I’ve recently been learning more about loan amortization. When you make a fixed monthly payment on a loan (say, a mortgage), at first most of the money in each payment just covers interest and very little actually goes towards paying off the principal. Towards the end of the loan period, most of each month’s payment goes towards the principal and very little goes to interest.
There is actually a pretty straightforward reason for this: early on in the loan period, the principal is still very high. At the end of each month, you’ll be charged a large amount of interest on that large principal. Maybe that month you are charged $600 for interest. If your monthly payment is $700, most of that just pays off the $600 of interest, while only $100 is left to reduce the principal.
However, with a (slightly) smaller principal next month, you’ll also owe slightly less interest next month, so slightly more of your payment can be put towards reducing the principal (maybe $103 this month).
Towards the end of the loan period, when there is much less principal left to pay off, the interest on that principal is a smaller dollar value (even though it is still the same percent), so your fixed payment will cover a very small interest payment and much more of the principal.
Whenever you take out a loan, the based on the principal, the interest rate, and the length of the loan, the bank must calculate a monthly payment such that if you pay exactly that amount each month, there will be exactly a zero balance on the loan at the end of the loan period.
It turns out that geometric series play a big part in this calculation and form the basis of the formulas bankers use to do these calculations.
Exploring this in more depth with your students provides a cool opportunity to connect geometric series with a very concrete application. It is also a good opportunity to help your students practice using spreadsheets. Finally, at the end of the lesson, have your students call a bank and see if the students’ numbers match the bank’s!
Here are some more details about the math: