Question

Q: In a defined benefit pension plan for public employees of a state: (a) Employees work for 30 years earning wages that increase at a real rate of 1.0% per year. (b) They retire with a pension equal to 60% of their final salary. This pension decreases at the real of rate of 1% per year. (c) The pension is received for 18 years. (d) The pension fund assets earn a real rate of 2.5%.

Find the percentage of an employee's salary that must be contributed to the pension plan if it is to remain solvent.

Answer 1

Employee's annual salary currently = X {Assumption}

Employee's annual salary after 30 years, on retirement = X*(1+0.01)^30 {Final Salary also known}

= X(1.01^30)

The pension required after retirement = 60% of final salary

= X(1.01^30)*0.60

Now this pension decreases at the rate of 1% per year.

woulbe something like:-

Year	Pension
1	X(1.01^30)*0.60
2	X(1.01^30)*0.60*(0.99)
3	X(1.01^30)*0.60*(0.99^2)
4	X(1.01^30)*0.60*(0.99^3)
...	...
18	X(1.01^30)*0.60*(0.99^17)

{Above we have assumed that at the year of retirement person one year after that the person will receive his first pension. So assume he retire in 2050 (same year he will receive his last salary) then he will receive his first pension at the end of 2051}

{Also we are assuming that the pensions are received at the end of the year only, not the monthly payments will be made after retirement}

Now to find the retirement corpse needed to facilitate the pensions payments. This corpse would be equal to the PV of pension discounted at the rate of dsicounted rate.

Here discounted rate will be the 2.5% as it is the earned interest in the pension funds after retirement.

Now one thing to note is that these earn 2.5% rate so discounting them at rate of 2.5% means they will come back to same future number.

So, PV of pension payments = X(1.01^30)*0.60 + X(1.01^30)*0.60*(0.99) + X(1.01^30)*0.60*(0.99^2) + ...+ X(1.01^30)*0.60*(0.99^17)

= X(1.01^30)*0.60*[ 1 + 0.99 + 0.99^2 + ... + 0.99^17 ]

= X(1.01^30)*0.60*[ 1*(1 - 0.99^18)/(1 - 0.99) ] {Sum of GP formula is used here}

Now we know the FV of the annual annuity required towards the retirement corpse:-

FV of annuity = PMT*((1+r)^t - 1)/r

The question ask for the ((PMT/X)*100%). We need to arrange the boev equation in such a manner that we have the required fraction.

X(1.01^30)*0.60*[ 1*(1 - 0.99^18)/(1 - 0.99) ] = PMT*((1.025)^30 - 1)/0.025

PMT/X = (1.01^30)*0.60*[ 1*(1 - 0.99^18)/(1 - 0.99) ] / (((1.025)^30 - 1)/0.025)

= 13.3830/43.9027

PMT/X = 30.48%