If you have 300000 saved for retirement. What if your account earns 7 interest. How much will you be able to pull out each month if you want to be able to take withdrawals for 25 years?

Answer 1

You have not said if the 7% is APR, how it is accrued - yearly or monthly - and if you want to touch the capital invested or not.

Assuming it is 7% APR compounded accrued monthly, then:

Let p be the APR (p = APR% / 100)

Let r be the multiplier for each month; it is found as: (1 + p)^(1/12)

Let D be the amount you withdraw.

Let C be the amount you invest.

Assuming you withdraw at the end of each month after the interest is paid:

At the end of the first month you will have Cr - D

At the end of the second month you will have (Cr - D)r - D = Cr² - Dr - D

At the end of the third month you will have (Cr² - Dr - D)r - D = Cr³ - Dr² - Dr - D

At the end of the nth month you will have:

Crⁿ - Drⁿ⁻¹ - Drⁿ⁻² - ... - Dr - D

= Crⁿ - D(rⁿ⁻¹ + rⁿ⁻² + ... + r + 1)

Now, the sum of a GP is given by:

Sn = rⁿ⁻¹ + rⁿ⁻² + ... + r + 1 = (rⁿ - 1)/(r - 1)

→ At the end of month n you will have Crⁿ - D(rⁿ - 1)/(r - 1) left in you account.

25 years = 25 × 12 month = 300 months

→ n = 300

You have an APR of 7% → r = 1.07^(1/12)

C = 300,000

You cannot take out more than is left in your account

→ Crⁿ - D(rⁿ - 1)/(r - 1) ≥ 0

→ D(rⁿ - 1)/(r - 1) ≤ Crⁿ

→ D ≤ Crⁿ(r - 1)/(rⁿ - 1)

→ D ≤ 300,000 × (1.07^(1/12))^300 × (1.07^(1/12) - 1)/((1.07^(1/12))^300 - 1)

→ D ≤ 2,079.36 (approx)

Which means you can take out up to 2,079.36 per month and it will last 25 years. (This will leave a few pennies as the real figure is slightly more than this but less than 2,079.37)

If you take out less than this figure, it will last 25 years and you will still have some capital left.

If you take out 300,000 × (1.07^(1/12) - 1) ≈ 1,696.24 each month, this is the monthly interest gained and so the capital will still be there after 25 years (even though it will be worth a lot less due to inflation - at 7% inflation prices double every 10 years, so if you get 7% interest, inflation is likely to be higher which means after 25 years your 300,000 will only be worth about 90,000 in today's money).