Question

You have just turned 22 received your degree and accepted your first job. You must decide how much to put in your retirement plan. The plan works as follows. Every dollar in the plan earns 6.5 percent per year. You may not make withdrawals until you retire at 65. After that you can withdrawal money as needed. You believe you will live to 100 and work until 65, and believe you need 95000 per year to be comfortable starting at the end of the first year of retirement and will end on your 100^th birthday at the end of the year. You contribute the same amount to the plan at the end of every year you work. How much will you need to contribute each year to fund your retirement

Answer 1

Part A

PV of Annuity:

Annuity is series of cash flows that are deposited / withdrawn at regular intervals for specific period of time at the end of the period

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period
n - No. of periods

first withdrawl made end of the year.

he retired at 65th year , so first withdrawl made at the end of 66 th year . end of 66th year to end of 100th year is 35 years

Particulars	Amount
Cash Flow	$          95,000.00
Int Rate	6.5000%
Periods	35

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
= $ 95000 * [ 1 - [(1+0.065)^-35]] /0.065
= $ 95000 * [ 1 - [(1.065)^-35]] /0.065
= $ 95000 * [ 1 - [0.1103]] /0.065
= $ 95000 * [0.8897]] /0.065
= $ 1300260.89

if we have $ 1300260.89 at the time of retirement we can withdraw $ 95000 over the next 35 years

Part B

Future value of annuity

Annuity is series of cash flows that are deposited at regular intervals for specific period of time at the end of the period

Particulars	Amount
FV of Annuity	$   1,300,260.89
Int Rate	6.5000%
Periods	43

FV of Annuity = Cash Flow * [ [(1+r)^n ] - 1 ] /r
$1300260.89 = Cash Flow * [ [ ( 1 + 0.065 ) ^ 43 ] - 1 ] / 0.065
$1300260.89 = Cash Flow * [ [ ( 1.065 ) ^ 43 ] - 1 ] / 0.065
$1300260.89 = Cash Flow * [ [ ( 14.998 ] - 1 ] / 0.065
$1300260.89 = Cash Flow * [ 13.998 ] / 0.065
Cash Flow = $ 1300260.89 * 0.065 / 13.998
Cash Flow = $ 6037.79

if deposit $ 6037.79 every year from end of 22 nd birthday year to end of 65th birthday (i,e 23 years) we can have

$ 1300260.89 , then we can recive $ 95000 over the next 35 years