Compound interest is said to have originated in the Old Babylonian period (about 2000-1600 BCE). Babylonians termed it şibt şibtim ("interest on interest") in Akkadian and even solved math problems using it.

Compound interest can be defined as "interest on interest." It will increase the growth of a sum quicker than simple interest, which is computed just on the principal amount. Compound interest can also be called compounding interest.

Compound interest is calculated on a loan or deposit depending on the initial Principal and the gained interest from prior periods. Compound interest vastly differs from simple interest. The previously accrued interest is not added to the principal amount of the current month, resulting in no compounding.

The simple annual interest rate is calculated by multiplying the interest amount per period by the number of periods per year. The nominal interest rate is another term for the introductory annual interest rate.

To calculate compound interest, you must first understand the following:

1. The quantity of your first investment
2. The interest rate offered by your investor
3. How many times is your interest compounded per year
4. The number of years you want to invest for

Once you have this data, you can easily calculate how much you will receive from a compounding interest investment.

In general, the types of compound interest are.

Periodic Compounding

Under this method, the interest rate is applied at intervals and generated. This interest is added to the Principal. Periods here would mean annually, bi-annually, monthly, or weekly.

The total value, including the principal sum P and compounded interest I, is given by:

A=P(1+rn)nt

Where: 

A= final amount,

P= Principal sum,

r = nominal annual interest rate,

n = Compounding frequency

t = Overall length of time the interest is applied (expressed using the same time units as r, usually years).

Continuous Compounding

It is the mathematical limit that compound interest can reach if it's calculated and reinvested into an account's balance over a theoretically infinite amount of periods. 

The amount after t periods of continuous compounding, the amount can be expressed in terms of the initial amount P as

P(t)=Pert

The Benefits of Compound Interest

When it comes to savings and investments, compound interest is your best friend. You stand to make far more from the interest due if you invest. However, compound interest will be your biggest adversary when applied to your loan or other debt. You'll end up paying a large amount of interest on your loan.

Compound interest on fixed deposits is an excellent technique to increase the return on your investment. Compound interest on long-term deposits yields significantly bigger rewards. Compounding interest monthly, quarterly, and semi-annually can raise your interest rate even further. The following are some of the benefits of compound interest:

• Reinvestment 
• The higher value of the deposit 
• Long-term savings 
• Increased Earnings 

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Difference between CI and SI

The difference between compound interest and simple interest is as follows:

Compounding frequency

It is the number of times each year (or, in some instances, another unit of time) that the collective interest is paid out or capitalized (added to the account). The interval might be yearly, half-yearly, quarterly, monthly, weekly, or indefinitely (or not until maturity).

For example, monthly capitalization with interest represented as an annual rate indicates that the compounding frequency is 12, and the periods are measured in months.

The impact of compounding is determined by:

The nominal interest rate and the frequency of interest are compounded.

Annual equivalent rate

The nominal rate cannot be easily compared between loans with various compounding frequencies. Both the nominal rate of interest and the compounding frequency must be provided.

To compare interest-bearing financial instruments,

Many nations require financial institutions to report the yearly compound interest rate on similar deposits or advances to help consumers compare retail financial products more equitably.

In different markets, annual equivalent interest rates are referred to as effective annual percentage rate (EAPR), annual equivalent rate (AER), effective interest rate, effective annual rate, annual percentage yield, etc. The effective annual rate is the total cumulative interest payable up to the end of one year divided by the principal amount.

Solved questions

Q1: Find the amount of Rs 200000 invested at 10% p.a. for five years.

Solution: Using the formula:

A=P(1+R/100)^n

A=20000(1+0.1/100)^5

Solving the above equation, we get 32210.2

Q2: Find the CI if Rs 2000 was invested for two years at 10% p.a. compounded half-yearly.

Solution: Interest is stated to be compounded twice a year. As a result, the interest rate will be cut in half, and the duration will be doubled.

CI = P [1+(R/100)]^n- P

CI = 2000 [1+(5/100)]^4- 2000

On Solving, we get CI = Rs. 431.0125.

Q3: The Compound Interest of Rs 37 in 3 years is Rs 27. Find the rate of interest.

Solution: We know that A = CI + P

A =  37 + 27 = 64

Now going by the formula:

A = P [1+(R/100)]^n

64 = 27 [1+(R/100)]^3

64/27= [1+(R/100)]^3

Since 64 is the cube root of 4 and 27 is the cube root of 3

Therefore,

(4/3)^3 = [1+(R/100)]^3

4/3 = [1+(R/100)]

4/3 - 1 = R/100

On solving, R = 4.1%

Q4:  Money is put on Compound Interest for two years at 20%. It will fetch Rs 483 more if the interest is paid half yearly than annual. Find the sum.

Solution: Let the Principal = Rs 100. When compounded annually,

A = 100 [1+20/100]^2

When compounded half-yearly,

A = 100[1+10/100]^4

Difference,

146.41 - 144 = 2.41

If the difference is 2.41, then Principal = Rs 100. If the difference is 483,

Principal = 100/2.41 × 483

P = Rs 20000.33

Q5: Bhubesh invested a sum of money at compound Interest. It amounted to Rs 2421 in 3 years and Rs 2663 in 3 years. Find the rate per cent per annum.

Solution: Last year interest = 2663 - 2421 = Rs 242

Therefore, Rate% = (242 * 100)/(2420 * 1)

R% = 10%

Essential Formula: To find the difference between SI and CI for two years, we use the formula Difference = P[R/100]^3

Q6: The difference between Simple and Compound Interest for two years @ 20% per annum is Rs 8. What is the principle?

Solution: Using the formula:

Difference = P (R/100)^2

8 = P[20/100]^2

On Solving, P = Rs 200