Suppose we examine our bank statements, we note that some interest is credited each year to the same principal amount. For the same principal amount, this interest is different each year. The bank charges compound interest, known as CI because the interest increases for successive years. Since interest increases for successive years, it is not simple interest. CI is a tool that allows one to determine price changes by compounding every year, half-yearly, quarterly, etc.

Through the examples given based on real-life applications, one can also understand why compound interest yields more than simple interest. In contrast to simple interest, the compound interest formula is calculated based on both the principal and interest accumulated over some time. However, compound interest does not have the same impact on the principal as simple interest. Among the many popular uses of compound interest are in the banking and finance sector, as well as in other areas. It can be used for things like Increase or decrease in population, bacteria growth, or rise or decrease in the value of goods.

## The formula for compound interest:

To calculate compound interest, we need to know the amount and principal. It is the difference between amount and principal. To calculate compound interest, we can use the compound interest formula, which is:

**Compound
Interest = Amount – Principal**

Here, Amount (A) = P (1+r/n) nt

In this, A = amount, P = principal, r = rate of interest, n = number of times interest is compounded per year, t = time (in years)

As noted above, this is the general formula for discovering how many times the principal will compound in a given year.

**Using
the annual compound interest method, this amount is as follows:**

A= P (1+R/100)t

If the interest rate is compounded annually, half-yearly, quarterly, monthly, daily, and so forth, the compound interest rate formula can be expressed differently.

## How to derive the formula for compound interest?

We know that SI for one year is equal to CI (compound interest when compounded annually). So we can use the simple interest formula to calculate compound interest.

For example, let P be the principal amount, n years be the period, and R be the interest rate

**Simple
interest for year 1:**

SI 1 = P×R×T/ 100

Amount after year 1= P + SI 1

= P + P×R×T/100

= P (1+R/100) = P 2

**Simple
interest for year 2:**

SI 2 = P×R×T/ 100

Amount after year 2= P 2 + SI 2

= P 2 + P 2 ×R×T/100

= P 2 (1+R/100)

= P (1+R/100) (1+R/100)

= P (1+ R/100) 2

### Compound interest compounded half-yearly:

- The principal amount will shift after the first six months, for which the interest for the next six months will be calculated on the amount after six months.
- Compound interest is calculated on a half-year basis. Using the general formula for compound interest, the half-yearly rate is divided by two and the time is multiplied by two.

### Compound interest compounded quarterly:

- In this case, since interest is compounded quarterly, the principal amount will change after the first three months (first quarter) and the interest and charges for the next three months (second quarter) will be based on the balance after the first three months.
- Likewise, interest will be calculated on the remaining balance after the first 6 months for the third quarter and over the remaining balance after the first 9 months for the fourth quarter.

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