Question

You are 22 now and would like to retire at age of 67. After you retire, you assume you will live for another 20 years until 87 years old. Suppose you want to withdraw $5,000 at the beginning of each month from your retirement plan after you retire, and the monthly return of your retirement plan is 0.5%.

a. In order to support your monthly withdrawal after retirement, how much do you need to accumulate when you are 67?

b. In order to accumulate what you need (the number you get in part 1), how much do you need to save at the end of each month from now?

Answer 1

Part A:

PV of Annuity Due:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here cash flows are happened at the begining of the period. PV of annuity is current value of cash flows to be received at regular intervals discounted at specified int rate or discount rate to current date.

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

Particulars	Amount
Cash Flow	$            5,000.00
Int Rate	0.500%
Periods	240

PV of Annuity Due = [ Cash Flow + Cash Flow * [ 1 - [(1+r)^-(n-1)]] / r ]
= [ $ 5000 + $ 5000 * [ 1 - [(1+0.005)^-239] ] / 0.005 ]
= [ $ 5000 + $ 5000 * [ 1 - [(1.005)^-239] ] / 0.005 ]
= [ $ 5000 + $ 5000 * [ 1 - [0.3036] ] / 0.005 ]
= [ $ 5000 + $ 5000 * [0.6964] ] / 0.005 ]
= [ $ 5000 + $ 696393.38 ]
= $ 701393.38

Part B:

FV of Annuity :

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here deposits are made at the end of the period. FV of annuity is future value of cash flows deposited at regular intervals grown at specified int rate or Growth rate to future date.

FV of Annuity = CF [ (1+r)^n - 1 ] / r
r - Int rate per period
n - No. of periods

Particulars	Amount
FV of Annuity	$      701,393.38
Int Rate	0.5000%
Periods	540

FV of Annuity = Cash Flow * [ [(1+r)^n ] - 1 ] /r
$701393.38 = Cash Flow * [ [ ( 1 + 0.005 ) ^ 540 ] - 1 ] / 0.005
$701393.38 = Cash Flow * [ [ ( 1.005 ) ^ 540 ] - 1 ] / 0.005
$701393.38 = Cash Flow * [ [ ( 14.78 ] - 1 ] / 0.005
$701393.38 = Cash Flow * [ 13.78 ] / 0.005
Cash Flow = $ 701393.38 * 0.005 / 13.78
Cash Flow = $ 254.5