Why do we divide by the number of compounding periods when calculating compound interest?

Question

I know this is a very, very basic, but I am attempting to fully understand the implications of the equation that calculates compound interest and can't find anywhere that explains why the equation is the way it is - most sites and textbooks simply provide the formula with no explanation. In its simple form: $$A=P(1+\frac{r}{n})^{ny}$$ I understand why we raise the product to the power of ny but do not quite understand why we divide the rate of interest by the number of compounding periods. How does using the rate of interest as the divisor in relation to the number of periods, plus one, give us any meaningful value that we can use for the rest of the equation?

Answer 1

Suppose that I am to earn 10% every year. If I get one big interest payment at the end of the year, then I should get 10% of my principle in one lump sum. That is, if your initial deposit was $P$, then at the end of the year you should have $$ P + 0.1 P = P(1 + 0.1). $$ Now suppose that you get two payments in a year. Since you are still earning 10% per year, you shouldn't expect to get paid 10% of your principle twice. Instead, you should get half of the interest at the 6 moth mark, then half of the interest at the end of the year. In notation, you get paid $0.05 P$ at the 6 month mark, meaning that you have $P(1+0.05)$ after 6 months, then you paid 5% of that after another 6 months, i.e. you will have $$ \underbrace{P(1+0.05)}_{\text{amt after 6 mo}} + \underbrace{[P(1+0.05)](0.05)}_{\text{interest}} = P(1+0.05)(1+0.05) = P\left( 1+\frac{0.1}{2}. \right)^2 $$ If you earn interest every month, then you should expect to get $\frac{10}{12}\%$ every month. That is, you expect to earn about 0.833% interest 12 times over the course of the year. Generalizing the computation above, this means that at the end of the year, you should have $$ P\left( 1 + \frac{0.1}{12} \right)^{12}. $$ Each time you earn interest, the new amount of money in your account is $A\left(1+\frac{0.1}{12}\right)$ (where $A$ is the amount in your account), and you earn interest 12 times in a year.

Generalizing a bit, suppose that you get $n$ payments over the course of the year, and you are earning $r$ percent (rather than 10%) every year. Then each payment should be $\frac{r}{n}$ percent, which means that if you have $A$ in your account just before you get paid, you will have $$ A\left(1 + \frac{r}{n}\right)$$ in your account just after interest is credited. As this happens $n$ times per year, you should have $$ P\left( 1 + \frac{r}{n} \right)^n $$ in your account at the end of the year. Replace $n$ in the exponent with $ny$ and you have your total after $y$ years.

Answer 2

Because that is what interest rates mean in real life.

For example, if you go to a bank and take out a loan with a 16% interest rate compounded quarterly, you will be charged 4% interest every three months.