Questions tagged [compound-interest]

Compound interest takes into account the simple interest on the original principal amount plus the simple interest at the same rate on the interest that has already been earned.

It is often contrasted with simple interest since in compound interest, the interest increases with time unlike in simple interest where the interest remains fixed since it is calculated only on the original principal amount.

Compound interest depends on $2$ factors:

$1.$ The simple interest rate/nominal annual interest rate applied

$2.$ The frequency of compounding

To better understand the $2^\text{nd}$ example, say the simple interest rate is $r\text{%}$ per annum. Then,

• If compounding occurs annually, the simple interest rate rate will be $r\text{%}$ per compounding.
• If compounding occurs half-yearly$($,i.e., $2$ times a year$)$, the simple interest rate will be $\frac{r}2\text{%}$ per compounding.
• If compounding occurs quarterly$($,i.e., $4$ times a year$)$, the simple interest rate will be $\frac{r}4\text{%}$ per compounding.

Generalizing, if compounding occurs $n$ times a year, the simple interest rate will be $\frac{r}n\text{%}$.

Hence, using this statement, the expression for the net amount$(A)$ via compound interest can be derived in terms of the principal amount$(P)$, nominal annual interest rate $(r)$, frequency of compounding in a year$(n)$ and time in years$(t)$:

$$A=P\left(1+\frac{r}{n}\right)^{tn}$$

It is interesting to note that as frequency of compounding increases(tends to infinity, i.e., gets arbitrarily large), the net amount also increases but approaches a particular value and does not grow arbitrarily large. The limiting ratio of the net amount and the principal amount as the frequency of compounding approaches infinity is defined to be Euler's constant $e$:

$$e=\lim_{n\to\infty}\left(1+\frac1{n}\right)^n$$

Albert Einstein has called compound interest the $8^\text{th}$ wonder of the world.