Question

Your uncle Abdallah is celebrating his 33th birthday today and wants to start saving for his retirement at the age of 63. He wants to be able to withdraw AED 100,000 from his saving account on each birthday for 20 years following his retirement. The first withdraw will be on his 64th birthday. Your uncle intends to invest his money in a local bank in Abu Dhabi that offers 7% interest rate per year. He wants to make equal payments on each birthday into the account established in the local bank for his retirement fund.

1. If your uncle starts making these deposits on his 33th birthday and continues to make deposits until he is 63, what amount must he deposit annually to be able to make the desired withdrawals at retirement?
2. If your uncle has just inherited a large sum of money, so instead of making equal payments, he has decided to pay one lump sum payment on his 33th birthday to cover his retirement needs. What amount does he have to deposit?
3. If your uncle’s employer informs your uncle that he will contribute AED 1,000 to your uncle account every year. Also, if your uncle expects AED100,000 from another investment on his 53th birthday, which he will also put into the retirement account. What amount must he deposit annually now to be able to make the desired withdrawals at retirement.

Answer 1

we can use the present value of an annuity formula to determine how much money your uncle will need when he retires at 63:

PV = annual distribution x annuity factor

• annual distribution = $100,000
• PV annuity factor, 7%, 20 periods = 10.594

PV = $100,000 x 10.594 = $1,059,400

1) the present value of your uncle's retirement account at 63 will become the future value of his contributions, but this time we need to use the future value of an annuity due. He will make 31 deposits in total starting at age 33 and ending at age 63:

$1,059,400 = annual contribution x 102.07304 (FV annuity due factor, 7%, 31 periods)

annual contribution = $1,059,400 / 102.07304 = $10,359.25

2) we should now use the present value formula:

PV = FV / (1 + i)ⁿ

PV = $1,059,400 / (1 + 0.7)³⁰ = $138,907.59

3) the future value of your uncle's employer contributions = $1,000 x 102.07304 = $102,073.04

his $100,000 investment will be worth = $100,000 x (1 + 0.07)¹⁰ = $196,715.14

that means that your uncle still needs to save $1,059,400 - $102,073.04 - $196,715.14 = $760,611.46

his annual contribution will be:

annual contribution = $760,611.46 / 102.07304 = $7,451.64