Question

You wish to retire in 18 years , at which time , you want to have accumulated enough money to receive an annuity of $500 monthly for 20 years of retirement. During the period before retirement you can earn 4% annually,while after retirement you can earn 6 percent on your money. What monthly contributions to the retirement fund will allow you to receive the 500 dollars annuity?

Answer 1

First we need to find the present value to be required to withdraw of $500 monthly for 20 years at the rate of 6%.

We can find by using the present value formula

Pmt = Periodic payment = $500

i = interest rate per period = 6% = 6/100 = 0.06

compounded monthly so r = 0.06/12 = 0.005

n = Number of remaining payment payments = 20 years = 20 x 12 = 240

and present value = $?

we can use below formula

PV = pmt x [(1 - 1 / (1 + i)ⁿ)] / i

PV = 500 x [(1 - 1 / (1 + 0.005)²⁴⁰)] / (0.005)

PV = 50 X [(1-1/(1.005)²⁴⁰)]/( 0.005)

PV = 500 X [1-1/(3.3102)]/( 0.005)

PV = 500 X [1-0.302]/( 0.005)

PV = 500 X [0.698]/( 0.005)

PV = 500 X 139.6

PV = 69800

So present value to be required of annuity will be $69800

Now the present value amount to be accumulated in 18 years with monthly contributions which he is going to retire in 18 years.

Formula for ordinary annuity

FV = C x [ ( 1 + r )ⁿ-1 ] / ( r )

        FV = Future value =$69800

        c = Periodic deposit =?

        r = rate of interest = 4% = 4/100 = 0.04

and compounded monthly so r = 0.04/12 = 0.003333

And n = number of payments = 18 x 12 = 216

                        69800 = Cx [ ( 1 + 0.003333 )²¹⁶ – 1 ] / (0.003333)

                69800 = C x [ ( 1.003333 )²¹⁶ – 1 ] / (0.003333)

                69800 x 0.003333 = C x [( 1.003333 )²¹⁶ – 1]

                232.6434 = Cx [2.05182 – 1]

                232.6434 = C x (1.05182)

                232.6434 / 1.05182 = C

                221.1817 = C

                C = $221.19

So we need to contribute the monthly payment of $221.19 in 18 years to withdraw $500 in every month upto 20 years post retirement.