0
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – GEdgar Nov 26 '14 at 1:58
0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ So normally if I were compounding I would expect the compounded amount be greater than 110 because I'm not paying anything and it continues to accumulate. However, since I am paying off an amount each period and compounding the total interest accumulating is decreasing, so it lowers the total paid to 105.48? Is this the amount that is meant when the term 'APR' is used? $\endgroup$ – user195519 Nov 26 '14 at 3:41
  • $\begingroup$ You must have been using an APR calculator and not a compound interest calculator. If you used compound interest, the total amount paid would be $110.47. APR is a similar but different beast and has a much different formula, which is why the total paid is less than the simple interest. The wikipedia page is insightful, though this is definitely a more advanced topic in interest theory compared to compound interest. I would also check out Khan Academy for visual explanations. $\endgroup$ – mxmrt Nov 26 '14 at 4:11
  • $\begingroup$ Thanks. So APR is different from simple interest and compound interest? Why do so many sites describe it as simple interest then (APR) and then define compound as APY? $\endgroup$ – user195519 Nov 26 '14 at 13:51
  • $\begingroup$ @user195519 APY is annual percentage yield. Informally, it is a way of comparing compound interest wih different parameters. If bank A compounds quarterly at a rate of 2% and bank B compounds monthly at a rate of 1.9, who has the better deal? 2%>1.9% but maybe compounding more often off sets this? APY helps us compare the two. If you just compare rates, it's like telling someone they are tall. If you compare APY its similar to saying you are tall for your age. $\endgroup$ – mxmrt Nov 26 '14 at 22:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.