You decide to open a retirement account at your local bank that pays 10%/year/mo

Question

You decide to open a retirement account at your local bank that pays 10%/year/month (10% per year compounded monthly). For the next 20 years, you will deposit $300 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 20 years?

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

Explanation / Answer

Monthly (nominal) interest rate = 10%/12 = 0.83%

Number of months in 20 years = 20 x 12 = 240

Future worth (FW) of deposits ($) = 300 x F/A(0.83%, 240) = 300 x 755.4413** = 226,632.38

(1) Monthly withdrawal for 20 years ($) = FW / P/A(0.83%, 240) = 226,632.38 / 103.9098** = 2,181.05

(2) Monthly withdrawal (infinite) ($) = FW x Monthly interest rate = 226,632.38 x 0.0083 = 1,881.05

**F/A(r%, N) = [(1 + r)N - 1] / r

F/A(0.83%, 240) = [(1.0083)240 - 1] / 0.0083 = (7.2702 - 1) / 0.0083 = 6.2702 / 0.0083 = 755.4413

**P/A(r%, N) = [1 - (1 + r)-N] / r

P/A(0.83%, 240) = [1 - (1.0083)-240] / 0.0083 = (1 - 0.1375) / 0.0083 = 0.8625 / 0.0083 = 103.9098

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