Suppose that I have a loan value $x$ and interest rate $r$. The simple interest is then $x\cdot(1+r)$. If I take out a loan compounded annually and paid monthly for $12$ months the amount at the end of the year would be the same. Why is it then that loan calculators give a different value? For example, if I were to borrow $100$ at $10\%$ I would owe $110$ in simple interest and so $110/12$ each month, yet using an online calculator I get that I owe $5.48$ in interest for a total of $105.48$. Why the discrepancy?
2026-03-27 22:30:37.1774650637
Loan Interest Discrepancy
166 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
The reason you are getting a discrepancy is because you are using compound interest calculators to calculate something that is NOT compound interest. You are saying if I divide the total amount by 12, and then add all 12 of them back together, it should be the same as not breaking it apart:
$$x(1+r) = 12 \left(\dfrac{ x(1+r)}{12}\right)$$ Which is clearly true. But compounding annually and breaking apart into 12 payments is very different from compounding monthly.
Consider the formula you provided alongside the formula for compound interest: $$\underbrace{x(1+r)}_{\text{simple}} \neq \underbrace{x\left(1+\tfrac{r}{n}\right)}_{\text{compound}} \biggr|_{n=12,\ 0<r<1,\ 0<x}$$
These are clearly unequal. That is because with compound interest, you are paying a fraction of the annual interest on the currently remaining balance.