You decide to open a retirement account at your local bank that pays 8%/year/month (8% per year compounded monthly). For the next 20 years you will deposit $400 per month into the account, with all deposits and withdrawls occurring at the end of the month. 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.

a) What monthly withdrawal can you make if you want the account to last 15 years?

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

a.)

First, find the Future value of the annuity deposits. Using a financial calculator, input the following;

Number of months; N = 20*12 = 240

Monthly rate; I/Y = 8%/12 = 0.667%

PV = 0

Recurring payment; PMT = - 400

then compute future value; CPT FV = 235,725.317

Next find FV of at the end of first 12 months after retirement;

235,725.317 (1 + 0.00667) ^12 = 255,300.546

Next, use $255,300.546 as the PV of withdrawal annuity of 15 years to find annual PMT;

PV = - 255,300.546

N = 15*12 = 180

I/Y = 0.667%

FV = 0

then CPT PMT = $2,440.38

b.)

Infinite withdrawals means that they are perpetual hence referred to as Perpetuity.

Since we have the amount you will have saved by the end of 20 years (240 months) as $255,300.546, use that as the Present value (PV) of your perpetuity.

PV = PMT / rate

PMT is the recurring withdrawal

$255,300.546 = PMT / 0.667%

PMT = 0.667% * $255,300.546

PMT = $1,702.85.

Therefore, you will make a monthly withdrawal of $1,702.85.