Annuity Math Question?

Question

A retiree receives a payout from his Roth IRA where the taxes have already been paid.
He wants to take out 10% of the original amount at the beginning of each year for living expenses. The remaining money will earn 6% interest. How long before the money runs out?
For example, he starts with $400,000 and takes out $40,000 each year. The first year the interest would equal $360,000 * 6% or $21,600. So the second year would start with $381,000.
If you have a formula to put in a spreadsheet, it would help.

The question does not say 10% of current value, it says 10% of original value. You have to read the whole thing.

Answer 1

payout money=$m

after 1 year
(m - m/10)(1.06)
=0.9*1.06m

after 2 years
(0.9*1.06m - m/10)*1.06
=(0.9*1.06-0.1)*1.06m

after n years
1.06^n - 0.1 sigma(sub n) 1.06^n

money considered as "run out" when there's not enough to provide for living expenses at the beginning of the year.
thus

1.06^n - 0.1 sigma(sub n) 1.06^n < 0.1
1.06^n < 0.1[sigma (sub n) 1.06^n] + 0.1
10(1.06^n) - 1 < sigma (sub n) 1.06^n
10(1.06^n) - 1 < 1.06[1.06^(n-1)-1] / [1.06-1]
0.6(1.06^n) - 0.06 < 1.06^n - 1.06
0.4(1.06^n) > 1
1.06^n > 2.5
use log on both sides
n=16

after 15 years, the money will run out, as at the beginning of the 16th year, the retiree will only have less than 10% of his original payout.

Answer 2

Each year, the principal is reduced to 0.9 of its original value, but that amount generates interest, growing to 1.06 times its original amount. Therefore, each year the principal is reduced to (0.9)*(1.06) = 0.954 of its initial amount.

The money won't "run out" completely for a long, long time--in fact, after 250 years there is still $3.08 left in the account. However, the money will *effectively* run out after about 75 months, or just over 6 years. After this, the money is generating less than $12,000 per year; most people would find it difficult to live on only $1,000 per month.

It will "run out" when the amount left <= his living expenses, whatever that amount might be.

Answer 3

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