# Compound Interest

When we observe our bank statements, we generally notice that some interest amount is credited to our account every year. This interest varies with each year for the same principal amount. We can see that interest increases for successive years. Hence, we can conclude that the interest charged by the bank is not simple interest, this interest is known as compound interest or CI.

## Compound Interest Definition

Compound interest is the interest calculated on the principal and the interest accumulated over the previous period. It is different from simple interest, where interest is not added to the principal while calculating the interest during the next period. Compound interest finds its usage in most of the transactions in the banking and finance sectors and other areas. Some of its applications are:

1. Increase or decrease in population.
2. The growth of bacteria.
3. Rise or Depreciation in the value of an item.

### Compound Interest in Maths

To understand the compound interest we need to do its Mathematical calculation using the formula. To calculate compound interest we need to know the amount and principal. It is the difference between amount and principal.

## Compound Interest Formula

The compound interest formula is given below:

Compound Interest = Amount – Principal

Where the amount is given by:

Where,

A= amount

P= principal

R= rate of interest

n= number of times interest is compounded per year

It is to be noted that the above formula is the general formula for the number of times the principal is compounded in a year. If the interest is compounded annually, the amount is given as:

$A = P \left (1 + \frac{R}{100} \right )^t$

Thus, the compound interest rate formula can be expressed for different scenarios such as the interest rate is compounded yearly, half-yearly, quarterly, monthly, daily, etc.

Try out: Compound Interest Calculator

Let us see, the values of Amount and Interest in case of Compound Interest for different years-

 Time (in years) Amount Interest 1 P(1 + R/100) $\frac{PR}{100}$ 2 $P\left (1+\frac{R}{100} \right )^{2}$ P(1 + R/100) (R/100) 3 $P\left (1+\frac{R}{100} \right )^{3}$ P(1 + R/100)2 (R/100) 4 $P\left (1+\frac{R}{100} \right )^{4}$ P(1 + R/100)3 (R/100) n $P\left (1+\frac{R}{100} \right )^{n}$ P(1 + R/100)n-1 (R/100)

This data will be helpful in determining the interest and amount in case of compound interest easily.

 NOTE From the data, it is clear that the interest rate for the first year in compound interest is the same as that in simple interest. PR/100. Other than the first year, the interest compounded annually is always greater than that in simple interest.

## Derivation of Compound Interest Formula

Let, Principal amount = $P$, Time = $n$ years, Rate = $R$

Simple Interest (S.I.) for the first year:

$SI_1$ = $\frac{P~×~R~×~T}{100}$

Amount after first year = $P~+~SI_1$ = $P ~+~ \frac{P~×~R~×~T}{100}$ = $P \left(1+ \frac{R}{100}\right)$ = $P_2$

Simple Interest (S.I.) for second year:

$SI_2$ = $\frac{P_2~×~R~×~T}{100}$

Amount after second year = $P_2~+~SI_2$ = $P_2 ~+~ \frac{P_2~×~R~×~T}{100}$ = $P_2\left(1~+~\frac{R}{100}\right)$ = $P\left(1~+~\frac{R}{100}\right) \left(1~+~\frac{R}{100}\right)$
= $P \left(1~+~\frac{R}{100}\right)^2$

Similarly if we proceed further to $n$ years, we can deduce:

$A$ = $P\left(1~+~\frac{R}{100}\right)^n$

$CI$ = $A~–~P$ = $P \left[\left(1~+~ \frac{R}{100}\right)^n~ –~ 1\right]$

## Compound Interest when the Rate is Compounded half Yearly

Let us calculate the compound interest on a principal, P for 1 year at an interest rate R % compounded half-yearly.

Since interest is compounded half-yearly, the principal amount will change at the end of the first 6 months. The interest for the next six months will be calculated on the total amount after the first six months. Simple interest at the end of first six months,

$SI_1$ = $\frac{P~×~R~×~1}{100~×~2}$

Amount at the end of first six months,

$A_1$ = $P~ + ~SI_1$ = $P ~+~ \frac{P~×~R~×~1}{2~×~100}$ = $P \left(1~+~\frac{R}{2~×~100}\right)$ = $P_2$

Simple interest for next six months, now the principal amount has changed to $P_2$

$SI_2$ = $\frac{P_2~×~R~×~1}{100~×~2}$

Amount at the end of 1 year,

$A_2$ = $P_2~ +~ SI_2$ = $P_2 ~+~ \frac{P_2~×~R~×~1}{2~×~100}$ = $P_2\left(1~+~\frac{R}{2~×~100}\right)$ = P(1 + R/ 2×100)(1 + R/2×100) = $P \left(1~+~\frac{R}{2~×~100}\right)^2$

Now we have the final amount at the end of 1 year:

$A$ = $P\left(1~+~\frac{R}{2~×~100}\right)^2$

Rearranging the above equation,

$A$ = $P\left(1~+~\frac{\frac{R}{2}}{100}\right)^{2~×~1}$

Let $\frac{R}{2}$ = $R'$; $2T$ = $T’$, the above equation can be written as, [for the above case $T$ = $1$ year]

$A$ = $P\left(1~+~\frac{R’}{100}\right)^{T’}$

Hence, when the rate is compounded half-yearly, we divide the rate by 2 and multiply the time by 2 before using the general formula for compound interest.

### Compound Interest Quarterly Formula

Let us calculate the compound interest on a principal, P kept for 1 year at an interest rate R % compounded quarterly. Since interest is compounded quarterly, the principal amount will change at the end of the first 3 months(first quarter). The interest for the next three months (second quarter) will be calculated on the amount remaining after the first 3 months. Also, interest for the third quarter will be calculated on the amount remaining after the first 6 months and for the last quarter on the remaining after the first 9 months. We can also reduce the formula of compound interest of yearly compounded for quarterly as given below:

$A=P(1+\frac{\frac{R}{4}}{100})^{4T}$

CI = A – P

Or

$CI =P(1+\frac{\frac{R}{4}}{100})^{4T}-P$

Here,

A = Amount

CI = Compound interest

R = Rate of interest per year

T = Number of years

## How Compound Interest is Calculated

Let us understand the process of calculating compound interest with the help of the below example.

Example: What amount is to be repaid on a loan of Rs. 12000 for one and half years at 10% per annum compounded half yearly.

Solution:

For the given situation, we can calculate the compound interest and total amount to be repaid on a loan in two ways. In the first method, we can directly substitute the values in the formula. In the second method, compound interest can be obtained by splitting the given time bound into equal periods.

This can be well understood with the help of the table given below.

## Compound Interest vs Simple Interest

Now, let us understand the difference between the amount earned through compound interest and simple interest on a certain amount of money, say Rs. 100 in 3 years . and the rate of interest is 10% p.a.

Below table shows the process of calculating interest and total amount.

### Formula for Periodic Compounding Rate

The total accumulated value, including the principal P plus compounded interest I, is given by the formula:

P’ = P[1 + (r/n)]nt

Here,

P = Principal

P’ = New principal

r = Nominal annual interest rate

n = Number of times the interest is compounding

t = Time (in years)

In this case, compound interest is:

CI = P’ – P

## Compound Interest Examples

Let us solve various examples to understand the concepts in a better manner.

Increase or Decrease in Population

 Examples 1: A town had 10,000 residents in 2000. Its population declines at a rate of 10% per annum. What will be its total population in 2005? Solution: The population of the town decreases by 10% every year. Thus, it has a new population every year. So the population for the next year is calculated on the current year population. For the decrease, we have the formula A = P(1 – R/100)n Therefore, the population at the end of 5 years = 10000(1 – 10/100)5 = 10000(1 – 0.1)5 = 10000 x 0.95 = 5904 (Approx.)

The Growth of Bacteria

 Examples 2: The count of a certain breed of bacteria was found to increase at the rate of 2% per hour. Find the bacteria at the end of 2 hours if the count was initially 600000. Solution: Since the population of bacteria increases at the rate of 2% per hour, we use the formula A = P(1 + R/100)n Thus, the population at the end of 2 hours = 600000(1 + 2/100)2 = 600000(1 + 0.02)2 = 600000(1.02)2 = 624240

Rise or Depreciation in the Value of an Item

 Examples 3: The price of a radio is Rs. 1400 and it depreciates by 8% per month. Find its value after 3 months. Solution: For the depreciation, we have the formula A = P(1 – R/100)n. Thus, the price of the radio after 3 months = 1400(1 – 8/100)3 = 1400(1 – 0.08)3 = 1400(0.92)3 = Rs. 1090 (Approx.)

## Compound Interest Problems

Illustration 1: A sum of Rs.10000 is borrowed by Akshit for 2 years at an interest of 10% compounded annually. Calculate the compound interest and amount he has to pay at the end of 2 years.

Solution:

Given,

Principal/ Sum = Rs. 10000,  Rate = 10%, and Time = 2 years

From the table shown above it is easy to calculate the amount and interest for the second year, which is given by-

Amount($A_{2}$) = $P\left (1+\frac{R}{100} \right )^{2}$

$A_{2}$= $= 10000 \left ( 1 + \frac{10}{100} \right )^{2} = 10000 \left ( \frac{11}{10} \right )\left ( \frac{11}{10} \right )= Rs.12100$

Compound Interest (for 2nd year) = $A_{2} – P$ = 12100 – 10000 = Rs. 2100

Illustration 2: Calculate the compound interest (CI) on Rs.5000 for 2 years at 10% per annum compounded annually.

Solution:

Principal (P) = Rs.5000 , Time (T)= 2 year, Rate (R) = 10 %

We have, Amount, $A = P \left ( 1 + \frac{R}{100} \right )^{T}$

$A = 5000 \left ( 1 + \frac{10}{100} \right )^{2} = 5000 \left ( \frac{11}{10} \right )\left ( \frac{11}{10} \right ) = 50 \times 121 = Rs. 6050$

Interest (Second Year) = A – P = 6050 – 5000 = Rs.1050

OR

Directly we can use the formula for calculating the interest for the second year, which will give us the same result.

Interest (I1) = $P\times \frac{R}{100} = 5000 \times \frac{10}{100} =500$

Interest (I2) = $P\times \frac{R}{100}\left (1 + \frac{R}{100} \right ) = 5000 \times \frac{10}{100}\left ( 1 + \frac{10}{100} \right ) = 550$

Total Interest = I1+ I2 = 500 + 550 = Rs. 1050

Illustration 3: Calculate the compound interest to be paid on a loan of Rs.2000 for 3/2 years at 10% per annum compounded half-yearly?

Solution: Principal, $P$ = $Rs.2000$, Time, $T’$ = $2~×~\frac{3}{2}$ years = 3 years, Rate, $R’$ = $\frac{10%}{2}$ = $5%$, amount, $A$ can be given as:

$A = P ~\left(1~+~\frac{R}{100}\right)^n$

$A = 2000~×~\left(1~+~\frac{5}{100}\right)^3$

= $2000~×~\left(\frac{21}{20}\right)^3 = Rs.2315.25$

$CI = A – P = Rs.2315.25~ –~ Rs.2000$ = $Rs.315.25$

For detailed discussion on compound interest, download BYJU’S -The learning app. Students can also use a compound interest calculator, to solve compound interest problems in an easier way.

## Frequently Asked Questions on Compound Interest – FAQs

### What is Compound interest?

Compound interest is the interest calculated on the principal and the interest accumulated over the previous period.

### How do you calculate compound interest?

Compound interest is calculated by multiplying the initial principal amount (P) by one plus the annual interest rate (R) raised to the number of compound periods (nt) minus one. That means, CI = P[(1 + R)^nt – 1]
Here,
P = Initial amount
R = Annual rate of interest as a percentage
n = Number of compounding periods in a given time

### Who benefits from compound interest?

The investors benefit from the compound interest since the interest pair here on the principle plus on the interest which they already earned.

### How do you find the compound interest rate?

The compound interest rate can be found using the formula,
A = P(1 + r/n)^{nt}
A = Total amount
P = Principal
r = Annual nominal interest rate as a decimal
n = Number of compounding periods
t = Time (in years)
Thus, compound interest (CI) = A – P

### What is the formula of compound interest with an example?

The compound interest formula is given below:
Compound Interest = Amount – Principal
Where the amount is given by:
A = P(1 + r/n)^{nt}P = Principal
r = Annual nominal interest rate as a decimal
n = Number of compounding periods
t = Time (in years)
For example, If Mohan deposit Rs. 4000 into an account paying 6% annual interest compounded quarterly, and then the money will be in his account after five years can be calculated as:
Substituting, P = 4000, r = 0.06, n = 4, and t = 5 in A = A = P(1 + r/n)^{nt}, we get A = Rs. 5387.42

### What is the compounded daily formula?

The compound interest formula when the interest is compounded daily is given by:
A = P(1 + r/365)^{365 * t}

Quiz on Compound Interest