Question

You deposit​ $5,000 per year at the end of each of the next 25 years into an account that pays​ 8% compounded annually. How much could you withdraw at the end of each of the 20 years following your last deposit if all withdrawals are the same dollar​ amount? (The twentyminusfifth and last deposit is made at the beginning of the 20minusyear period. The first withdrawal is made at the end of the first year in the 20minusyear ​period.)A.​$18,276B.​$27,832C.​$37,230D.​$43,289

Answer 1

Answer:

The correct answer is A. $18,276

Explanation:

First you have to calculate how much you'd end up having at the end of the 25 years period in your savings account.

You calculate the total amount saved for each year, using the formula:

$S_{n} = S_{n-1} *(1+r)+D$

Where

$S_{n}$ is the total amount in the savings account for this period.

$S_{n-1}$ is the total amount in the savings account from the previous period.

$r$is the interest rate.

$D$are the annual deposits being made into the savings account.

Therefore for the first year you'd do:

$S_{1} = S_{0} *(1+r)+D$

$S_{1} = 0*(1+0.08)+5000=5000$

For the second year:

$S_{2} = S_{1} *(1+r)+D$

$S_{2} = 5000*(1+0.08)+5000=10400$

And so on. You can help yourself calculate the value of this series using programs like Excel.

I have attached an Excel file that has a table with the savings values for each of the 25 years.

So, the 25th year you’ll have $365,529.70 in your savings account. Now you simply divide this number by 20 (that will be the number of years you’ll be withdrawing the same dollar amount from your savings account):

$Withdrawals = 365,529.70/20=18,276.485$

In conclusion, you’d be able to withdraw $18,276.485 each year for the following 20 years after the 25th deposit, if all withdrawals are the same dollar amount.