abstract

This paper mainly studies the methods and modes of pension system reform for retired employees in enterprises. In the research process, we combine the knowledge of probability theory, mathematical statistics, least square method, linear regression and computer, and use MATLAB mathematical software to study the important factor of pension insurance fund balance-substitution rate according to the actual situation of pension insurance system reform for retired employees. The replacement rate of our research is 42.9%, which is close to the balance of pension funds in Shandong Province.

Question 1: Make a scatter diagram of the average population wage from 1996-20 10 through statistical data with MATLAB; As can be seen from the scatter plot, the mathematical model is approximately a straight line Y = KX+B. Finally, we decided to use the least square method to estimate the unknown parameters K and B in the model to test, predict and control the regression model. From the established linear regression model, it can be seen that the sum of F is obviously too large, indicating that the regression straight line has a high degree of fitting to the sample data, which can be found in the F distribution table (1, 13). It shows that the linear regression effect of regression equation is remarkable, and the final comprehensive answer is 75992.2 yuan.

Question 2: Using MATLAB software to calculate the total income and average wage of enterprises respectively, we can calculate the ratio of the wages of employees of all ages to the average wage of enterprises in 2009. That is, the ratio of employees aged 20-24 to employees aged 55-59 to the average wage of the enterprise is 0.669236, 0.804927, 0.9825 14, 1.066668, 1. 172806 and 656, respectively.

If these proportions are taken as the reference value of the employee contribution index, considering that the employees of this enterprise began to pay pension insurance at the age of 30 and 40 respectively from 2000 to retirement (55, 60 and 65), the pension replacement rate under different conditions is calculated according to = =.

Question 3: Calculate the remaining personal wages, personal contributions and interest deposited in the bank after the social pooling fund account from 2000 to 2024 according to the known data. Get the funding gap under different conditions.

Question 4: Based on all the data, if we want to achieve the target replacement rate and maintain the balance of pension funds, we can divide it into three situations.

1. If we want to approach the target replacement rate, we need to increase the proportion of "social pooling" and "personal account".

2. If the proportion of "social overall planning" is slightly increased, it will be closer to the target replacement rate.

3. If the proportion of "social pooling" and "personal account" is reduced, it is necessary to extend the retirement age of employees, and the time for employees to enjoy pension benefits will be shorter.

Keywords: pension personal account pension replacement rate balance of payments social pooling fund

I. Background and restatement of the problem

1. 1 background of the problem

Pensions are used to protect the basic needs of employees after retirement. At present, China's endowment insurance fund is paid to the social pooling fund account at 20% of the total wages of employees, and then paid to the personal account at 8% of the personal wages. After retirement, according to the ratio of the monthly or annual payment wages of employees to the average social wage, taking into account the average social wage in the year before retirement and other factors, funds are allocated from the social pooling account, and a certain proportion of funds in the individual wage account are used as the monthly pension after retirement. When an employee dies, the funds in the social pooling account will not be returned to the employee, and the balance in the personal account can be inherited. The deposit amount in personal account bears interest according to the one-year deposit rate published by the bank at that time, and the interest rate is uniformly set at 3%.

Pension payment is related to many factors when employees are on the job. With the rapid development of China's economy, it is expected to reach the level of moderately developed countries in the middle of this century. At present, the goal of China's endowment insurance management is to balance the income and expenditure of the endowment insurance fund, and an important factor affecting the income and expenditure balance of the endowment insurance fund is the replacement rate, and the future target replacement rate is determined to be 58.5%.

Restatement 1.2 problem

Question 1: Through the prediction and analysis of China's economic development and wage growth in the next 30 years, it is concluded that the unknown parameters k and b in the model can be estimated by the least square method, and the regression model can be tested, predicted and controlled, and the average annual wage of employees in Shandong Province from 20 1 1 to 2035 can be predicted by hypothesis.

Question 2: If we want to calculate the proportion of the wages of employees of all ages in the enterprise to the average wage of the enterprise in 2009, we should first consider the wages of employees of all ages and the average wage of the enterprise, and finally determine their proportion. If we continue to take these proportions as the reference value of the employee contribution index, consider that the employees of this enterprise began to pay pension insurance at the age of 30 and 40 from 2000 to retirement (55, 60 and 65), and sort out the data through Excel to calculate the pension replacement rate in various situations.

Question 3: How to adjust the payment ratio of "social pooling" and "individual account" so that it can not only achieve the target replacement rate, but also maintain the balance of pension funds.

Question 4: What measures do you think can be taken to achieve the target replacement rate and maintain the balance of pension funds? Please give your reasons.

Second, the model hypothesis.

2. 1 Assuming that there will be no large-scale casualties or migration caused by war or natural disasters during the forecast period, China has made remarkable progress in environmental protection, domestic demand-driven, narrowing the income gap and giving play to the leading role of market mechanism, that is, China's economy will grow steadily after 20 13 years.

2.2 Due to political, economic and cultural factors such as 1978- 1996, it is assumed that the average wage of employees is predicted from 1996.

2.3 Assume that the salary growth rate of employees continues after 20 13, and the growth rate is 1996-20 10.

2.4 It is assumed that all benefits and bonuses of employees' total wages are reflected in individual wages.

Third, the symbol description

X: year

average wage

A: Basic pension

Personal account pension

D: salary before retirement

W: Employees' pensions when they just retired.

: national/provincial/local social average wage of the insured before retirement 1 year, 2 years.

: 1 year, 2 years, ... m years before retirement.

M: The number of years that enterprises and employees actually pay the basic old-age insurance premium.

N: Number of months of personal account pension.

: contribution index of the first year before retirement, I = 1, …, m.

I index the average monthly salary.

: substitution rate

Fourthly, the analysis and establishment of the model

4. 1 question 1

In the next 30 years, under normal circumstances, assuming that China's economic growth rate will maintain an average annual rate of 8%, the United States' economic growth rate will be 2%, and Japan's economic growth rate will still be 2%, then by 2040, China's GDP will be about 50.3 1 trillion US dollars, the United States will be about 27. 17 trillion US dollars, and Japan will only be about 9.24 trillion US dollars. In other words, 30 years later, China's GDP will be at least 1.4 times that of the United States and 5 times that of Japan. Although there is still a huge gap between China, the United States and Japan in terms of per capita development, China has an absolute advantage over the United States in terms of national resources that can be upgraded. From this point of view, the future development prospect of China is bright.

Through the analysis of China's future, we predict that the annual average wage model of Shandong employees in 201-2035 is similar to the population growth model. So we can use some knowledge of population growth model to solve it, and use MATLAB to make a scatter diagram of the average salary of Shandong employees of 1996-20 10.

1) in the MATLAB running window input:

x =；

Multivalued (p, x)

ans =

Column 1 to 5

305 14. 120000000 1

32409.04

34303.96

36 198.8800000004

38093.8000000003

Column 6 to 10

38093.8000000003

39988.7200000002

4 1883.640000000 1

43778.560000000 1

45673.48

Column 1 1 to 15

47568.4000000004

49463.3200000003

5 1358.2400000002

53253. 160000000 1

55 148.080000000 1

Column 16 to 20

57043

58937.9 199999999

60832.8400000003

62727.7600000002

64622.6800000002

Columns 2 1 to 25

665 17.600000000 1

684 12.52

70307.4399999999

72202.3600000003

74097.2800000003

Column 26

75992.2000000002

Therefore, according to the above data, it can be predicted that the average annual salary of employees in Shandong Province from 201kloc-0/to 2035 is as follows:

4.2 Question 2

According to the existing materials and data (attachment), if we want to calculate the proportion of the wages of employees of all ages in this enterprise to the average wages of this enterprise in 2009, we need to make various assumptions about the fact that an employee of this enterprise began to pay pension insurance at the age of 30 in 2000 until he retired (55, 60 and 65 years old), and began to receive pension from retirement until he died at the age of 75. The specific age groups are classified as follows:

Through the application software, the total income and average wage of the enterprise are calculated respectively, so as to calculate the proportion of the wages of employees of all ages in the enterprise to the average wage of the enterprise in 2009. The ratio of the wages of employees aged 20-24 and employees aged 55-59 to the average wage of the enterprise is as follows:

0.669236、0.804927、0.9825 14、 1.066668、 1. 172806、 1.266624、 1.208664、 1. 155042

The pie chart is as follows:

If these proportions are taken as the reference values of the employee contribution index, it is considered that the employees of this enterprise began to pay pension insurance at the age of 30 and 40 respectively from 2000 to retirement (55, 60 and 65), so as to calculate the pension replacement rate under various circumstances.

Replacement rate: = = =

Basic pension: a=( +S)/2*m* 1%

Personal account pension: b = [(+...+) * 8%] * [1+]/n

Pension for employees just after retirement: W = A+B.

Combined with the above related data:

So the replacement rate: =

Through the application software, the pension replacement rate can be calculated under various circumstances, and the results are as follows:

All information is obtained in the same way: (See Annex I for details)

Age group (years)

substitution rate

30-55

0.42023 1

30-60

0.5488 13

30-65

0.64476

40-55

0.257756

40-60

0.36862

40-65

0.429344

4.3 Question 3

4.3. 1 Calculate the pension fund gap

Suppose an employee began to pay pension insurance at the age of 30 in 2000 until he retired (55, 60, 65), and received pension insurance after retirement until he died at the age of 75. The so-called gap refers to the balance of payments when the endowment insurance fund cannot make ends meet. In order to calculate the gap, it is necessary to calculate the balance of income and expenditure of the endowment insurance fund.

4.3.2 According to the known conditions in the topic:

1) The pension paid by enterprise employees before retirement consists of two parts: the enterprise pays the social pooling fund account according to a certain proportion (20%) of the total wages of employees, and the employees pay the personal account according to a certain proportion (8%).

2) The retirement pension of the employees of this enterprise consists of two parts:

Pension = basic pension+personal account pension

Personal account pension = personal account savings ÷ months (see table 1 for data).

Basic pension = (last year's average monthly salary of employees in the province+my average monthly payment salary) ÷2× payment period × 1%

My indexed monthly average payment salary = last year's average monthly salary of employees in the province × my average payment index.

4.3.3 Calculate the endowment insurance fund that employees need to pay before retirement. Suppose an employee retires at 55 and dies at 75. For example, the average salary in 2000 was 8772, and the remaining personal salary after being deposited in the social fund-raising account was 8772 * (1-20%) = 7017.600.

① Individual contribution is 7017.600 * 8% = 561.41.

② The interest deposited in the bank is 561.41* =11.

Then, according to the above method, the personal salary, personal contribution and interest paid in the bank after the social pooling fund account from 2000 to 2024 can be calculated as follows:

Enter the edit field through the software (the interest rate is uniformly set at 3%).

The total social balance in 2025 is SUM(C2:C26)*20%= 160707.

The balance of personal account is sum (E3: F27) =119376.64.

The balance of personal account after interest is sum (E3: F27) =119376.64.

The basic pension is (c27/12+5236.565)/2 * 25 *1%*12 =14748.36.

Personal account pension is $ H $28/170 *12 = 8426.59 (unchanged).

According to the above table, we can get the social overall balance, personal account balance, personal account balance after interest, basic pension and personal account pension from 2025 to 2044 (when employees died at the age of 75).

The following figure

1) has been receiving a pension since retirement and died at the age of 75. Calculate the gap of endowment insurance fund. According to the meaning gap, it is a loss. The loss after 2025 is whether the sum of the social pooling fund account balance and the individual account balance is greater than or equal to 0 after interest. If it is less than 0, it means that it has lost money, otherwise it means that there is no loss. As can be seen from the above figure, there was no loss from 2025 to 2037, and there was a loss from 2038 to 2044.

2) Calculate the age at which employees receive pensions, so that the pension insurance funds they pay and the pensions they receive can make ends meet. According to the calculation results in the table, you can draw the following chart to know the age when employees receive pensions, and achieve the balance between the pension funds they pay and the pensions they receive.

As can be seen from the above figure, when an employee receives a pension until he is 73 years old, the pension fund he pays and the pension he receives will break even.

3) Suppose employees retire at the age of 60 and die at the age of 75.

Combine the above methods as shown below:

The available loss (gap) is that there is no loss in 2030-2040 and there is a loss in 2042-2044.

Whether the pension insurance fund paid and the pension received have reached the balance of payments, see the following figure:

According to the chart, when an employee receives a pension until he is 74 years old, the pension fund he pays and the pension he receives will break even.

4) Suppose the employee retires at the age of 65 and dies at the age of 75.

According to the above method, the balance of payments and the number of years are shown in the table below.

The available loss (gap) is from 2035 to 2044, and all are not lost.

The pension insurance fund paid by it and the pension it receives are balanced. As can be seen below:

As can be seen from the chart, the employee's pension is not lost from retirement to death, and his pension fund and his pension can't make ends meet.

4.4 Question 4

4.4. 1 If we want to achieve the target replacement rate of 58.5% and maintain the balance of pension funds, according to the model data we have established, the problems faced by enterprise employees in achieving the target replacement rate and maintaining the balance of pension funds are as follows:

Analysis of the situation of reaching the target replacement rate at each retirement age

30-55-75

20% lower

8% lower

Below the target replacement rate of 58.5%

30-60-75

20% lower

8% ping

Close to the target replacement rate of 58.5%

30-65-75

20% higher

8% higher

58.5% higher than the target

Note: 30-55-75 means paying endowment insurance at the age of 30, retiring at the age of 55 and dying at the age of 75. 20% is low, that is, the enterprise will pay 20% of the total wages of employees to the social pooling fund account according to a certain proportion, and 8% is low, that is, the individual wages of employees will be paid to individual accounts according to a certain proportion, and the replacement rate below the target means that the replacement rate of 30-55-75 is lower than the basic old-age insurance stipulated by the state.

30-60-75 series and 30-65-75 series can refer to 30-60-75 series.

4.4.2 Analysis of measures taken (1)

Series 30-55-75: If the replacement rate is close, it needs to be increased by 20% and 8%, and the retirement age needs to be delayed to get close to the replacement rate.

30-60-75 series: close to the replacement rate, which needs to be slightly increased by 20%. This series is close to the replacement rate.

Series 30-65-75: If the replacement rate is close, it needs to be reduced by 20% and 8%. Moreover, this series has a larger retirement age, more payment and shorter time to enjoy pension benefits, so it is necessary to extend the length of service.

Age group (years)

substitution rate

30-55

0.42023 1

30-60

0.5488 13

30-65

0.64476

40-55

0.257756

40-60

0.36862

40-65

0.429344

Take measures analysis (2)

Three problems in the reform of the basic old-age insurance system for employees;

1) There is no corresponding fund in the employee's basic old-age insurance account, and the employee's basic old-age insurance personal account is only credited with the money transferred from individual contributions and enterprise contributions according to regulations, which has been used to pay retirees' pensions, and the personal account is actually virtual. Therefore, the preservation and appreciation of individual accounts of employees' basic old-age insurance plays a certain role in achieving the target replacement rate.

2) According to the current personal account amount, it is potentially impossible to calculate the interest by referring to the bank's statistical deposit interest rate every year, and it is difficult to ensure the realization of the pension replacement rate target through the preservation of personal account funds.

3) The birth rate in China is declining, the population with working ability is gradually decreasing, people's life expectancy is increasing, and social endowment insurance is getting heavier and heavier. For example, in question 4, the replacement rate of 30-55-75 series is lower than the target replacement rate by 58.5%, partly because of the early retirement age, while the replacement rate of 30-65-75 series exceeds the target replacement rate, partly because of the late retirement age.

Because the rate of maintaining and increasing value is linked to the average wage growth rate, we can use reverse thinking and reverse algorithm to realize the target replacement rate, and adopt the replacement rate formula T=Y/J( 1).

Geometric progression's general term A = A (1+K) The sum of the first n terms in the proportional series is a * (R- 1)/R- 1.

(t is the pension replacement rate, y is the monthly pension when employees retire, j is the monthly salary income when employees retire, a is the multiple of the monthly salary income when employees retire, a is the monthly salary income when employees are on the job, k is the expected average salary growth rate, and n is the average working years of employees).

Therefore, we first determine the value-added rate of individual accounts according to the ratio greater than the average wage growth rate of social workers and the value-added rate of individual accounts in the model, and realize "virtual value-added", which is used as the standard for determining the account pension when employees retire, so as to get the idea that the pension "replacement rate" reaches the target replacement rate. Second, individual account funds can maintain and increase the value in the true sense according to the measures stipulated by the state, and improve the operational efficiency of pension funds. Third, the state has taken measures and promulgated regulations to improve China's old-age insurance system.

Evaluation and improvement of verb (abbreviation of verb) model

The advantage of the above model lies in: in the process of modeling, combining with China's current national conditions and the prediction of future development, applying the knowledge of probability theory, mathematical statistics, least square method, linear regression and computer, and using MATLAB mathematical software, the substitution rate, an important factor in the reform and balance of pension system for retired employees, is studied. The replacement rate of our research is 42.9%, which is close to the balance level of pension funds in Shandong Province.

This helps us to build a model; In order to explain the problem more vividly in the solution of the model, the specific position of this point is calculated effectively by using MATLAB and Excel software, and the best result is calculated for the calculation of complex functions, so that the problem we solved can be answered satisfactorily.

The models established in this paper are few and relatively simple, so they are concise and to the point. As a student, I don't know much about the endowment insurance fund, which affects our lack of modeling. It takes three days to build the model first, which is a bit hasty. Some steps are not perfect, and there is not enough time to think deeply. After that, we need to continue to study the model and process the data, add more factors that affect the pension fund, and establish a more complex model.

refer to

1 piece of jewelry. Mathematical modeling in mathematical experiments. Beijing: Higher Education Press, 2005.

2 jie feng et al. Principles and Cases of Mathematical Modeling. Beijing: Science Press, 2007.

3 Jiang Qiyuan et al. Mathematical Model (3rd Edition). Beijing: Higher Education Press, 2003.