Question

13. You have just turned 50 and you plan to save for retirement. You plan to retire in 12 years. Once you retire you would like to have an income of $50,000 per year for the next 12 years. Determine the amount you must deposit at the beginning of each year to finance your retirement income. Use the following assumptions to determine this annual deposit:

a. All savings compound at a rate of 5% per year.

b. You make the first deposit today (when you turn 50) and the last deposit on the day you turn 61 (12 payments).

c. You make the first withdrawal when you turn 62 and the last withdrawal when you turn 73 (12 withdrawals).

Answer 1

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

Amount required in Account at 62 Years:

Particulars	Amount
Cash Flow	$          50,000.00
Int Rate	5.000%
Periods	12

PV of Annuity Due = [ Cash Flow + Cash Flow * [ 1 - [(1+r)^-(n-1)]] / r ]
= [ $ 50000 + $ 50000 * [ 1 - [(1+0.05)^-11] ] / 0.05 ]
= [ $ 50000 + $ 50000 * [ 1 - [(1.05)^-11] ] / 0.05 ]
= [ $ 50000 + $ 50000 * [ 1 - [0.5847] ] / 0.05 ]
= [ $ 50000 + $ 50000 * [0.4153] ] / 0.05 ]
= [ $ 50000 + $ 415320.71 ]
= $ 465320.71

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	$              465,320.71
Int Rate	5.000%
Periods	12

Cash Flow = [ FV of Annuity Due * r] / [ ( 1+ r) * [ [ ( 1 + r )^n ] - 1 ] ]
= [ $465320.71 * 0.05 ] / [ ( 1 + 0.05 ) * [ [ ( 1 + 0.05 ) ^ 12 ] - 1 ] ]
= [ $465320.71 * 0.05 ] / [ ( 1.05 ) * [ [ (1.05 ) ^ 12 ] - 1 ] ]
= [ $23266.04 ] / [ ( 1.05 ) * [ [ 1.7959 ] - 1 ] ]
= [ $23266.04 ] / [ ( 1.05 ) * [ 0.7959 ] ]
= [ $23266.04 ] / [ 0.8356 ]
= $27841.87

Annual deposit to be made is $ 27841.87