Compound Interest

“What would you rather have: Rs 10 Lakhs right now or 1 Paise doubled every day for 30 days?”

Rs 10 lakhs seems an enormous amount and sounds very tempting. 

However, getting your money doubled every day for the next 30 days is even a bigger deal.

To your surprise, that 10 lakhs will still remain the same at the end of the month while that single paise would turn into an amount more than 50 lakhs.

compound interset increases over time

Let's see how it works, while we go through the topics compound interest formula, compound interest formula examples, and more.

We are about to realize the power of compound interest in a short while by solving the above riddle.

Compound Interest Calculator

You can find the compound interest in a much easier way with the following compound interest rate calculator.

Enter all the required data like principal amount, rate of interest, time period, frequency of compounding, and then press GO, to get the compound interest on that principal amount.

Lesson Plan 

What Is Meant by Compound Interest?

Compound interest is the interest paid on both principal and interest, compounded in regular intervals. 

At regular intervals, the interest so far accumulated is clubbed with the existing principal amount and then the interest is calculated for the new principal.

New principal = Initial principal + interest accumulated so far.

The compound interest is calculated at regular intervals like annually(yearly), semi-annually, quarterly, monthly, etc;

It is like, re-investing the interest income from an investment makes the money grow faster over time!

It is exactly what the compound interest does to the money.

Banks or any financial organization calculate the amount based on compound interest only. 


What is the Compound Interest Formula?

The amount calculated by the compound interest for a principal amount P, with the rate of interest r, over a period of n years is given by:

\begin{equation}
A=P\left(1+\frac{r}{n}\right)^{n t}
\end{equation}

The compound interest is given by:

\begin{equation}
I=P\left(1+\frac{r}{n}\right)^{n t}-P
\end{equation}

Where,

  • P is the principal amount
  • r is the rate of interest
  • n is frequency or no. of  times the interest is compounded annually
  • t is the overall tenure.

How to Derive a Compound Interest Formula?

The amount calculated by the compound interest for a principal amount P, with the rate of interest r, over a period of n years is given by:

Derivation:

Let the principal is P and the rate of interest be r.

At the end of the first compounding period,

the simple interest on the principal is \(P \times \frac{r}{100}\).

And hence, the amount is \(P+(P \times \frac{r}{100})=P\left(1+\frac{r}{100}\right)\) 

At the end of the second compounding period, 

the simple interest on the principal is \(P\left(1+\frac{r}{100}\right) \times \frac{r}{100}\)
and hence the amount is \(P\left(1+\frac{r}{100}\right)+P\left(1+\frac{r}{100}\right) \times \frac{r}{100}=P\left(1+\frac{r}{100}\right)^{2}\)

Continuing in this manner for n compounding periods, \begin{equation}\text { Amount at the end of the } n^{t h} \text { compounding period is } P\left(1+\frac{r}{100}\right)^{n}\end{equation}

Let us look at an example and understand,

Year Simple growth with 10% compound growth with 10%
Principal Interest Principal Interest
0 10,000 - 10,000 -
1 10,000 1,000 11,000 1,000
2 10,000 1,000 12,100 1,100
3 10,000 1,000 13,310 1,210
4 10,000 1,000 14,641 1,331
5 10,000 1,000 16,105 1,464
Returns after 5 years   5,000   6105.1

You can observe that the compound interest is the same as simple interest for the first interval.

But, over a period of time, there is a remarkable difference in returns.


What Is the Compound Interest Formula When Rate is Half-Yearly?

If the rate of interest is for half-yearly, the interest is calculated for every six months, the amount is compounded twice a year. 

The amount calculated by the compound interest for a principal amount P, with the rate of interest r, semi-annually or half-yearly, i.e., over a period of 6 months is given by:

\begin{equation}
A=P\left(1+\frac{r / 2}{100}\right)^{2 t}
\end{equation}

What Is the Compound Interest Quarterly Formula?

If the rate of interest is quarterly, the interest is calculated for every three months, the amount is compounded 4 times a year. 

The amount calculated by the compound interest quarterly for a principal amount P, with the rate of interest r, is given by:

\begin{equation}
A=P\left(1+\frac{r / 4}{100}\right)^{4 t}
\end{equation}

Now let us come back to our interesting question:

“Which one will you pick: Rs 10 Lakhs right away or 1 Paise doubled every day for 30 days?”

simple interest vs compound interest

If option B is picked, at the end of the 31st day, we will get Rs 1.07 crore.

If we choose one paise doubled for 31 days, the result might seem low and disappointing in the initial days.

However, if we keep on going, the final amount we receive at the end of the month will be worth the patience.

This is magic in the compound interest.

Note: Here every day the amount is depending on the previous day's amount, not on the original amount of 1 paise.

 
important notes to remember
Important Notes
  1. Compound interest depends on the amount accumulated at the end of the previous tenure but not on the original principal. 
  2. Banks, insurance companies, etc. generally levy compound interest.
  3. If the interest is compounded annually, the formula of amount is given by:\begin{equation}A=P\left(1+\frac{r / 4}{100}\right)^{4 n}\end{equation}
  4. While calculating the compound interest, the rate of interest, and each time period must be of the same duration.

Solved Examples

Example 1

 

 

Noah lends $4000 to Emma at an interest rate of 10% per annum, compounded half-yearly for a period of 2 years.

Can you help him find out how much amount he gets after a period of 2 years from Emma?

Solution

Let us identify the data given to us:

The principal amount 'P' is $4000.

The rate of interst 'r' is 10% per annum.

\begin{equation}
\begin{aligned} \text { Conversion period } &=\text { Half year } \\ \therefore \text { Rate of interest per half year } &=\frac{10}{2} \% \\ &=5 \% \end{aligned}
\end{equation}

The time period 't' is 2 years.

Time Period Principal (₹) Interest (₹) Amount (₹)
1st half year \($4000\) \(\begin{align}5 \% \text{ of } $4000 = $200 \end{align}\) \($4200\)
2nd half year \($4200\) \(\begin{align}5 \% \text{ of } $4200 = $210 \end{align}\) \($4410\)
3rd half year \($4410\) \(\begin{align}5 \% \text{ of } $4410 = $220.5 \end{align}\) \($4630.5\)
4th half year \($4630.5\) \(\begin{align}5 \% \text{ of } $4630.5= $231.53 \end{align}\) \($4862.03\)

\(\begin{align}\text{The total interest to be paid over 2 years } &= $200 + $210 + $220.5 + $231.53 \\& = $862.03 \end{align} \)

Total Amount = P + I=$4000 + $862.03 = $4862.03

\(\therefore\) Total Amount 'A' is $4862.03
Example 2

 

 

Let us solve the above problem using the compound interest formula.

Solution

The principal amount 'P' is $4000.

The rate of interst 'r' is 10% per annum.

\begin{equation}
\begin{aligned} \text { Conversion period } &=\text { Half year } \\ \therefore \text { Rate of interest per half year } &=\frac{10}{2} \% \\ &=5 \% \end{aligned}
\end{equation}

The time period 't' is 2 years.

The compounding frequency 'n' is 2. 

Let us substitute the given data in the compound interest formula:

\begin{eqnarray}
\text{A}&=&\text{P}\left(1+\frac{r / 2}{100}\right)^{2 n}\\\\
&=&\text{4000}\left(1+\frac{(10 / 2)}{100}\right)^{2(2)}\\\\
&=& $4862.03
\end{eqnarray}

\(\therefore\) The final amount is $4862.03.

The compound interest formula makes the solution simple.

Example 3

 

 

Compare the returns when the interest compounded yearly, semi-annually, quarterly, and monthly if \(\$ 30000\) is invested at \(10 \%\) for 2 years.

Solution

The amount of return for compound interest formula is: \begin{equation}A=P\left(1+\frac{r}{n}\right)^{n t}\end{equation}

Here, P = \(\$ 30000\), r = \(10 \%\), t = 2.

Annually Semi-Annually Quarterly Monthly
\begin{equation}
A=P\left(1+{r}\right)^{t}
\end{equation}
\begin{equation}A=P\left(1+\frac{r / 2}{100}\right)^{2 t}\end{equation} \begin{equation}A=P\left(1+\frac{r / 4}{100}\right)^{4 t}\end{equation} \begin{equation}A=P\left(1+\frac{r / 12}{100}\right)^{12 t}\end{equation}
\begin{equation}
A=30000\left(1+{0.1}\right)^{2}
\end{equation}
\begin{equation}A=30000\left(1+\frac{10 / 2}{100}\right)^{2(2)}\end{equation} \begin{equation}A=30000\left(1+\frac{10 / 4}{100}\right)^{4(2)}\end{equation} \begin{equation}A=30000\left(1+\frac{10 / 12}{100}\right)^{12(2)}\end{equation}
\(\$ 36,300\) \(\$ 36,465.19\) \(\$ 36,552.09\) \(\$ 36,611.73\)
\(\therefore\) The returns increases if the number of compound cycles increases.
 
tips and tricks
Tips and Tricks
  1. The rule of 72: It is a quick method to know how long it will take for your money to double.
  2. Doubling Time \(=\frac{72}{\text { Interest Rate }}\)
  3. Using the rule of 72, we can find the number of years to double your money by simply dividing 72 by the rate of interest.
    • For example, at an 8% compounded interest rate your money will double in \(72 \div 8 = 9\)
  4. The time duration over which an interest rate is applicable is referred to in many different terms.
    • Sometimes it is called “per annum” or “annual” or “per year”. All of these mean you’ll get the given rate of interest over a period of 1 year.
    • Semi-annual is 6 months.
    • While quarterly is 3 months duration.

Interactive Questions

Here are a few activities for you to practice.Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of compound interest. The math journey around compound interest started with what a student already knew and went on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. 

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Frequently Asked Questions (FAQs)

1. Is interest compounded daily better than monthly?

The interest compounded daily has 365 compounding cycles a year. It will generate more money compared to interest compounded monthly, which has only 12 compounding cycles per year.

2. What are the main disadvantages of compound interest?

If we miss a payment by a day also, towards the end of tenure it may incur a huge loss.
Compound interest is actually designed to help the lenders but not the borrowers.

3. What is the difference between simple and compound interest?

Simple interest is the interest paid only on the principal, whereas, compound interest is the interest paid on both principal and interest compounded in regular intervals.

  
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