Question

Jack has saved $225 at the end of every month in his bank account for 6 years with the rate of interest being 3.6% p.a. compounding monthly, then he decided to go to school for 5 years when he could not contribute to his account. After his studies ended (at the end of the 5th year at school), he moved his balance to another account and started to withdraw equal amounts of money from his account at the end of every quarter for 10 years. Find the size of his quarterly withdrawals if interest has been 4% p.a. compounding quarterly.

Answer 1

FV of Annuity at the end of 6 Years:

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
Cash Flow	$               225.00
Int Rate	0.300%
Periods	72

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 225 * [ [ ( 1 + 0.003 ) ^ 72 ] - 1 ] / 0.003
= $ 225 * [ [ ( 1.003 ) ^ 72 ] - 1 ] / 0.003
= $ 225 * [ [1.2407] - 1 ] / 0.003
= $ 225 * [0.2407] /0.003
= $ 18052.58

FV after 11 Years ( At the end of 5th year in School):

Future Value:

Future Value is Value of current asset at future date grown at given int rate or growth rate.

FV = PV (1+r)^n
Where r is Int rate per period
n - No. of periods

Particulars	Amount
Present Value	$          18,052.58
Int Rate	0.3000%
Periods	60

Future Value = Present Value * ( 1 + r )^n
= $ 18052.58 ( 1 + 0.003) ^ 60
= $ 18052.58 ( 1.003 ^ 60)
= $ 18052.58 * 1.1969
= $ 21607.04
Amount can be with drawn each quarter:

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here cash flows are happened at the end 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 = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period
n - No. of periods

Particulars	Amount
PV Annuity	$          21,607.04
Int Rate	1.0000%
Periods	40

Cash Flow = PV of Annuity / [ 1 - [(1+r)^-n]] /r
= $ 21607.04 / [ 1 - [(1+0.01)^-10]] /0.01
= $ 21607.04 / [ 1 - [(1.01)^-10]] /0.01
= $ 21607.04 / [ 1 - 0.6717 ] /0.01
= $ 21607.04 / [0.3283 / 0.01 ]
= $ 21607.04 / 32.8347
= $ 658.06

Quarterly withdrawable amount for 10 years is $ 658.06