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?
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2 Answers
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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 $A=P((1+\frac{r}{n})(1+\frac{r}{n})(1+\frac{r}{n})...(1+\frac{r}{n}))^{t}=P(1+\frac{r}{n})^{nt}$
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$\begingroup$ @user153001 Because Silynn just proved it? $\endgroup$ May 23, 2014 at 21:44
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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.
