Why is the interest formula $A = P(1 + \frac{r}{m})^{tm}$ and not $A = P(1 + r)^{tm}$?

Question

Sample question: We invest $100,000 in an account earning interest at a rate of 7.5% for 54 months. How much money will be in the account if interest is compounded quarterly?

To me "compounded quarterly" means we apply the interest rate to the current balance every three months, so $100000 \cdot (1 + .075)$ after three months, $100000 \cdot (1 + .075)^2$ after six months, etc.

But apparently that's not correct at all and I don't understand why.

Answer 1

If the interest rate is $7.5$ percent per year, then we don't want it to be $7.5$ percent per quarter. The idea is to take the interest rate, and split it up, applying one quarter of it each quarter (or $1/12$ of it each month, etc.).

If you get $\frac14$ of $7.5$ percent after one quarter, then you're on pace to earn $7.5$ percent per year, but you go ahead and get a partial payment after one quarter.

This is because the given interest rate is an "APR" - an "Annual Percentage Rate". It's not a QPR, if that makes sense.

Answer 2

7.5% is the APR (annual percentage rate). The monthly percentage rate is $\dfrac rm$ where $m$ is the number of times per year that the interest is applied. Note that ther actual percent of principal paid per year is therefore more than 7.5%.

Here is what $\color{red}{\text{Wikipedia}}$ has to say about it.

Answer 3

The interest rate of $ 7.5 $ percent is an annual rate.

Thus for a quarterly rate you need to divide it by $4.$

The new rate is $1.875$ percent and you need to compound it for $18$ quarters.

The final result is $$100000(1.01875)^{18} =139706.6862 $$