Question

You decide to open a retirement account at your local bank that pays 9%/year/month (9% per year compounded monthly). For the next 20 years, you will deposit $600 per month into the account, with all deposits and withdrawals occurring at month’s end. On the day of the last deposit, you will retire. Your expenses during the first year of retirement will be covered by your company’s retirement plan. As such, your first withdrawal from your retirement account will occur on the day exactly 12 months after the last deposit.

What monthly withdrawal can you make if you want the account to last 15 years? $ What monthly withdrawal can you make if you want the account to last forever (with infinite withdrawals)? $

Answer 1

Monthly withdrawal if the account is to last for 15 years: $ 4,430.30

Monthly withdrawal if the account must last forever : $ 3,275.99

Solution:

n = 20 x 12 = 240

i = 9 % / 12 = 0.75 % or 0.0075

FVA _{0.75%, n=240} = [ { ( 1.0075) ²⁴⁰ -1 } / 0.0075 ] = 667.8869

Future value of the monthly deposits at retirement = $ 600 x 667.8869 = $ 400,732.14

Future value one year later = $ 400,732.14 x 1.09 = $ 436,798.03

Future value of $436,798 is the present value for subsequent monthly withdrawals for 15 years.

n = 15 x 12 = 180

i = 0.0075

PVA _{0.75%, n=180} = [ { 1 - ( 1 / 1.0075) ¹⁸⁰ } / 0.0075 ] = 98.5934

Monthly withdrawal for 15 years = $ 436,798 / 98.5934 = $ 4,430.30

Present value of a prepetuity = Perpetuity / r

or Perpetuity = $ 436,798 x 0.0075 = $ 3,275.99