concepts about equity in contributions to benefits will be presented, the next topic will be the routines of the calculations developed through computational resources and dealt with their impact in the calculation of social security.

### 4. METHODOLOGY

The first step of the methodical phase, in addition to the previous study, which was presented in the chapter of the literature review, is the structuring of the computing resources used. The program that was used in this thesis wa*r Matlab*.

The simulation program in *MATLAB *was developed and structured to ask the user four pieces of information: gender of the person (man or woman), starting age, retirement age and salary or first income of entry into the social security system.

Some simplifying hypotheses have been taken into account: Contributions are Initerruptas, without the effects of income tax and inflation on the calculation of contributions and receipts from benefits. Figure 02 shows the input data paste screen right on the command prompt screen when the program is run.

Based on Afonso and Lima (2011) and Giambiagi and Afonso (2009), six simulation events of social security calculations were grouped by gender, three variations for men and three variations for women were developed. All simulation parameters were the same for both sexes with the exception of income (1,200, 2,500 and 5,000).

Figure 2: Matlab prompt*sbilds*chirm with the first data entries for simulation calculations.

From these four input pieces of information, as already shown in Figure 2, the calculations of the RGPS and the actuarial system of social security are carried out one after the other according to the program or the calculation routine.

The next chapter will be the results and discussions, this chapter shows the tables and graphs with the equations presented in methods, as well as the mathematics of the RGPS for the calculation of contribution and retirement versus the calculations of the social security system, according to the rules actuarial science.

### 5. RESULTS AND DISCUSSIONS

### 5.1 INPUT PARAMETERS AND CONSTANTS

The routine of social security calculations is structured into data input and output, as is the behavior of mathematical functions, the routine works from the previously configured constants, thus changing the input data when you change the output data.

Table 1 contains the data entry and constants for the social security calculations of the RGPS and the actuarial pension system, both of which are performed in sequence according to the *in mat*lab available routine can be calculated.

The constants were taken as average values for the initial situation. Undoubtedly, the constants are the crucial part of the work, as well as the most complicated part of justifying their assumed values, hence the assumed values can be justified according to the works of Giambiagi and Afonso (2009); Plamondon, et al. (2011); Gremaud and Patrick (2004) and Afonso and Lima (2011) evaluate macroeconomic figures: growth rate, discount rate, wage progression rate.

The values of the input parameters and the values of the constants are shown in Table 2.

Table 2 - Entry and constant parameters of social security.

Input data | Constants | ||

Entry age (woman) | 30 years old | Economic growth | 1% per year |

Entry age (man) | Discount rate | 6% per year | |

Retirement age (woman) | 60 years old | ||

Retirement age (man) | 65 years old | ||

Starting salary (both) | R $ 1,200.00 | Wage progression rate | 0.25% per year |

R $ 2,500.00 | 0.5% per year | ||

R 5,000.00 | 1.2% per year |

*Source - Prepared by the authors.*

The entry and exit age data were selected according to the age rules according to contribution time according to the information provided by the Insss. The starting salary was set at three levels for both sexes, as the key here is to test actuarial fairness under the same macroeconomic circumstances, with only biometric (gender), entry and retirement age and starting salary electrifying, generally the Class of the individual in low (1,200.00 R), B2 (2,500.00) and high (5,000.00) Auds.

### 5.2 SOCIAL SECURITY CONTRIBUTIONS TO RGPS

Table 3 shows the average contribution flows that result from the social security calculations of the RGPS as a function of retirement age, entry age and starting income. The average contribution hums up significantly due to the initial difference in income between the same-sex wage level.

Table 3 - Average contributions paid to the RGPS by gender and starting income of taxpayers.

man | ||||

Retirement age | Entry age | Initial Income (R / month) | Average posts (R / month) | Grand Total (R) |

65 years old | 30 years old | 1.200,00 | 122,00 | 52.703,68 |

30 years old | 2.500,00 | 300,58 | 129.850,26 | |

30 years old | 5.000,00 | 848,02 | 366.342,67 | |

Mrs | ||||

Retirement age | Entry age | Initial Income (R / month) | Average posts (R / month) | Grand Total (R) |

60 years old | 30 years old | 1.200,00 | 117,99 | 43.891,86 |

30 years old | 2.500,00 | 288,61 | 107.363,77 | |

30 years old | 5.000,00 | 797,28 | 296.587,50 |

*Source - Prepared by the authors.*

The total number differs considerably from man to woman, as the contribution period is shorter for women as they retire 5 years earlier than men. It has not been observed that the application of actuarial concepts in the contributions to the RGPS, which, according to Penafieri and Afonso (2013), represents an actuarial imbalance that jeopardizes the fairness of the contributions to the RGPS.

### 5.3 SOCIAL SECURITY

The calculation of the social security factor was carried out according to equation 2.If the calculations were carried out according to the guidelines already set out in the methodology for the RGPS, the necessary requirements for retirement through contribution period are met, the social security factor is not considered to be a reduction in the value of the old-age pension received by the taxpayer work, while the taxpayer retires before the legal and necessary conditions are met, the social security factor reduces the value of the old-age pension.

The initial income does not affect the value of the social security factor, as suggested in Table 4. The difference in the value for men equal to 1.0000 and for women equal to 0.6561 is reduced to the contribution period by 5 years for women and also due to biometric differences, since men have a lower life expectancy compared to women, according to the IBGE 2017 survival plan.

Table 4 - Social security factor, segmented by gender.

man | ||

Retirement age | Entry age | Social security factor |

65 years old | 30 years old | 1,0000 |

Mrs | ||

Retirement age | Entry age | Social security factor |

60 years old | 30 years old | 0,6561 |

*Source - Prepared by the authors.*

In practice, the decline in social security for women in the RGPS has been predicted as a way to ensure greater equity between the sexes, according to Penafieri and Afonso (2013), but some disagree, according to Afonso and Lima (2011 ) the lower social security factor continues to punish women, exacerbating the historical income gaps between men and women. It is salutary to check whether the social security factor has been worked out partly on the basis of biometric criteria, as it takes into account the life expectancy disclosed by ibGE, although the controversial social security factor discourages early retirement and the longer life expectancy of women with the shorter life expectancy balanced by men.

### 5.4 RGPS SOCIAL SECURITY BENEFITS

The results of the calculations of the advantages according to the parameters in Table 2, generated from Equation 3, and the results shown in Table 5.

Table 5 - Flow of RGPS benefits to taxpayers, broken down by gender and starting income.

man | |||

Retirement age | Entry age | Initial Income (R / month) | advantages (R / month) |

65 years old | 30 years old | 1.200,00 | 1.568,19 |

2.500,00 | 3.450,88 | ||

5.000,00 | 8.062,30 | ||

Mrs | |||

Retirement age | Entry age | Initial Income (R / month) | advantages (R / month) |

60 years old | 30 years old | 1.200,00 | 989,68 |

2.500,00 | 2.160,28 | ||

5.000,00 | 4.931,19 |

*Source - Prepared by the authors.*

It is important to note that men will receive a higher pension value than their own starting income considering that starting income will grow as a result of Equation 5. In the case of women, they receive less old-age benefits than they started to participate in the RGPS. The difference between the behavior of the benefits for men and women is due to the social security factor, since, according to Table 4, men have a social security factor of 1.0000 and women 0.6561.

Given the funds revealed, the behavior of the contributions to the RGPS does not suggest that the actuarial rules are followed, since the contribution is calculated by multiplying the taxpayer's income by the percentages given in Table 1. In relation to the benefits that the RGPS pays to retirees, there is evidence of the existence of actuarial science concepts seeking balance and equity between male and female taxpayers in the thesis, as their biometric properties crystallized in the IBGE 2017 Survival Board are.

The issues of contributions are just as sensitive to public opinion as the benefits obtained through the contribution period because they affect the taxpayer's quality of life in the delicate moment of life when the years weigh down the days of youth, hence the loss the ability to work directly on the income of the person.

### 5.5 THE RGPS COMPARED TO THE ACTUARIAL SOCIAL SECURITY REGIME

This section is devoted to the behavior of social security values from the perspective of actuarial science and is compared to the social security impact values of the RGPS.

Contributions from the perspective of actuarial science are shown in the last column according to Table 6.

Table 6 - Difference between the contributions to the RGPS for the actuarial segregation by sex.

man | ||||

Retirement age | Entry age | Initial Income (R / month) | Average Posts - RGPS (R / month) | Actuarial average contribution (R) |

65 years old | 30 years old | 1.200,00 | 122,00 | 88,43 |

30 years old | 2.500,00 | 300,58 | 206,47 | |

30 years old | 5.000,00 | 848,02 | 483,74 | |

Mrs | ||||

Retirement age | Entry age | Initial Income (R / month) | Average Posts - RGPS (R / month) | Actuarial average contribution (R) |

60 years old | 30 years old | 1.200,00 | 117,99 | 87,30 |

30 years old | 2.500,00 | 288,61 | 190,92 | |

30 years old | 5.000,00 | 797,28 | 438,19 |

*Source - Prepared by the authors.*

There is a noticeable increase in the average contribution values as the initial contribution income increases for both the RGPS and the actuarial system, and it is also clear that the average contribution to the RGPS for the actuary is significant. The considerable difference between the contributions is the result of the principles postulated by actuarial science, in which the contribution rate according to Table 6 is significantly lower in relation to the RGPS rate.

Table 6 - RGPS contribution rate compared to the social security actuarial rate.

Contribution salary (R) | rate | General tariff: employee + company | Actuarial Rate Man | Actuarial Rate Women |

Up to 1,659.38 | 8% | 8%+20% = 28% | 16% | 13% |

From 1,659.39 to 2,765.66 | 9% | 9%+20% = 29% | 17% | 14% |

From 2,765.67 to 5,531.31 | 11% | 11%+20% = 31% | 17% | 14% |

*Source - Prepared by the authors.*

It is evident that, according to Table 6, the actuarial rate is lower than the RGPS rates for both the sexes and the starting income brackets. Giambiagi and Afonso (2009) came to this conclusion.

The contribution system to the RGPS differs greatly in all its aspects from the actuarial regime, as can be seen when looking at the contribution values that have accumulated over the years. The first figure shows the relationship between the contributions to the RGPS and the actuarial contribution regime. The behavior of the curves is markedly divergent, where the RGPS tends to keep increasing its contribution values over time, in contrast to the actuarial regime showing that the contribution values are decreasing. In addition, an important aspect is possible at the interface between the two charts, where from the age of 11 the actuarial contribution scheme becomes financially more advantageous for taxpayers compared to the RGPS.

Figure 1: Contribution to social security systems RGPS and actuarial social security system as a function of time for men with an initial income of 1,200.00 R.

The different behavior of the curves in Figure 1 is justified by the lack of biometric considerations in the accumulation system to the RGPS, in which the actuarial contribution regime obviously shows the presence of actuarial assumptions, as there is a tendency to decrease the taxpayer's probability of survival over time, such as presented in the theoretical framework.

Similar behavior in Figure 1 can be shown in Figure 2. In Figure 2, the woman's initial contribution income is almost R 2,000.00 and R500.00 near the 30th year of contribution and the women in Figure 2, but the difference is in the total insurance contribution, which is higher, taking into account the 1st contribution to the total value, is between R 2,000.00 and R 2,500.00 near the middle of the two values (R 2,250.00).

Figure 2: Contribution to social security systems RGPS and actuarial social security system as a function of time for women with an initial income of 1,200.00 R.

Another difference is in Figure 1 in relation to Figure 2 in the time of the intersection between the two curves, in Figure 1 the point of intersection is close to the 11 contribution years, in which in Figure 2 there are almost 9 years.

The difference in the actuarial contribution period of Figures 1 and 2 also follows the same logic as Figures 3 and 4. The initial income of 2,500.00 R. men and women follows the same characteristics of the actuarial curve of Figures 1 and 2. Figure 3 below .

Figure 3: Contribution to social security systems RGPS and actuarial social security system as a function of time for men with an initial income of 2,500.00 R.

The difference is in the total contributions, as the higher the starting income the contributions will be, as men have a lower life expectancy and retire later, so the numbers 1, 2, 3 and 4 show that men are in actuarial System contribute higher values each year in relation to women.

Figure 4: Contribution to social security systems RGPS and actuarial social security system as a function of time for women with an initial income of 2,500.00 R.

Figures 5 and 6 behave similarly to the other figures, but the starting salary range is R5,000.00 for both genders in Figures 5 and 6.

The impact of Initial Income on Aturais Contributions shows that the higher the Initial Income, the higher the actuarial contributions. Men who contribute R5,000.00 contribute a little more than R10,000.00 in the first year, and women a total of just under R10,000.00 in the 1st year.

Figure 5: Contribution to social security systems RGPS and actuarial social security system as a function of time for men with an initial income of 5,000.00 R.

Regarding the behavior of the intersection between the contributory system of the RGPS and actuarial mathematics in Figures 5 and 6, Figure 5 of the men, the value of the intersection is almost 9 years and how much of the female figure 6 is almost 5 years. This divergence in the behavior of the intersection can be justified given the actuarial characteristics between men and women.

The man shown in Figure 5 differs from the men shown in Figures 1 and 3 because the macroeconomic characteristics that reflect the contribution and time to the contribution system become beneficial relative to the RGPS. If the numbers 5 and 3 in the initial earnings of R 2,500.00 and R 5,000.00 do not have a large difference in the time of the intersection, the difference becomes more noticeable as the income in the time of the intersection increases. This suggests that the higher the income of an individual regardless of gender, the less it will be for the RGPS

Figure 6- Contribution to social security systems RGPS and actuarial social security system as a function of time for women with an initial income of 5,000.00 R.

[/ caption] Figure 6 corresponds to the graph of women with an initial income of R 5,000.00. Observation of this graph is evident that the actuarial system for women is in the range of RG5,000.00 the chart with the RGPS in the Time intercepts close to 5 years, this is the lowest subject of interception between the curves detected in all numbers from 1 to 5. The actuarial contribution system is therefore more advantageous than the RGPS for women who earn an initial income of R 5,000.00 from the 5 years of contribution.

The graphs suggest the same trend, the contribution to the RGPS is in contrast to the actuarial regime, in cases where there has been a strange behavior, the contributions to the RGPS start at a lower level with respect to the actuary as the time the Contributions to the RGPS increased while contributions to the actuarial regime decreased. This behavior is justified in view of the characteristics of each individual system.

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