Compound interest coumpounded n time per year formula. $A=P\left(1+\frac{r}{n}\right)^{nt}$ intuition behind it.

Question

I know that the compound interest formula for the interest compounded annually is given by $$A=P(1+r)^t$$ I know the intuition behind it. But why the compound interest formula for the interest compounded n time per year is: $$A=P\left(1+\frac{r}{n}\right)^{nt}$$ What's the intuition behind it and why is it true?

Answer 1

So the intuition behind it is that compounding interest multiple times in a year is the same as compounding at a rate $\frac{r}{n}$, $n$ times.

So, we have $A=P((1+\frac{r}{n})(1+\frac{r}{n})(1+\frac{r}{n})...(1+\frac{r}{n}))^{t}=P(1+\frac{r}{n})^{nt}$

Answer 2

Let's work through this by example. Suppose that our interest compounds $4$ times per year.

• After zero months, $A=P\left(1 + \frac{r}{4} \right)^{4(0)}=P.$
• After three months, $A=P\left(1 + \frac{r}{4} \right)^{4(1/4)}=P\left(1+\frac{r}{4}\right).$
• After six months, $A=P\left(1 + \frac{r}{4} \right)^{4(1/2)}=P\left(1+\frac{r}{4}\right)\left(1+\frac{r}{4}\right).$
• After nine months, $A=P\left(1 + \frac{r}{4} \right)^{4(3/4)}=P\left(1+\frac{r}{4}\right)\left(1+\frac{r}{4}\right)\left(1+\frac{r}{4}\right).$
• After one year, $A=P\left(1 + \frac{r}{4} \right)^{4(1)}=P\left(1+\frac{r}{4}\right)\left(1+\frac{r}{4}\right)\left(1+\frac{r}{4}\right)\left(1+\frac{r}{4}\right).$

Notice how many times $r/4$ has compounded at different times of the year. After one year $r/n$ will have compounded $n$ times.