# Compound Interest

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## Introduction to Compound Interest

Compound interest is the natural evolution of simple interest. Simple interest always is paid on the original fixed amount. Compound interest is when interest is paid on the original fixed amount plus any accumulated interest. So, we end up getting interest on the interest. Essentially, with compound interest you get back a lot more as compared to simple interest.

## The Big Idea: What is Compound Interest?

• In Compound Interest the interest amount is added to the Principal after the passage of a time period. Basically, one can call it interest on interest. This is the key area where compound interest differs from simple interest. When calculating compound interest, the number of compounding periods makes a lot of difference; more the number of compounding periods, more the interest.

• As far as terminologies go, compound interest retains everything from simple interest. The principal is the original amount of money (P), the interest rate is the rate at which loan is charged (R) and the time period is the duration for which the money is loaned/borrowed (T).

The formula for the amount obtained from compound interest is:

\begin{align}\text{A = P}\left(\frac{1+\text{R}}{100}\right)^n\,\end {align}  when money is compounded annually. Both R and n need to be on the same base. For example: R = x per year; n = number of years.

When money is compounded monthly, the formula changes to:
\begin{align}\text{A = P}\left(\frac{1+\text{R}}{m}\right)^{mT} \,\end {align} where m is number of months.

If you want to obtain the interest from the amount, just subtract the principle:

\begin{align}\text{I = P}\left(\frac{1+\text{R}}{100}\right)^n-\text{P = P}\left[\left(\frac{1+\text{R}}{100}\right)^n-1\right]\end{align}

A — Amount,  I — Interest,  P — Principal Amount,  R — Rate of interest,  n — Time period (In years)

### How it works:

It’s quite straight forward. If a principal amount has been deposited for n years, at a rate of interest of R, then after each year, the interest generated gets added to the principal amount deposited and is then considered the new principle for the upcoming year.

Taking an example where P= Rs. 10,000, R=10% and time period=2 years, let us calculate both, simple and compound interest:

\begin{align} & \text{SI}=\frac{10,000\times10\times2}{100}=\text{Rs.}\;2,000\\ & \text{A}\rm{_S}{_I}=\text{Rs.}\;12,000\\ & \text{CI}=10,000\left[\left(\frac{1+10}{100}\right)^2-1\right]=10,000\left[1.21-1\right]=10,000\times0.21=\text{Rs.}\;2,100\\ & \text{A}\rm{_C}\rm{_I}=\text{Rs.}\;12,100\end{align}

### The working of Compound Interest:

Amount after 1 year\begin{align} =10,000+\left(\frac{10,000\times10\times1}{100}\right)=\text{Rs.}\;11,100\end{align}

Amount after 2 years\begin{align} =11,000+\left(\frac{11,000\times10\times1}{100}\right)=\text{Rs.}\;12,100\end{align}

It might look like a meagre amount in such a short amount of time, but  in the long run, compound interest increases drastically, especially if there are more compounding periods. This happened because after each time period passes, C.I also takes into consideration the interest previously acquired which when added over enough time can even double the investment (Investment Doubling Time is an actual term which you will learn later on if you pursue this field in the future).

## Why Compound Interest? Why is it important?

Compound interest works in favor of those who are willing to put in time because short term gains are minimal. In the long run though, there are exponential gains. Compound interest are used for home loans or any kind of long-term loans.

### Be careful:

The interest rates are not always given on an annual basis. Since time periods are of utmost importance in compound interest, they are more likely to be quoted biannually, quarterly or even monthly. As an example, if an investment earns 5% compounded monthly and the initial investment is Rs. 3000 for a period of 5 years, then the compound interest will be calculated as such:

R = 0.05 and the number of times it is compounded each year is, m = 12

P = Rs. 3000 and T = 5 years

Using the formula above:

\begin{align}\text{A}=\text{P}\left(\frac{1+\text{R}}m\right)^{mT}=3000\left(\frac{1+0.05}{12}\right)^{12\times5}=\text{Rs.}\;3850.07\end{align}

## Important Formulas and Shortcuts

• When the interest is compounded Annually: \begin{align}\text{Amount}=\text{P}\left(\frac{1+\text{R}}{100}\right)^n\end{align}

• When the interest is compounded Half-yearly: \begin{align}\text{Amount}=\text{P}\left(\frac{1+\left({\displaystyle\frac {\text{R}}{2}}\right)}{100}\right)^{2n}\end{align}

• When the interest is compounded Quarterly: \begin{align}\text{Amount}=\text{P}\left(\frac{1+\left({\displaystyle\frac {\text{R}}{4}}\right)}{100}\right)^{4n}\end{align}

• When the rates are different for different years, say R1%, R2% and R3% for 1 year, 2 years and 3-year respectively. Then, \begin{align}\text{Amount}=\text{P}\left(\frac{1+\text{R}_1}{100}\right)\left(\frac{1+\text{R}_2}{100}\right)\left(\frac{1+\text{R}_3}{100}\right)\end{align}

• If a certain sum becomes “x” times in n years, then the rate of compound interest will be \begin{align}\text{R}=100\left(\text{x}^\frac1n-1\right)\end{align}

• If a sum of money P amounts to A1 after T years at CI and the same sum of money amounts to A2 after (T + 1) years at CI, then \begin{align}\text{R}=\frac{\left(\text{A}_2-\text{A}_1\right)}{\text{A}_1}\times100\end{align}

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