Interest and Annuity

Question

A man starts working a day after he celebrates his 23rd birthday. He wants to have a fund that could, from the date of his retirement at age 58, give him a benefit payment of $6000 at the end of each month until his 72nd birthday. It is assumed that the money can be invested at an annual interest rate of 3%.

Calculate the amount of the fund required on the date of retirement.

Answer 1

He wants 6,000 per month for 14 years or 168 months. The interest rate is 3 percents compounded monthly. the monthly or period rate is .03/12=.25%

The Present Value (PV) of an annuity is

PV + PMT*K where PMT = payment and k is a multiplier given as

K = (1/i)(1- (1/(1+i))^N) = (1/.0025)(1-(.6573)) = 137.04

so, PV = 6,000*(137.04 = 822,260.91. This is what he needs on his 58th Birthday.

The lump sum amount he would have to put on deposit on his 23rd birth day is the present value of 822,260.91 discounted back to his 23rd birthday or 35 years. 35 years is 35*12 months = 420 deposits he will have to make. So discount 822,260.91 back 420 months at 0.25% interest we get.

PV= (1/(1.0025))⁴²⁰*822,260.91 = 288,117.41

Before we had the payment and had to determine the present value. Now we have the present value and nee to determine the payment.

PV= PMT*K or PMT = PV/K we need to calculate the multiplier , k when i = .0025 and n= 420

K = (1/.0025)(1- (1/1.0025)⁴²⁰ = 400( 1-.3504) =259.84.

So, the size of the depostir = PV/K = 288,117.41/259.84 = 1,108.82

If he puts this much away every month for 35 year he can withdraw 6,000 a month from his 58th birthday to his 72nd birthday. Then hie will have to live off of his kids.

If you have any questions, Contact me. I have been discounting al typse of cash flows for more years than I care to mention.

Best,

Lenny