Compound Interest

Compound interest is the interest calculated on both the initial principal (the original amount) and the accumulated interest from the previous periods. In simple words, it means you earn "interest on interest," so the money grows faster compared to simple interest (which is only calculated on the principal).

It is used in the banking and finance sectors and is also useful in other sectors. A few of its uses are:

• Growth of the population of a country
• Value of investment over a period of time.
• To find the inflated costs and the depreciated value of any article.
• For predicting the growth of any institution or country.

Compound Interest Formula

Compound interest is calculated by finding the total amount accumulated over a period of time, based on the initial principal, the rate of interest, and the frequency of compounding.

Where,

• P = Initial Principal Amount
• r = Annual Interest Rate
• n = Number of Times Interest is Compounded
• t = Number of Years

Note : Compound Interest can be calculated yearly, half-yearly, quarterly, monthly, daily, etc; as per the requirement.

Derivation of Compound Interest Formula

We can derive the compound interest formula starting from the formula for simple interest.

Simple Interest (SI) = (P x R x T)/100

For the First Year

• SI₁ = (P x R x 1)/100
• Amount after Year 1: A₁ = P + SI₁ = P + (P x R/100) = P(1 + (R /100))

For the Second Year

• P₂ = A₁ = P (1 + (R/100))
• Interest for Year 2: SI₂ = (P₂ x R)/100
• Amount after Year 2: A₂ = P₂​ + SI₂ = P(1 + (R/100)) (1 + (R/100)).

Generalizing for n years

• A_n = P(1 +(R/100))ⁿ
• Compound Interest: CI = A_n​ − P = P(1 + (R/100))ⁿ - P

Final Formula: CI = P[1 + (R/100))ⁿ - 1]

Compound Interest Calculator

Compound Interest is calculated at regular intervals like annually(yearly), semi-annually, quarterly, monthly, etc. It is like re-investing the interest income from an investment makes the money grow faster over time! It is exactly what compound interest does to money. Banks or any financial organization calculate the amount based on compound interest only.

Compound Interest = Interest on Principal + Interest on Interest of Principal(From second year and onwards)

Try it yourself:

What will $10000 be worth in 10 years?

Suppose you have $10,000 in a savings account with an annual interest rate of 10%, compounded yearly. Assuming you don’t withdraw any money for 10 years, your investment will have grown to:

Your savings of $10,000 at a fixed interest rate of 10% for 10 years has grown to $25,937.43. This means your initial savings have increased approximately 2.5 times, showcasing the power of compound interest.

The Same Amount would have compounded to $67,275 after 20 years.

General Compound Interest Formulas

If P is compounded n times per year at an annual interest rate r, the interest rate r is divided by n and applied n times per year. So, after t years, the formula becomes:

A = P(1 + r/n)^nt

Where,

• r/n- interest rate per compounding period.
• nt - total number of compounding periods over t years.

Half-yearly Compound Interest Formula

The principal is compounded half-yearly; the principal will be changed at the end of 6 months, and interest earned till then will be added to the principal, and then this becomes the new principal. Similarly, the final amount is calculated as:

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

Quarterly Compound Interest formula

If the principal is compounded quarterly, the principal will be changed at the end of 3 months, and interest earned till then will be added to the principal, and then this becomes the new principal. Similarly, the final amount is calculated as:

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

Monthly Compound Interest Formula

If the interest is compounded monthly then the number of times compounding will be 12 and the interest each month will be 1/12 of the annual compound interest. Hence, the Compound Interest Formula is given as:

A = P[1 + (R/1200)]^12t
CI = A - P

Daily Compound Interest Formula

If interest is compounded daily, then. The new Rate of interest will be R/365% and n = 365

Hence, the Daily Compound Interest Formula is given as:

A = P[1 + (R/36500)]^365t
CI = A - P

Rule of 72

Rule of 72 is the formula that is used to estimate how many years our money gets double 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 double.

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

N = 72 / r

Where,

• N - Approximate number of years it takes for our money to double.
• r - Rate at which our money is compounded annually.

Example: Suppose Kabir has invested 10,00,000 dollars in a debt fund that gives an 8% return. In how many years will his money double if it is compounded annually?

Using above formula:
N = 72/8 = 9 years

Thus, it takes 9 years for Kabir's money to get doubled.

Compound Interest of Consecutive Years

If we have the same sum and the same rate of interest. The C.I. of a particular year is always more than C.I of the 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 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 

Key Results

When we have the same sum and the 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

Continuous Compounding Interest Formula

Continuous Compounding Formula is used in Finance to calculate the final value of an investment that undergoes continuous compounding over different periods, and the value is added over time. The formula for continuous compounding is given as

Final Value = Present Value × e^rt

Where,

• e - Euler's number
• r - Rate of interest
• t - Time

Interesting Fact:

Suppose you have $100 and you put it in a saving account with the fixed interest rate of 10%, so after 100 years, this amount will increased to $1378061.23​.

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

Simple Interest vs Compound

The difference between Compound Interest and Simple Interest can be understood in the following table:

Solved Examples of Compound Interest

Example 1: Find the Compound Interest when principal = $6000, rate = 10% per annum, and time = 2 years.

Solution: 

Interest for first year = (6000 × 10 × 1)/100 = 600
Amount at the end of first year = 6000 + 600 = 6600
Interest for second year = (6600 × 10 × 1) / 100 = 660
Amount at the end of second year = 6600 + 660 = 7260

Compound Interest = 7260 - 6000 = $1260

Example 2: What will be the compound interest on $8000 in two years when the rate of interest is 2% per annum?

Solution:

Given,
Principal P = 8000
Rate r = 2%
Time = 2 years

by formula

A = P (1 + R/100)ⁿ
A = 8000 (1 + 2/100)² = 8000 (102/100)²
A = 8323

Compound interest = A - P = 8323 - 8000 = $323

Example 3: Sam deposited $4000 with a finance company for 2 years at an interest rate of 5% per annum. What is the compound interest that John gets after 2 years?

Solution: 

Given,
Principal P = 4000
Rate r = 5%
Time = 2years

By formula,

A = P (1 + R/100)ⁿ
A = 4000 (1 + 5/100)²
A = 4000 (105/100)²
A = 4410

Compound Interest = A - P = 4410 - 4000 = $410

Example 4: Find the compound interest on $2000 at the rate of 4 % per annum for 1.5 years. When is interest compounded half-yearly?

Solution:

Given,

Principal p = 2000
Rate r = 4%
Time = 1.5 ( i.e 3 half years )

by formula ,

A = P (1 + R/200)²
A =  2000 (1 + 4/200)³
A = 2000 (204/200)³
A = 2122

Compound Interest = A - P = 2122 - 2000 = 122

Related Articles:

• Complete Guide to Compound Interest
• Tricks to Solve Compound Interest Questions
• Compound Interest - Aptitude Questions and Answers

Suggested Quiz

10 Questions

An amount of Rs.9000 is invested for 2 years at interest rate of 15% per annual and compounded annually. At the end of 2nd year how much amount will be obtained as interest?

• A

  Rs. 2902.50

• B

  Rs. 2900.50

• C

  Rs. 2899.50

• D

  Rs. 2899

Explanation:

Amount = P[1+ (R/100)]ⁿ 
where P = principal, R = rate of interest and n = time(years)
Amount= Rs.9000 × [1 + (15/100)]ⁿ = 9000 × (23/20)ⁿ = 23805/2 = Rs.11902.5
The amount obtained by the way of interest in compound interest = Amount - principal = Rs.(11902.5 - 9000) = Rs.2902.50

Ram deposits Rs.2000 each on 1^st January and 1^st July of a year at the rate of 8% compound interest calculated on half-yearly basis. At the end of the year how much amount he would have?

• A

  Rs.4215.50

• B

  Rs.4182.40

• C

  Rs.4243.20

• D

  Rs.4280.40

Explanation:

We can break this problem into two parts: Rs. 1500 invested for 1 year (Jan to Dec) and Rs. 2000 invested for 6 months (Jul to Dec)
When interest is compounded Half-yearly:
Amount = P[1+ (R/2)/100 ]²ⁿ
The total amount for the investment on 1^st Jan is: Amount1 = Rs. 2000 × [1+ (8/2)/100]^2×1 = Rs. 2000 × [1 + (4/100)]² = Rs. 2000 × [26/25]²
The total amount for investment on 1^st july is: Amount2 = Rs. 2000 × [1+ (8/2)/100][2 × (1/2)] = Rs. 2000 × [1+ 4/100 ] = Rs. 2000 × [26/25]
The total amount at the end of the year = amount1 + amount2 = 2000 × [26/25]² + 2000 × [26/25] = 2000 × [26/25] × [(26/25) + 1] = 2000 × 26/25 × 51/25 = 4243.20

Find the interest returned for an investment of Rs.5,000 after 2 years, if the rate of interest for the 1st year is 5% and for the 2nd year is 10%.

• A

  772

• B

  775

• C

  622

• D

  820

Explanation:

Amount = (Principal + Compound interest) = P(1 + R1/100)(1 + R2/100)(1 + R3/100)
Amount = 5000(1 + 5/100)(1 + 10/100)
= 5000 × (21/20)(11/10)
= 5000 × (231/200)
=5775
Interest = 5775 - 5000 = 775

What sum will be amount to Rs.30000 at CI in 3 years, if the rate of interest for 1^st, 2^nd and 3^rd year being 10%, 20% and 30% respectively?

• A

  17482.5

• B

  20145

• C

  16524

• D

  17000

Explanation:

Let Rs.P be the required sum.
30000= p(1 + 10/100)(1 + 20/100)(1 + 30/100)
= p (110/100) × (120/100) × (130/100)
p = 30000 × 100 × 100 × 100 / (110 × 120 × 130)
p = 17482.5

An amount of Rs.9600 lent out at a rate of 4.5 % per annum for a 1 year and 9 months. At the end of the period, the amount he earned was:

• A

  10450.69

• B

  10368.7

• C

  10465.69

• D

  10555.69

Explanation:

Amount = 9600 × (1+(4.5)/100)^7/4 = 10368.7103188.

So, option (B) is correct.

An amount of Rs.1500 is invested at 10% per annum for one year. If the interest is compounded half-yearly, then what amount will be received at the end of the year?

• A

  1652.20

• B

  1642.50

• C

  1652.7

• D

  1653.75

Explanation:

Amount = 15000[1+(10/2)/100]²
= 1500[105/100]²
= 1500 X 1.1025
= 1653.75

Find the compound interest on Rs. 15,000 for 9 months at 16% per annum compounded quarterly.

• A

  1872.96

• B

  1972.96

• C

  2072.96

• D

  2172.96

Explanation:

CI = 15000[1+(16/4)/100]^4X(9/12) -15000
= 15000[104/100]³ - 15000
= 15000 [1.125-1]
= 15000 X 0.125
= 1872.96

The least number of complete years in which a sum of money put out at 20% compound interest will be more than doubled is:

• A

  6

• B

  4

• C

  5

• D

  3

Explanation:

Let the Principal be P.
Then, P(1 + 20%)^n > 2P
Or, P(1 + 1/5)^n > 2P
Or, P(6/5)^n > 2P
Or (6/5)^n > 2.
Now, (6/5)^3 = 1.728 (6/5)^4 = 2.074
Therefore, in 4 years.

A sum of money is borrowed and paid back in two annual installments of Rs. 882 each allowing 5% compound interest. The sum borrowed was:

• A

  1540

• B

  1640

• C

  1580

• D

  1680

Explanation:

Principal = Present worth of the installment at the end of first year + Present worth of the installment at the end of second year .
Principal = 882/(1 + 5%) + 882/(1 + 5%)²
Principal = 882/(1 + 5/100) + 882/(1 + 5/100)²
Principal = 882/(105/100) + 882/(105/100)²
Principal = 882/(21/20) + 882/(21/20)²
Principal = 882 × 20/21 + 882 × 400/441
Principal = 42 × 20 + 2 × 400
Principal = 840 + 800 = 1640.

The difference between compound interest and simple interest on an amount of Rs 15,000 for 2 years is Rs 96. What is the rate of interest per annum?

• A

  8

• B

  9

• C

  10

• D

  None of these

Explanation:

Simple Interest = 15000 × R/100 × 2
Compound Interest = 15000 × (1 + R/100)² - 15000
We have, [ 15000 × (1 + R/100)² - 15000 ] - [15000 × R/100 × 2] = 96
15000 [ (1 + R/100)² - 1 - 2R/100 ] = 96
15000 [ 1 + R²/10000 + 2R/100 - 1 - 2R/100 ] = 96
15000 [ R²/10000 ] = 96
R² = 96 × 10000/15000 =
96 × 2/3 = 64 = R²
R = 8.

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