I've come to realize that, while I understand the mathematics behind compound interest, I don't quite understand the principles behind it in the context of debt.

I understand that compound interest means interest is charged not only on the principal (the original loan amount), but also on the interest already paid on the loan. What I'm confused about is why I should owe interest on money that I didn't actually borrow. Is there something I'm missing or misunderstanding about this?

What I'm confused about is why I should owe interest on money that I didn't actually borrow.

You're not paying on what you "didn't actually borrow". You pay interest on the outstanding principal balance only every year or every term or every whatever. It's just that you spread the payments over many years in order to reduce the monthly/yearly payments. It's as if you borrow every year the remaining principal because you didn't pay it all back except it's the same lender that is relending you the money.

I understand that compound interest means interest is charged not only on the principal (the original loan amount), but also on the interest already paid on the loan.

I think this is wrong. You don't pay interest on interest "already paid," but rather, you pay interest on interest not already paid.

I can think of one way in which one would pay interest on interest on a loan: A loan on which interest was calculated more frequently than interest payments on the loan were made.

Very simple example:

You borrow $1000.00, to be paid back (including interest owing) at the end of 12 months, in one "lump" payment. Interest on the loan is calculated once every six months, i.e., two times over the period of the loan. Let's say the six-month interest rate is 1%.

So at the end of the first six months, you owe $1000.00 plus six months' interest. That's $1000.00 plus $10.00, i.e., $1010.00. Now, if you paid the bank the $10.00 interest for the first six months, you'd still owe $1000.00 plus interest (another $10.00) for the next six months. At the end of one year, you would owe have paid the bank $1020.00.

But since you don't pay the interest, you now owe $1010.00 plus interest for the second six months.

At the end of the second six months, you owe $1010.00 plus interest, i.e., $1010.00 plus $10.10, i.e. $1020.10. The extra $0.10 is the interest on the first six months' interest.

Also, a note on why people would engage in this kind of arithmetic is in order. It's not just meanness - rather, the ratio between the future value and present value of money can be itself extrapolated to the future. For example, if $1.00 today is worth $1.10 a year from now, then $1.00 a year from now is worth $1.10 two years from now. The key is to realize that these facts are dependent - if $1.00 now is worth $1.10 a year from now, then it is worth $1.21 two years from now. This is called the time-value of money.

Thanks for the responses, everyone. They helped me work things out in my head.

Compound interest on a debt is simply the interest charged on the principal over the compounding period (typically every month). So essentially, it's the price one pays for keeping (part of) the principal for that span of time. If that interest isn't paid off by the next compounding period, it's added to the principal (i.e. becomes part of the outstanding debt). Only then could it be said that interest is charged on existing interest. I think that's only fair, since the existing interest wasn't paid when it was due.