Simulation of pension plan supplemental cost based on withdrawal rate interpolation and different benefit

One factor that affects the supplemental cost is the probability that a participant will survive in employment to the retirement age. The probability depends on the mortality rate, the disability rate, the withdrawal rate, and the normal retirement rate. All of the rates are presented in one-year interval except the withdrawal rate. This paper uses the cubic spline interpolation to get the withdrawal rates at one year intervals, whereactuarial calculation method used is Projected Unit Credit (PUC), and pension benefits are calculated using two assumptions, namely the final average and the carrier accrued salary. The interpolation results show that first, the probability of withdrawal at the age of 25 for participants entering the pension program at the age of 20 differs from ages of 21, 22, 23, and 24. Likewise the probability of withdrawal at 30 years of age for participants entering a pension program at the age of 25 is different from ages 26, 27, 28 and 29. The same applies to other ages. The younger the age enters the pension program, the greater the unfunded liabilities that can arise. The unfunded liabilities for the final salary benefit assumption is smaller than the carrier average salary benefit assumption. The unfunded liabilities for entry at a 1-year interval are greater than 5-year intervals.


Introduction
The Projected Unit Credit (PUC) method is one of actuarial method in pension funding. Contribution of participants in the PUC method consists of normal cost (NC) and supplementary cost (SC). NC is the difference between accrued liability at the beginning and end of the year, while SC is an amortization of accrual liability that are not funded. SC appears because the present value of benefits is greater than the present value of the normal cost, and can arise either before the establishment of the system or after the system starts. Before establishment of the system, SC can arise due to actuarial cost assumptions, whereas after the system starts, SC can arise due to changes in benefits, changes in actuarial assumptions, or actuarial losses.
The SC is influenced by the interest rate, the benefit, and the probability that a participant will survive in employment to the retirement age. Simulation of SC as well as simulation of actuarial liability (AL) from various perspectives and methods are useful for pension funds in making their investment decisions.Stock is one of the investment instruments chosen by many pension funds. According to [3], significant factor that influence return on investmenton stock is Bank Indonesia (BI) interest rates. [2] who uses fixed interest rates and market interest rates to simulate AL for Netherlands Pension Fund show that the values of AL are not different significantly. The different SC can be found using the different benefit assumptions, namely 1) the final salary at one year before retirement, 2) the average salary since entering the pension program until reaching retirement age. The probability that a participant will survive in employment to the retirement age probability depends on the mortality rate, the disability rate, the withdrawal rate, and the normal retirement rate. All the rates are given in a one-year age unless the withdrawal rate is given at a five-year interval.The presentation of the rate in a one-year age can provide a more detailed calculation in pension funds, especially for simulation of SC. Interpolation can be used to obtain the withdrawal rate in a one-year age. The purpose of this paper is to make a simulation of SC based on the different benefit and the withdrawal rate interpolationin a one-year age.This simulation results can provide an overview of the effects of the entry age -withdrawal rate in the pension program on SC that have not been done by [1] and [2].

Methodology
Let be the benefit coefficient of additional costs for payment of unfunded liabilitiesat , bethe amount of benefits received by participants at age , − be the present value of the interest rate during the participation period − , ( ) − be theprobability that a participant age will survive in employment to age , and be thelifetime annuity since a pensioner aged . The supplemental cost of pension funds for the participant who valuated at age for retirement atage for the PUC methods is defined as([6]), In this paper, supplemental cost simulations are based on 2 benefit calculation assumptions, namely the final salary at age − 1 , and the average salary during work − years (carrier average salary).
Let the final salary at age − 1 is denoted as −1 . For proportion of , the benefit based on the final salary isdefined as, For the salary at age is , the assumption of salary increase per year is , the salary at age + is defined as, So the benefit based on the average salary is defined as, Let the final average salary is formulated as So the benefit based on the final average salary is defined as, Each of + ′ . in (5)   The values of ′ for 1-year intervals, i.e. when = 21, ⋯ ,24, 26, ⋯ , 29, ⋯ , 4949 can be obtained using the cubic spline interpolation which is formulated as a solution of the following linear equation system, Simulation in this paper is based on the retirement age is 56 years, the salary when the initial period of participation is Rp 3,000,000 per month, an interest rate of 7.2587 % (interest rate average of Bank Indonesia in years 2006-2017), a salary growth of 8% and unfunded liabilitiesare evaluated at the age of 50 years.  Table 1, the probability of withdrawal at the age of 25 for participants entering the pension program at the age of 20, is the same as the participant who enters the pension program at the ages of 21, 22, 23, 24 and 25. The interpolation results shown in Table 2 show that the probability of withdrawal at the age of 25 years for participants entering the pension program at the age of 20 is different from the ages of 21, 22, 23, 24, and 25.   Table 3.From Table 3, some information is obtained, namely, first, the younger the age enters the pension program, the greater the unfunded liabilities that can arise. Second, the unfunded liabilities for the final salary benefit

Conclusion
Application the cubic spline interpolation shows thatthe probability of withdrawal at the age of 25 for participants entering the pension program at the age of 20, is different from the ages of 21, 22, 23, and 24. Likewise for other ages. The younger the age enters the pension program, the greater the unfunded liabilities that can arise. The unfunded liabilities for the final salary benefit assumption is smaller than the carrier average salary benefit assumption. Thus, the unfunded liabilities for entry at a 1-year interval are greater than 5-year intervals.