Question

Justin is planning on retirement investment, if invested at 6 percent per year (6%p.a. is APR, interest compounded monthly), he wish to have $3,500 of monthly income for 30 years. To date, he has saved $100,000, but he still has 20 years until he retires.

How much money does he need to contribute per month to reach his goal? (Give your answer in 2 decimal places, provide your answer as a positive number.)

Answer 1

Amount Required after 20 Years:

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
Cash Flow	$            3,500.00
Int Rate	0.5000%
Periods	360

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
= $ 3500 * [ 1 - [(1+0.005)^-360]] /0.005
= $ 3500 * [ 1 - [(1.005)^-360]] /0.005
= $ 3500 * [ 1 - [0.166]] /0.005
= $ 3500 * [0.834]] /0.005
= $ 583770.65
Future value of $ 100000 after 20 Years:

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	$       100,000.00
Int Rate	0.5000%
Periods	240

Future Value = Present Value * ( 1 + r )^n
= $ 100000 ( 1 + 0.005) ^ 240
= $ 100000 ( 1.005 ^ 240)
= $ 100000 * 3.3102
= $ 331020.45

Future value of annuity required:

= $ 583770.65 - $ 331020.45

= $ 252750.20

Monthly deposit Required:

Particulars	Amount
FV of Annuity	$      252,750.20
Int Rate	0.5000%
Periods	240

FV of Annuity = Cash Flow * [ [(1+r)^n ] - 1 ] /r
$252750.2 = Cash Flow * [ [ ( 1 + 0.005 ) ^ 240 ] - 1 ] / 0.005
$252750.2 = Cash Flow * [ [ ( 1.005 ) ^ 240 ] - 1 ] / 0.005
$252750.2 = Cash Flow * [ [ ( 3.3102 ] - 1 ] / 0.005
$252750.2 = Cash Flow * [ 2.3102 ] / 0.005
Cash Flow = $ 252750.2 * 0.005 / 2.3102
Cash Flow = $ 547.03

Monthly deposit required is $ 547.03