Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 576213, 7 pages

http://dx.doi.org/10.1155/2014/576213

Research Article

Optimization of the Actuarial Model of Defined Contribution Pension Plan

Yan Li,¹ Yuchen Huang,¹ and Yancong Zhou²

¹School of Insurance and Economics, University of International Business and Economics, Beijing 100029, China

²School of Information Engineering, Tianjin University of Commerce, Tianjin 300134, China

Received 17 March 2014; Accepted 20 May 2014; Published 12 June 2014

Academic Editor: Xiaochen Sun

Copyright © 2014 Yan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The paper focuses on the actuarial models of defined contribution pension plan. Through assumptions and calculations, the expected replacement ratios of three different defined contribution pension plans are compared. Specially, more significant considerable factors are put forward in the further cost and risk analyses. In order to get an assessment of current status, the paper finds a relationship between the replacement ratio and the pension investment rate using econometrics method. Based on an appropriate investment rate of 6%, an expected replacement ratio of 20% is reached.

1. Introduction

China’s current social security system was established July 1, 1997, which was updated in 2005. When it comes to supplemental pension plan, it is used as a way to provide more benefits over the social security system which is discussed in [1]. The supplemental pension plan functions in the following two fields. The first one is to complement the insufficient social pension insurance treatment. The second one is to stabilize employees and keep enterprises employees focus on their work. Retirement plans of supplemental pension are divided into defined benefit and defined contribution plans. In a defined benefit plan, the funding is intended to ensure that sufficient pension funds will be available to pay promised benefits. In a defined contribution plan in [2], the benefit received equals the contributions to the plan, adjusted for possible investment.

Though China supplemental pension plan has been carried out commonly, there are some large troubles. In 2011, the replacement ratio of whole pension plan was only 42.9%. When reaching 80%, we will judge that the protection of retirement employees is adequate. Some researchers expect that the average replacement ratio of developed supplemental pension system will reach 20%, which is appropriate.

Foreign researches on supplemental pension plan are wide. Leime and Seling regarded supplemental pension plan as a deferred survival annuity. Within the prescribed period of time, the insured is to pay a fee every year to the insurer, which will be maintained until the insured retires. If the insured is still survival at retirement, the insured can achieve a certain amount from the insurer each year until death. The actuarial present value of the insured’s contribution equals the actuarial present value of the insurer’s future payments.

In this paper, application is given priority. We try to compare different types of defined contribution pension plans and test their effectiveness under the consideration of costs and risks. All of these try to give some explanations of the actual application of the pension plan and provide some suggestions for China pension market.

2. Data and Assumption

The sample data used in our analyses is contained in Table 1, simply the basic information of employees, which comes from a certain governmental institution. The institution has 15 executive management level employees.

Table 1: Sample employees information.

ID
Name
Gender
Present age
Entry age
Present salary

per month

1
A
F
54
46
29190

2
B
M
47
45
39591

3
C
M
46
32
65100

4
D
M
44
32
29190

5
E
F
42
34
26625

6
F
F
46
38
23000

7
G
M
40
31
39591

8
H
F
36
26
15954

9
I
M
40
28
27800

10
J
F
50
36
5023

11
K
M
44
37
27957

12
L
F
50
32
4556

13
M
F
37
30
27957

14
N
M
49
39
26625

15
O
F
47
30
16751

Resource: government institution sample data.

The China life insurance industry experience life table (2000–2003) is also used in the spreadsheet model.

As discussed by Bills and Lyon, the following assumptions are also used in the model in [3].

(i)In this paper, all of the pension plans are based on mechanism of defined contribution model.(ii)The present time is 7/1/2013.(iii)The data of annuity factor in life table is used for calculations.(iv)The retirement age for male is 60, for female 55.(v)The retirement benefits are paid until death, with a discount rate of 80%. This assumption presents the possibility of deducting benefits and potential risks.(vi)The fixed rate of monthly salary contribution is 10%, which is a general level.(vii)The future salary increase rate per year is related to future services year as Table 2 shows.(viii)The interest rate is 5% per year, 0.41% per month. The risk-free rate varies from 5% to 6%. The minimum one, 5%, is first chosen in the model analysis. This assumption is performed in the following sensitivity analysis.(ix)The Investment return rate is 5%. The investment of China supplemental pension individual accounts has many limits. The percentage of fixed interest products (mainly the Treasury bond) must be greater than 50%. The percentage of equity products must be less than 30%. In particular, stocks must be less than 20%. In practice, the expected invested return is 5.5%. We assume that it is 5%.

Table 2: Assumption of salary increase rate.

Future service years
Salary increase rate

0 < Year ≤ 5
10%

5 < Year ≤ 10
8%

10 < Year ≤ 15
6%

Year > 15
5%

Resource: spreadsheet model assumption.

3. Model Description (The Spreadsheet Model Is Contained in Table 3)

Table 3: Summary.

ID
Gender
Present

age
Entry

age
Present

salary

per month
Original

DC
Modified

DC
Hybrid

plan
Modified

cost
Hybrid

cost

1
F
54
46
29,190
0.79%
5.72%
24.92%
342,531.00
1,409,256.00

2
M
47
45
39,591
8.02%
9.16%
9.47%
99,636.00
192,012.00

3
M
46
32
65,100
8.57%
19.40%
8.57%
1,569,581.00
0.00

4
M
44
32
29,190
10.39%
20.43%
12.56%
571,364.00
215,232.00

5
F
42
34
26,625
6.87%
11.43%
14.09%
312,432.00
746,028.00

6
F
46
38
23,000
4.56%
8.56%
17.40%
269,894.00
1,074,372.00

7
M
40
31
39,591
12.91%
19.86%
12.91%
536,557.00
0.00

8
F
36
26
15,954
10.52%
17.32%
19.84%
246,680.00
683,952.00

9
M
40
28
27,800
12.91%
22.95%
12.91%
544,157.00
0.00

10
F
50
36
5,023
2.68%
11.08%
98.89%
121,106.00
1,416,276.00

11
M
44
37
27,957
10.39%
15.52%
13.11%
279,672.00
258,828.00

12
F
50
32
4,556
2.68%
14.74%
109.03%
157,799.00
1,419,936.00

13
F
37
30
27,957
9.98%
14.38%
11.89%
279,672.00
233,820.00

14
M
49
39
26,625
6.89%
14.04%
15.83%
411,674.00
704,244.00

15
F
47
30
16,751
4.13%
15.30%
25.80%
532,804.00
1,222,500.00

The work is completed using a basic “defined contribution plan” spreadsheet model. The relative symbols are contained in Table 4. Three different defined contribution plans are tested. The analysis is performed using a spreadsheet model, entering the established assumptions as parameters into the model. Using the Input fields, various parameters related to the employees’ information are entered. The model is used to calculate replacement ratios at retirement age for an individual employee under the conditions of different plans.

Table 4: Symbol determination.

Employer contribution per month
EC

Annuity factor (life table)
AF

Years before retirement

Replacement ratio
RR

Initial account amounts
IA

Accumulated account balance
AA

Retirement benefits per month
RB

Accumulated value of annuity
Sn

Salary per month

Promised benefits
PB

Resource: model symbol determination.

Calculation Original Mechanism. See Figure 1.

Figure 1: Defined Contribution Plan Mechanisms. Resource: Model Calculation Analysis.

Symbol Description. See Table 4.

Calculation Basic Formula.

(i)Original defined contribution plan: ; .(ii)Modified defined contribution plan: ; .(iii)Hybrid plan: ; ; .

4. Analysis

4.1. Sample Human Demographic Analysis

Human demographic information is an important external factor for a pension plan. We do some analyses on the data of 15 executive management level employees and show some findings and graphics below.

(i)The average age of employees is high, 44.80. For men, the average age is 44.28; for women, the average age is 45.25. As the women will get retired 5 years earlier than men, the pressure of pension plan for women is greater.(ii)All employees are under the same social security system, which I don’t consider in my analyses. The defined contribution pension plan is used for supplemental pension plan.(iii)Some employees have high age with shorter future service years. 5 employees will retire in 10 years. Their salary is on the low side, compared to the younger. The monthly salary of two employees aged 50 is less than 5,500. The future accumulated years for individual accounts may not be enough.(iv)The salary of the employees who have a longer past services year (more than 10 years) is more different, 65,100 as the maximum and 4,556 as the minimum. The salary for shorter service years’ employees is more stable.

4.2. Defined Contribution Pension Plan

4.2.1. Original Defined Contribution Pension Plan

A defined contribution plan will provide many details to determine the annual contribution made by employer on behalf of each employee as well as any relevant regulations. In analysis, contribution rate is assumed. The formula determines how much an employee receives from each plan.

Based on the data, assumptions, and model above, the replacement ratio of all employees are calculated under original defined contribution plan. First, we calculate the expected accumulated individual account at retirement age for each employee. Then, with the annuity factor provided by life table, we estimate the future retirement benefits. Finally, the replacement ratio is obtained.

The results are summarized in Table 5. The results vary from 0.79% (the oldest one) to 12.91% (the youngest one). We can almost see that “the younger, the better”. The average replacement ratio is 7.49%, far less than the average replacement ratio of China, 20%. In fact, no employees will have a reasonable replacement ratio, over 20%.

Table 5: Replacement ratio for original defined contribution pension plan.

ID
Gender
Present

age
Entry

age
Present salary

per month
Original replacement

ratio

1
F
54
46
29,190.00
0.79%

2
M
47
45
39,591.00
8.02%

3
M
46
32
65,100.00
8.57%

4
M
44
32
29,190.00
10.39%

5
F
42
34
26,625.00
6.87%

6
F
46
38
23,000.00
4.56%

7
M
40
31
39,591.00
12.91%

8
F
36
26
15,954.00
10.52%

9
M
40
28
27,800.00
12.91%

10
F
50
36
5,023.00
2.68%

11
M
44
37
27,957.00
10.39%

12
F
50
32
4,556.00
2.68%

13
F
37
30
27,957.00
9.98%

14
M
49
39
26,625.00
6.89%

15
F
47
30
16,751.00
4.13%

Average
7.49%

Resource: model calculation results.

The accumulated period (future service years) of individual accounts is adequate, certainly including the sample company. The pure pension plan, defined contribution, can’t provide appropriate retirement benefits coverage.

4.2.2. Modified Defined Contribution Pension Plan

According to the results in Table 5, we can conclude that the supplemental retirement compensation, simply the defined contribution pension plan, is not enough for employees and more compensation expenses are necessary. Some solutions are proposed.

Considering the past services years for each employee, there should be an initial accumulated individual account. This amount will also be invested until retirement. We assume that the past salary is the present salary provided and the contribution rate is still fixed 10%. Therefore, the accumulated individual account at retirement age, the expected replacement ratio, will increase. The results are summarized in Table 6.

Table 6: Replacement ratio for modified defined contribution pension plan.

ID
Gender
Present

age
Entry

age
Present salary

per month
Replacement

ratio

1
F
54
46
29,190.00
5.72%

2
M
47
45
39,591.00
9.16%

3
M
46
32
65,100.00
19.40%

4
M
44
32
29,190.00
20.43%

5
F
42
34
26,625.00
11.43%

6
F
46
38
23,000.00
8.56%

7
M
40
31
39,591.00
19.86%

8
F
36
26
15,954.00
17.32%

9
M
40
28
27,800.00
22.95%

10
F
50
36
5,023.00
11.08%

11
M
44
37
27,957.00
15.52%

12
F
50
32
4,556.00
14.74%

13
F
37
30
27,957.00
14.38%

14
M
49
39
26,625.00
14.04%

15
F
47
30
16,751.00
15.30%

Average
14.66%

Resource: model calculation results.

From the results, we can see that the compensation expense is a huge expenditure for employer. Some employees have served for long years. The modified defined contribution plan may be easy to accept. The initial account is a big amount.

However, the replacement ratio is increased but it is still unexpected, almost still less than 20%, possibly because of the future increased salary and the inadequate investment return. The average replacement ratio of this plan is 14.66%, with an average increase of 7.17%. Employees 4, 7, and 9 may be satisfied, as their length of service is long and their salary is not too high. With a relatively low salary, the pressure of pension benefit will be much smaller.

4.2.3. Hybrid Pension Plan

The hybrid pension plan will guarantee the profits more effectively. In this plan, the benefit formula looks like a defined contribution plan, but the plan is really a defined benefit plan. The plan administrator keeps track of the monthly contributions into the account and guarantees a crediting pension level. If the employees earn less, the plan must fund the shortfall.

Our sample company is a governmental one, reasonably with better pension care. As some employees’ estimated monthly pension is not enough, the following hybrid pension plan is more direct. We assume that every employee should have a credit monthly pension of 8,000. Therefore, it is necessary to replenish future pension for those with less than certain 8,000. Thus the expected replacement ratio for certain employees will increase. The results of hybrid pension plan are summarized in Table 7.

Table 7: Replacement ratio for hybrid pension plan.

ID
Gender
Present

age
Entry

age
Present salary

per month
Replacement

ratio

1
F
54
46
29,190.00
24.92%

2
M
47
45
39,591.00
9.47%

3
M
46
32
65,100.00
8.57%

4
M
44
32
29,190.00
12.56%

5
F
42
34
26,625.00
14.09%

6
F
46
38
23,000.00
17.40%

7
M
40
31
39,591.00
12.91%

8
F
36
26
15,954.00
19.84%

9
M
40
28
27,800.00
12.91%

10
F
50
36
5,023.00
98.89%

11
M
44
37
27,957.00
13.11%

12
F
50
32
4,556.00
109.03%

13
F
37
30
27,957.00
11.89%

14
M
49
39
26,625.00
15.83%

15
F
47
30
16,751.00
25.80%

Average
27.15%

Resource: model calculation results.

The hybrid pension is better than the modified pension. The 8,000 guarantee per month is directly added to the future retirement benefits, more effectively. The average replacement ratio of hybrid pension plan is 27.15%, with an average increase of 19.66%.

The minimum replacement ratio is 8.57%, with no change, due to his maximum salary. The replacement ratio of employees 10 and 12 even exceed 95% because of their relatively low salary. Just due to the low salary, their initial account in modified defined contribution plan is small, ineffectively.

Up to now, three defined contribution pension plans have been analyzed above. The comparison graph is shown in Figure 2. It is obvious that, the replacement ratio is improved under the two improved plans, especially the hybrid plan.

Figure 2: The replacement ratios of three defined contribution pension plans. Resource: spreadsheet model calculation results.

4.3. Cost Analysis

The modified defined contribution pension plan and the hybrid pension plan can be regarded as improved defined contribution plan. There is an estimated compensation expense for this improvement.

In modified defined contribution pension plan, the sample employer must supplement the initial individual account value at once. Employees with longer past service years require more expenditure. Certainly, there is a time value for this compensation. In this paper, we just measure the present value of the required costs.

In hybrid pension plan, the compensation expenses only appear for the employees who need to be assisted. There is a measurable amount for employers. With this added amount, the monthly 8,000 pension benefits will be reached. In order to compare these two plans’ compensation expenses, they are both measured by the use of present value and completed in the model.

The detailed results are summarized in Table 8.

Table 8: Present value of compensation expenses.

ID
Gender
Present

age
Entry

age
Cost for modified defined contribution plan
Cost for hybrid

plan

1
F
54
46
342,531.00
1,409,256.00

2
M
47
45
99,636.00
192,012.00

3
M
46
32
1,569,581.00
0.00

4
M
44
32
571,364.00
215,232.00

5
F
42
34
312,432.00
746,028.00

6
F
46
38
269,894.00
1,074,372.00

7
M
40
31
536,557.00
0.00

8
F
36
26
246,680.00
683,952.00

9
M
40
28
544,157.00
0.00

10
F
50
36
121,106.00
1,416,276.00

11
M
44
37
279,672.00
258,828.00

12
F
50
32
157,799.00
1,419,936.00

13
F
37
30
279,672.00
233,820.00

14
M
49
39
411,674.00
704,244.00

15
F
47
30
532,804.00
1,222,500.00

Total cost
6,275,559.00
9,576,456.00

Resource: model calculation results.

According to the results, there is no doubt that more expenses make more benefits. As the average replacement ratio of hybrid plan increases more, the estimated compensation expenses for employer are much larger.

4.4. Risk Analysis

Every retirement pension plan presents several risks and the design of each plan determines how the risks are divided between employers and employees. Employers have the main responsibility on defined contribution plans.

One obvious risk is investment return. If the asset earns more than expected, the employee under this risk is happier. If the asset earns less, there may be a loss of benefits. In a word, the ultimate accumulated accounts depend on investment earnings to some degree. Investment returns help provide for promised benefits. Everyone expects to have an account which is overperformed. This risk is predominant in the actual operation.

Another risk is inflation rate. In this paper, we regard inflation as one component of investment return. The investment return will provide some protection. The inflation depreciates the value of future accumulated accounts.

The next risk is longevity. In this paper, we assume that the future pension benefits will be paid until death. Regardless of the possibility of ignoring this risk, it is traditionally considered that a defined benefit plan puts this risk on the employers and a defined contribution plan puts this risk on the employees. In our opinion, this conclusion is reasonable as a defined benefit plan/hybrid plan guarantees a promised benefit.

Finally, the salary increase risk is considered. The salary at retirement age is a predominant factor which needs to be monitored. If the salary is relatively higher than expected, there is more pressure and more contributions for the certain pension plan. The final influence is uncertain. In the assumption, the salary increase rate is predicted based on the future service years.

4.5. Sensitivity Analysis

In order to measure the potential risks for several retirement pension plans, the following sensitivity analysis is completed. As we assume that the pension benefits will be paid until death, longevity risk is not considered in this part. The results are summarized in Table 9.

Table 9: Results of sensitivity analysis (replacement ratio).

Tested parameter
Changes
Original defined contribution
Modified defined contribution
Hybrid plan

Investment

return
(+1%)
0.55%
1.45%
0.21%

(−1%)
−0.50%
−1.29%
−0.19%

Salary

increase
(+1%)
−0.50%
−1.27%
−1.89%

(−1%)
0.55%
1.46%
2.12%

Resource: model calculation results.

Based on the results in Table 9, it is evident that the employees are under some risks when the investment return rate declines by 1% annually. If the inflation rate increases by 1% annually, it will have a similar impact on the replacement ratios. The investment return risk of hybrid plan is not as severe as the other two pension plans. The modified defined contribution pension plan will be influenced most. This result is easy to understand, in a modified defined contribution pension plan, and the time value of the initial individual account at retirement age is the core of this plan.

The salary at retirement age will have a direct influence on the replacement ratio. The hybrid pension plan is most sensitive to this risk. Most employees have a fixed monthly pension benefit of 8,000 in this plan; the change of the denominator of replacement ratio is much more important. The modified defined contribution pension plan is still influenced by this risk. When the salary increases faster, the accumulated individual account, and thus estimated pension benefit, will be greater.

In conclusion, though the two improved pension plans will get a better result of replacement ratios, the potential risks will increase to some degree. The modified defined contribution pension plan is sensitive to both of the risk parameters. The hybrid pension plan is more sensitive to salary increase rate. When it comes to our traditional defined contribution pension plan, it is the steadiest one, with less fluctuation. As a result, both of the risks need to be monitored.

5. Forecast

In order to forecast the pension replacement ratio, we simplify defined contribution pension plan. The important parameters are kept retained, including investment income rate/interest rate , pension benefit , contribution rate , contribution years , pension paid years , present salary, and salary increase rate , as in [4].

Based on the parameters listed above, we can get the accumulated individual account. We assume that the contribution is made at the end of years and investment income rate is not equal to salary increase rate.

According to the formula (1), the formula of is obtained.

So,

Finally, the replacement ratio

According to the sample data, employee 7 is chosen for example. The future service year is 20 years, . The fixed contribution rate is 10%, . The salary increase rate is 5%, . According to the life table and China pension market, we assume that m equals 18.77 years. Up to now, we find a basic formula relationship between investment return rate and pension replacement ratio. As the investment return rate changes, the replacement ratio will change.

In order to forecast the replacement ratio, Table 10 is shown.

Table 10: Forecast relationship (partial).

Interest rate ()
3%
4%
6%
7%
8%
9%
10%
11%
12%
13%

Replacement ratio (RR)
11.81%
14.04%
19.77%
23.43%
27.73%
32.45%
38.71%
45.69%
53.86%
63.44%

Resource: model calculation results.

It is obvious that the replacement ratio increases as the interest rate and certainly investment return increase. With the method of econometrics, linear regression, the relation between them is presented by “Eviews 5.0”. To minimize the expected error as much as possible, we vary the form of interest rate, using quadratic and cube interest rate as explaining variables. The biquadrate interest rate is not significant, which is discarded.

Regression Formula:

It is clear that the regression formula fits this relationship well. Under the condition that the other factors are unchanged, with the improvement of investment return, the growth of enterprise annuity replacement rate is more significantly higher than the growth of investment return.

Certainly, the improvement of defined contribution pension plan design will increase the replacement ratio. If we devote ourselves to increasing the investment return rate, the replacement ratio will increase too. According to the corresponding linear regression relationship analysis, if the enterprise annuity replacement ratio level is about 25%, the annual investment yields need to exceed 7%.

The investment structure in China is not well developed now. If we want to have a better investment return, some limits must be cancelled. The investment income rate is around 6%, which will result in a replacement ratio of 20%.

In Table 11 there are the historical investment returns in China. The investment return is not satisfying during recent years. In a word, it is significant for us to think how to increase investment benefits as well as guarantee the security.

Table 11: Investment returns rate.

Year
2007
2008
2009
2010
2011
2012

Asset
8.86
35.572
132.795
227.158
326.579
401.98

Investment rate
7.66%
2.6%
4.36%
2.85%
2.53%
5.68%

Resource: the national enterprise annuity fund business data in 2012, Human resources social security fund supervision, 2013.4.

In the analysis, we just assume that 5% is the investment return rate. The actuarial assumption is used in the calculations, but it will never exactly reflect the plan’s experience this year. The difference between actual and assumed experience creates actuarial gains and losses. A gain means the actual experience was more favorable, and less costly, than assumed. A loss means the opposite. If we want to forecast the future replacement ratio, it is important to get reasonable interest rate estimation which is discussed in [5].

6. Conclusions and Recommendations

The traditional defined contribution plan cannot satisfy the employees’ demand, with shorter time to accumulate pension accounts. The initial amounts are not enough. According to the results of two improved plans proposed, modified defined contribution plan and the hybrid plan, both the replacement ratios become improved. The modified defined contribution pension plan considers the past service years of each employee while the hybrid pension plan presents the feature of defined benefit plan and guarantees a fixed pension benefit. The effectiveness of the latter one is better. The two improved defined contribution plans both mean a huge expenditure to employer. In particular, the cost of hybrid pension plan is much larger. In addition, the potential risks for retirement plans are analyzed, investment return, inflation rate, longevity risk rate, and salary increase rate. The relationship between investment return and replacement ratio is clear. If the investment return can be raised under a secure method, the pension market will have a great improvement as discussed in [6]. Much more consideration about the pension accumulated accounts investment strategy is necessary.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referee and the associate editor for their very useful suggestions and detailed comments. This paper is supported by the Fundamental Research Funds for the Central University in UIBE (13QNLG01).

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