Question

Today, you open a new savings account and plan to begin depositing equal amounts at the beginning of each year for 9 years, including the deposit you make today. There will only be these 9 deposits and no withdrawals. Assume the interest rate you will earn is 4%. If you want your account balance to be exactly $20,000 at the end of 9 years, what must be the amount of each deposit?

Answer 1

FV of Annuity Due:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here deposits are made at the begining 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 DUe = ( 1 + r ) * FV of Annuity

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

Particulars	Amount
FV of Annuity Due	$                20,000.00
Int Rate	4.000%
Periods	9

Cash Flow = [ FV of Annuity Due * r] / [ ( 1+ r) * [ [ ( 1 + r )^n ] - 1 ] ]
= [ $20000 * 0.04 ] / [ ( 1 + 0.04 ) * [ [ ( 1 + 0.04 ) ^ 9 ] - 1 ] ]
= [ $20000 * 0.04 ] / [ ( 1.04 ) * [ [ (1.04 ) ^ 9 ] - 1 ] ]
= [ $800 ] / [ ( 1.04 ) * [ [ 1.4233 ] - 1 ] ]
= [ $800 ] / [ ( 1.04 ) * [ 0.4233 ] ]
= [ $800 ] / [ 0.4402 ]
= $1817.17

Annual deposit to be made is $ 1817.17