Question

How much money should the department deposit at the end of each quarter in order to save enough to buy a new copy machine at the end of 5 years?

The Math department purchased a copy machine for
$ 12000. After 5 years, the machine will be worthless. How
much money should the department deposit at the end of each
quarter, if the money is worth 7% compounded quarterly, in order
to save enough to buy a new copy machine at the end of 5
years?

Answer 1

A monthly investment of `$167` will accumulate `$12013` in five years. Whether a replacement copier is still available at that price is another question.

Explanation:

This is a very practical compound interest problem. It is a bit trickier than a straight accumulation problem because the amount on which the interest is calculated changes with every payment.

There is a direct formula for the calculation, but to understand how it works better and to use it for personal situations I prefer to calculate each step individually. I use a spreadsheet program to make the repetitive calculations easier. Then, given a principle amount, an interest rate and payment schedule, the paydown can be calculated and the number of years or months for repayment determined.

The interest rate charged (or paid) on the principle depends on the time period. An annual rate, compounded monthly, means that `1/12` the rate is applied each month to the outstanding principle. Then any payment is deducted (or added), the principle + interest balance is reduced, and the cycle starts over.

The advantage of a spreadsheet program with individual entries instead of a straight time formula is that you can “play games” with the amounts to see how missed or late payments, or early/additional payments affect the payback period and the total interest paid. For this problem we have A = 12000, t = 5, r = 0.07 , and n = 4.

An amount is decreased by a fixed amount over a set time interval, and the new amount becomes the basis for the next interval interest calculation. The formula for compound interest is:
`A = P(1 + r/n)^(nt)`

Where
P = principal amount (the initial amount you borrow or deposit)
r = annual rate of interest (as a decimal) (or Monthly rate x 12)
t = number of years the amount is deposited or borrowed for.
A = amount of money accumulated after n years, including interest.
n = number of times the interest is compounded per year.
https://qrc.depaul.edu/StudyGuide2009/Notes/Savings%20Accounts/Compound%20Interest.htm

A monthly investment of `$167` will accumulate `$12013` in five years as found by setting the known values in the equation and adjusting the investment amount until the desired final amount is calculated.
(interest compounded quarterly - added at the end of each quarter)

Year Deposits Interest Total Deposits Total Interest Balance
1 $2,004.00 $77.18 $2,004.00 $77.18 $2,081.18
2 $2,004.00 $226.73 $4,008.00 $303.90 $4,311.90
3 $2,004.00 $387.03 $6,012.00 $690.93 $6,702.93
4 $2,004.00 $558.84 $8,016.00 $1,249.77 $9,265.77
5 $2,004.00 $743.01 $10,020.00 $1,992.78 $12,012.78
https://www.thecalculatorsite.com/articles/finance/compound-interest-formula.php