COMPARISON OF PROJECTED UNIT CREDIT AND ENTRY AGE NORMAL METHODS IN PENSION FUND VASICEK AND COX-INGERSOLL-ROSS MODELS

ABSTRACT


INTRODUCTION
Civil servants are citizens who are appointed as state civil apparatus with permanent employment status.Employees who will retire in government agencies are uncertain in number, someone can retire due disability, death, resignation and when reaching normal retirement age.The pension program managed by the Pension Fund is a form of compensation to improve employee welfare when facing risks at work.To anticipate risks in old age, the government needs to calculate periodic payments for employees by including them in pension program to get retirement benefits.
Actuarial calculation can help determine pension fund projections in the form of pension benefits to received and normal contributions to be paid by employees or participants.In general, there are two groups of actuarial valuations to determine the amount of normal contributions and actuarial liabilities, namely the Accrued Benefit Cost Method which includes the Projected Unit Credit and Projected Benefit Cost Method with the Entry Age Normal method.The Projected Unit Credit method divides the total pension benefit at normal retirement age by the total service period allocated to each year of service, while Entry Age Normal assumes that the projection of pension benefits when reaching retirement age begins at entry age.
Interest rates are something that needs to be considered in pension fund calculations because they fluctuate with the economy.Constant interest rates can be used in calculating normal contributions and actuarial liabilities of pension funds, but this article uses stochastic interest rates, namely the Vasicek and Cox-Ingersoll-Ross (CIR) interest rate models.The Vasicek interest rate model predicts future interest rates by looking at the movement of previous interest rates and has a mean reverting phenomenon, which always moves towards the equilibrium point [1].The Vasicek model does not limit the interest rate requirement, so it is possible to obtain negative interest rate results [2].Unlike the Vasicek model, the Cox-Ingersoll-Ross (CIR) interest rate has mean reversion properties or a tendency to return to the average and is positive [3].
There are several studies that discuss the use of Projected Unit Credit and Entry Age Normal methods with constant interest rate including [4], it was found that the amount of normal contributions using the Projected Unit Credit method continues to increase as salary increases while the Entry Age Normal results remain the same for each year.These results are in line with the results obtained by [5] by providing several examples of cases in civil servants.In addition, the amount of actuarial liabilities for each year is smaller when using the Projected Unit Credit method.In line with previous research, [6] shows that the greater the salary and the level of salary increase, the greater the normal contributions and actuarial liability to pension fund participants.A comparison of the Projected Unit Credit with other methods such as Attained Age Normal is given by [7], every year normal contributions with the Projected Unit Credit method show a considerable increase while the Attained Age Normal method is relatively slow.Furthermore, studies using Vasicek dan Cox-Ingersoll-Ross (CIR) interest rates are increasingly being studied because the interest rate follows a stochastic process [2], [8]- [11].Furthermore, in research [12] used the Projected Unit Credit method with stochastic interest rates, namely Vasicek and Cox-Ingersoll-Ross (CIR).Based on these conditions, this research focuses on the comparison of normal contributions and actuarial liabilities using the Projected Unit Credit and Entry Age Normal methods with the Vasicek and Cox-Ingersoll-Ross (CIR) interest rate models.

RESEARCH METHODS
The data used in this study is a retired civil servant at the Water Resources Management Office in Bulukumba Regency, namely age when registered as a civil servant, gender, normal retirement age, basic salary per month, percentage of salary given for retirement benefits, and accumulated salary for the last year before retirement.The data obtained from pension fund participants is used to determine the opportunity for participants to actively work and the amount of pension benefits that will be obtained based on the accumulated last salary before retirement.The Bank Indonesia monthly interest rate for 2018-2022 is used to determine the value of the Vasicek and Cox-Ingersoll-Ross (CIR) interest rate parameters.

Survival Function
The survival function describes an employee's life expectancy, specifically it can be used to determine the probability of a person still actively working to reach a predetermined normal retirement age.The probability of an employee aged  years old survives until the age of  year is expressed by [13]: with    : The probability that a person aged  years old can work until the age of ( + ) years  + : Number of people exactly aged  +  years   : Number of people exactly aged  years

Interest Rate Function
The interest rate is used to determine the deductibility of future payments which is referred to as the present value.If  denotes the interest rate in  year with  = 1,2, … ,  then the present value of a unit of money in  year is shown by [14]:   (2)

Life Annuity Function
A life annuity is a series of payments that a person makes as long as the person is alive, it can be expressed by [15]: with ̈ is the initial lifetime annuity.Form   and   are commutations symbols simplifying calculations in calculating mortality tables, expressed by: =   +  +1 +  +2 + ⋯ +   = ∑  + − =0 (5) with  is the highest age reached.Cash value of term life annuity  year is expressed by [9]:

Benefit Function
The benefit function is used to determine the amount of benefits that the company needs to pay for participants when entering retirement age.In this study, the benefit function of participants with retirement age is used  based on the last salary [8]: : Accumulated retirement benefits at age  years  : Percentage of salary that will go towards retirement benefits  : Normal retirement age  : Age of entry into the pension fund  −1 : Last year's salary before retirement

Present Value of Future Benefits (PVFB)
For example −   denoted the probability of an employee staying employed until the age of  years,   state the amount of retirement benefits at age  year, ( − ) state the expected cash value of payments at time ( − ) year, and ̈ denotes the initial lifetime annuity.Present Value of Future Benefit for someone aged  years old and with a normal retirement age  can be expressed [5] :

Vasicek and Cox-Ingersoll-Ross Interest Model
The Vasicek interest rate model is a stochastic model that has a tendency to return to an equilibrium point after experiencing a decrease or increase, expressed as follows [10]: : Interest rate at time  () : Wiener Process at time  Suppose  1 () denote the expectation of the cash value of a one-unit payment when  for interest rates following the Vasicek model: with (0) =  0 and () = : Interest rate at time  () : Wiener process at time  Suppose  2 () denote the expected cash value of a one-unit payment at time  for an interest rate that follows the Cox-Ingersoll-Ross (CIR) model: with (0) =  0 and  = √ 2 + 2 2 .
The value of interest rate parameter of the Vasicek and Cox-Ingersoll-Ross models is calculated using Bank Indonesia interest rate data for 2018-2022, obtained using the following equation [3] :

Normal Cost (NC)
Normal Cost (NC) is the contribution required in one year to fund part of the present value of pension benefits, allocated in the current year according to the actuarial calculation method used [17].The normal contribution equation that will be used in this study using the Projected Unit Credit method [8]:

Actuarial Liability
Actuarial Liability (AL) is the amount of pension plan funds accumulated to pay pension benefits in the future, the actuarial liability equation uses the Projected Unit Credit method [8]: The actuarial liability equation using the Entry Age Normal method is given as follows

RESULTS AND DISCUSSION
The application of the Projected Unit Credit (PUC) and Entry Age Normal (EAN) methods with the Vasicek and Cox-Ingersoll-Ross (CIR) interest rate models used in this study is a retired civil servant at the Bulukumba regency water resources management office.Aged when registered as a civil servant is 26 years ( = 26), male gender, normal retirement age 56 years ( = 56), and with a basic monthly salary of Rp2.200.000,−.It was assumed that the percentage of salary given for retirement benefits is equal to 2,25% and the accumulated salary during the last year before retirement is Rp26.400.000,− .

Life Annuity Value
Life annuity is a series of payments made periodically in monthly, quarterly, semi-annual, and annual periods.To calculate a life annuity, it is necessary to expect the cash value of payments of one unit at a certain time following the Vasicek interest rate using Equation ( 10) and the Cox-Ingersoll-Ross interest rate in Equation ( 12).Parameter values , , and  in Equation ( 13)-( 15) are calculated using monthly data on the Bank Indonesia interest rate for 2018-2022, the parameter estimation results are shown in Tabel 1 below:

Present Value Future Benefit (FVFB)
The present value of pension benefits is influenced by the life expectancy value obtained from the 2019 Mortality Table for males, the value of pension benefits with retirement age , the expected cash value of payments following the Vasicek and Cox-Ingersoll-Ross model, and the lifetime annuity.The PVFB value is calculated using the equation: The complete value of present value future benefit for the valuation age of  = 26 − 56 is given in Figure 3.

Normal Cost (NC)
To get the Present Value Future Benefit (FVFB), a pension fund participant must make normal cost.The amount of normal cost to be paid by participants is calculated using two methods, namely Projected Unit Credit (PUC) and Entry Age Normal (EAN), with each method using two interest rate model, namely Vasicek and Cox-Ingersoll-Ross (CIR).Normal cost using the Projected Unit Credit method are calculated using the equation The complete normalized cost with Vasicek and Cox-Ingersoll-Ross (CIR) interest rates are given in Figure 4. Based on Figure 4, it can be seen that in general, in the 0-2 years of services period, the normal cost using the Projected Unit Credit (PUC) method is smaller than the Entry Age Normal (EAN) method.In the third year, the normal cost with the CIR interest rate of the Projected Unit Credit (PUC) method is the highest and continues to increase until the working period reaches 30 years by 14.773.176every year.While the Projected Unit Credit (PUC) method of the Vasicek interest rate model is smaller than Entry Age Normal with the same interest rate until the working age reaches 17 years.The at 18 years of service, the Projected Unit Credit (PUC) method with the Vasicek interest rate continues to increase and the value is always higher than EAN method of the Vasicek model.It can be seen that using these two methods, the normal cost every year for the CIR interest rate is always greater than Vasicek.Normal cost using the Projected Unit Credit (PUC) method tend to always increase as the working period increases.Whereas with the Entry Age Normal (EAN method, the Vasicek model initially increases and then decreases as the normal retirement age approaches, and the CIR model initially decreases but increases again at the end of the working period with the amount of contributions equivalent to the initial contribution of the working period.

Actuarial Liability (AL)
The actuarial liability using the Projected Unit Credit (PUC) method is calculated using the equation The complete actuarial liabilities using the Projected Unit Credit (PUC) and Entry Age Normal (EAN) methods are given Figure 5.The actuarial liability for 0 years of services is 0, − because at that time participants had just started contributions so they did not have a benefit reserve.It can be seen that actuarial liabilities continue to increase every year for each method and interest rate.The actuarial liability of the Projected Unit Credit (PUC) method with the Vasicek interest rate is annually smaller than the Entry Age Normal (EAN) method as well as for the CIR model interest rate, but the same value at the end of the working period.In addition, the actuarial liability of the Entry Age Normal (EAN) method with the CIR interest rate has the highest value since the beginning of the year.

CONCLUSIONS
The calculation of normal cost and actuarial liabilities using the Projected Unit Credit (PUC) method with the Vasicek and Cox-Ingersoll-Ross (CIR) interest rate models increases as the length of service increases.Unlike the Projected Unit Credit (PUC) method, the normal cost of the Entry Age Normal (EAN) method of the Vasicek interest rate model initially increases and then decreases towards the initial contributions value of the working period.Meanwhile, the Entry Age Normal (EAN) method of the Cox-Ingersoll-Ross (CIR) model starts with a fairly high initial contribution then decreases but rises again towards the initial contribution of the working period.As for the actuarial liabilities of the Entry Age Normal (EAN) method for both interest rate models, it always increases every year.Thus, it can be concluded that the Entry Age Normal method with the Vasicek model interest rate is better for participants because it has a lower normal cost value.Meanwhile, for pension funds, the Projected Unit Credit method with the Cox-Ingersoll-Ross (CIR) model interest rate is more profitable.

Figure 5 .
Figure 5. Actuarial Liability ⌉ denote the amount of the initial term annuity for an employee with the age of  years old at the time of entry into the pension fund, and will retire normally at the age of  year.The normal cost equation using the Entry Age Normal method is expressed as follows [5]: ()  : Normal cost of pension fund participants aged  year   : Accumulated retirement benefits at age  years   ̈ : Discrete lifetime annuity starts in  year  − : Interest rate to ( − ) year −   : Chance of participants aged  years old live up to ( − ) years  ()  : Cash value of total retirement benefits at age  year  : Normal retirement age  : Age of entry into the pension fund Suppose ̈ :⌈−

Table 1 . Vasicek and Cox-Ingersoll-Ross (CIR) Interest Rate Parameter Estimation Results
Based on the data used, it is known that the age of becoming a civil servant is  = 26, normal retirement age  = 56, and states the age at the time of actuarial valuation.Expected cash value of payments based on the Vasicek interest rate model is calculated using Equation (10) and the Cox-Ingersoll-Ross (CIR) interest rate is calculated using Equation (12) and (0) = 0,055 the results obtained in
56()  =  56 56−   (56 − )̈5 6Vasicek interest rate model for valuation age  = 26, 27, and 28 years, the results are as follows: Normal cost using the Entry Age Normal method are calculated by the equation