I've been reading about loans and interest rates, and I've run across a mathematical sticking point. The crux is that, if my loan's outstanding balance is £100,000, I make no payments, and it has an annual interest of 8%, I expect that the outstanding balance after one year will be £108,000.
Sites like Investopedia give examples like the following:
The interest on a mortgage is compounded or applied on a monthly basis. If the annual interest rate on that mortgage is 8%, the periodic interest rate used to calculate the interest assessed in any single month is 0.08 divided by 12, working out to 0.0067 or 0.67%.
This example does not support my intuition though. If the balance increases by 0.67% per month and I make no payments, then after a year I will have an outstanding balance of £100,000 * (1 + 0.08/12)^12, which is around £108,300. The actual annual interest is closer to 8.3%, which is somewhat higher than 8%.
If we wanted the monthly interest rate that would cause the yearly interest to actually be 8%, then we should compute the 12th root of 1.08; the monthly interest ought to be around 0.643%.
I've looked around for answers to this discrepancy - a maths.stackexchange post clarifies how the interest rate works, but the root of my question is why the interest rate works like this. If it is indeed the case that an 8% yearly interest actually means that a loan's balance increases by 8.3% per year, what use is the number 8% here?