[FONT=q_serif]If you have $20,000 in an account earning 8% annually, what constant amount could you withdraw each year and have nothing remaining at the end of five years?[/FONT]
The way i did it was
20,000*0.08= 1600
1600*5= 8000
20,000+8000= 28,000
28,000/5= 5600
So my answer is 5600 but the answer i found almost everywhere on the internet is 5009 or 5008 but can't seem to find an understandable solution.
I know my solution is completely wrong but what am i doing wrong?
Thank you for the answers!
mmm has explained that the reason your method gives the wrong answer is that you are assuming that the same amount is earning interest each year. This would be true only if the amount withdrawn at the end of the year is the amount of interest earned during the year, but in that case you would still have 20,000 in the account after 5 years, not nothing (or close to nothing).
\(\displaystyle 20,000 + 20,000 * 0.08 - x = 20,000 * 1.08 - x * (1) =\)
the amount remaining to earn interest at the end of year 1.
Do you see that?
\(\displaystyle (20,000 * 1.08 - x) + (20,000 * 1.08 - x) * 0.08 - x = 20,000 * (1.08)^2 - x * (1 + 1.08) =\)
the amount remaining to earn interest at the end of the second year.
Similarly, \(\displaystyle \{(20,000 * 1.08^2 - x * (1 + 1.08)\} * 1.08 - x = \\
20,000 * 1.08^3 - x(1 + 1.08 + 1.08^2) =\)
the amount remaining to earn interest at the end of the third year.
You should see the pattern now. I'll let you work out the math for the fourth and fifth year, but here is where you should end up at the end of the fifth year, which is also nothing left, so
\(\displaystyle 20,000(1.08)^5 - x * (1 + 1.08 + 1.08^2 + 1.08^3 + 1.08^4) = 0 \implies \\
x = \dfrac{20,000 * 1.08^5}{1 + 1.08 + 1.08^2 + 1.08^3 + 1.08^4} \approx 5009.12.\)
Note that the answer is approximate, but let's check how close it is.
20,000 * 1.08 - 5009.12 = 16590.88. That's what's in the account after 1 year.
16590.88 * 1.08 - 5009.12 = 12909.03 after 2 years.
12909.03 * 1.08 - 5009.12 = 8932.63 after 3 years.
8932.63 * 1.08 - 5009.12 = 4638.12 after 4 years.
4638.12 * 1.08 - 5009.12 = 0.05 after 5 years.
However, if you work with 5009.13, you will overdraw the account and pay a fee. The exact amount is closer to 5009.13 than 5009.12, but we do not work with fractional pennies.
Of course, bankers and such peculiar people use formulas, but I have explained the math behind the formulas.
EDIT: I notice that denis posted the formula while I was scribbling. He can do the math, and I know the formula so either of us could have written the other's answer. (So yes, that makes both of us peculiar, a fact that many here will affirm.)
