# Compound Interest

**Compound interest** is the interest that is calculated against a loan or deposit amount in which interest is calculated for the principal as well as the previous interest earned. The common difference between compound and simple interest is that in compound interest, interest is calculated for the principal amount as well as for the previously earned interest whereas simple interest depends only on the principal invested.

**Amount: **The total sum of money that a person gets after a certain period of time including the interest is called the amount.

## Definition of Compound Interest

Compound interest is the interest calculated on the principal and the interest earned previously. It is denoted by C.I. it is very useful for investment and loan repayment purposes. It is also known as “interest on interest”.

Compound interest is very useful in the banking and finance sectors and is also useful in other sectors. A few of its use are:

- Growth of population of a country
- Value of investment over a period of time.
- For finding Inflated costs and the depreciated value of any article.
- For predicting the growth of any institution or country.

**Compound interest (C.I) = Amount – Principal**

## How to Calculate Compound Interest?

Compound interest is the interest paid both on principal as well as interest accumulated. The interest earned at each interval is added to the initial principal ad thus principal goes on increasing.

Use the following methods to find the compound interest.

**Step 1:** Note the Principal, rate, and time period given.

**Step 2:** Calculate the amount using the formula A = P(1 + r/100)^{n}

**Step 3:** Find the Compound Interest using the formula CI = Amount – Principal

## Compound Interest Formula

Compound interest is calculated, after calculating the total amount over a period of time, based on the rate of interest, and the initial principal. For an initial principal of P, rate of interest per annum of r, time period t in years, frequency of the number of times the interest is compounded annually n, the formula for calculation of CI is as follows.

CI = P ${(1+\frac{r}{100})}^{n}-P$

Where,

P = principal

r = rate of interest

n = number of times interest is compounded per year

t = time (in years)

**Compound Interest = A – P**

## Half-yearly Compound Interest formula (**rate = (R/2)%)**

## A = P(1 + R/200)^{2t}

## CI = A – P

## Quarterly Compound Interest formula (**rate = (R/4)%)**

**A = P(1 + R/400)**^{4t}

^{4t}

**CI = A – P**

## Periodic Compounding Rate

The total amount, including the principal P and compounded interest CI is given by:

**A = P[1 + (r/n)] ^{nt}**

where,

P = Principal

A = Final amount

r = annual interest rate

n = number of times interest is compounding

t = Time (in years)

Thus, compound interest is: **CI = A – P**

## Rule of 72

Rule of 72 is the formula that is used to estimate, **how many years our money gets doubled** if it is compounded annually. For example, if our money is invested at **r % **compounded annually then it takes 72/r years for our money to get doubled.

### Rule of 72 formula

The following formula is used to approximate the number of years for our investment to get doubled.

**N = 72 / r**

where,

N is approximate number of year our money get doubled

r is the rate at which our money is compounded annually

**Compound Interest of Consecutive Years**

If we have the same sum and at the same rate of interest. The C.I of a particular year is always more than C.I of Previous Year. (CI of 3rd year is greater than CI of 2nd year). The difference between CI for any two consecutive years is the interest of one year on C.I of the preceding year.

**C.I of 3rd year – C.I of 2nd year = C.I of 2nd year × r × 1/100 **

The difference between the amounts of any two consecutive years is the interest of one year on the amount of the preceding year.

**Amount of 3rd year – Amount of 2nd year = Amount of 2nd year × r × 1/100 **

When we have the same sum and same rate,

**C.I for nth year = C.I for (n – 1)th year + Interest for one year on C.I for (n – 1)th year**

**Some Other Applications of Compound Interest**

**Growth:** This is mainly used for growth if industries are related.

**Production after n years = initial production × (1 + r/100) ^{n}**

**Depreciation: **When the cost of a product depreciates by r% every year, then its value after n years is

**Present value × (1 + r/100) ^{n}**

**Population Problems: **When the population of a town, city, or village increases at a certain rate per year.

**Population after n years = present population × (1 + r/100) ^{n}**

## Difference between Compound Interest and Simple Interest

- CI is the interest that is calculated both on the principal and the previously earned interest. SI is the interest that is calculated only on the principal.
- For the same Principal, Rate, and Time period
**CI > SI .**For the same Principal, Rate, and Time period**SI < CI**