Question

You just turned 26. Now that you’re old, you want to be responsible and start a retirement fund. You plan to invest $445 every month for the next 39 years. You expect to earn a 7.9% annual return until you retire. Once you retire, you will shift your savings into safer investments that you expect will earn 4.8% per year. You plan to make monthly withdrawals for 21 years of retirement (and don't want to have any money left at the end). Under this plan, how much money will you withdraw from your retirement account each month? (NOTE: Round your answer to the nearest cent)

Answer 1

Part A

Invest $445 each month for the next 39 years

FV of Annuity :
Annuity is series of cash flows that are deposited at regular intervals for specific period of time .

FV of Annuity = CF [ (1+r)^n - 1 ] / r
r - Int rate per period = 7.9 % / 12 = 0.6583 % or 0.006583
n - No. of periods = 39 years * 12 = 468 months

Invest $445 each month for the next 39 years

Particulars	Amount
Cash Flow	$ 445.00
Int Rate	0.65833%
Periods	468

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 445 * [ [ ( 1 + 0.006583 ) ^ 468 ] - 1 ] / 0.006583
= $ 445 * [ [ ( 1.006583 ) ^ 468 ] - 1 ] / 0.006583
= $ 445 * [ [21.5614] - 1 ] / 0.006583
= $ 445 * [20.5614] /0.006583
= $ 1389843.58

Amount having at the time of retirement is $ 1389843.58

Part B

Calculation of monthly withdrawl amount for the next 21 years

PV of Annuity:  
Annuity is series of cash flows that are deposited / Withdrawn at regular intervals for specific period of time.  
  
PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r  

r - Int rate per period = 4.8 % / 12 = 0.4 % or 0.004
n - No. of periods = 21 years * 12 months = 252 months

Annuity amount = amount having at the time of retirement = $ 1,389,843.58

Particulars	Amount
PV Annuity amount	$ 1,389,843.58
Int Rate	0.4000%
Periods	252

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r  
Cash Flow = PV of Annuity / [ 1 - [(1+r)^-n]] /r
= $ 1389843.58 / [ 1 - [(1+0.004)^-252]] /0.004
= $ 1389843.58 / [ 1 - [(1.004)^-252]] /0.004
= $ 1389843.58 / [ 1 - 0.3657 ] /0.004
= $ 1389843.58 / [0.6343 / 0.004 ]
= $ 1389843.58 / 158.5793
= $ 8764.34

if we deposit $ 445 for the each month for 39 years we have $ 1,389,843.58 at the time of retirement , then we can withdraw $ 8764.34 each month for 21 years .

Please comment if any further explanation is required.