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.

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.

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 

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.

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.

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}

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?”

If option B is picked, at the end of the 31^st 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.

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.