# Loan Interest Discrepancy

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?

• You pay interest only on the portion not yet repaid. So each payment consists of: whatever interest is owed for that month, plus an installment on the principal. The next month, the principal is smaller, so the interest is smaller, and the payment toward the principal is larger. – GEdgar Nov 26 '14 at 1:58

$$x(1+r) = 12 \left(\dfrac{ x(1+r)}{12}\right)$$ 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: $$\underbrace{x(1+r)}_{\text{simple}} \neq \underbrace{x\left(1+\tfrac{r}{n}\right)}_{\text{compound}} \biggr|_{n=12,\ 0<r<1,\ 0<x}$$