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.
Table of Content
- What is Compound Interest?
- Compound Interest Formula
- How to Calculate Compound Interest?
- Compound Interest Formula – Derivation
- Half-yearly Compound Interest Formula
- Quarterly Compound Interest formula
- Monthly Compound Interest Formula
- Daily Compound Interest Formula
- Periodic Compounding Rate Formula
- Rule of 72
- Compound Interest of Consecutive Years
- Continuous Compounding Interest Formula
- Some Other Applications of Compound Interest
- Difference between Compound Interest and Simple Interest
- Compound Interest Examples
- Compound Interest – Practice Questions
What is Compound Interest?
Compound Interest is interest on the principal amount as well as the interest earned on the principal amount. The word “Compound Interest” is composed of two words “Compound” which means composed of two or more and “Interest” means money earned on lending the amount. Hence, compound interest is the money earned on lending and it is composed of two types of interest which are:
- Interest on Principal Amount
- Interest on Interest Earned on Principal Amount over the Period
Compound Interest Definition
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”.Â
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 the 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
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 + r/100)^{n }– P
The above formula for calculating compound interest is added in the image form below:
Where,
- P = Principal
- r = Rate of Interest
- n = Number of Times Interest is Compounded Per Year
- t = Time (in years)
We can write the formula for compound interest as:
Compound Interest = A â€“ P
Where,
- A = Total Amount of Money after Compounding
- P = Initial Principal Amount
Compound Interest = P(1 + r/n)^{nt} -P
Where,
- P = Initial Principal Amount
- r = Annual Interest Rate
- n = Number of Times Interest is Compounded
- t = Number of Years
Compound Interest can be calculated yearly, half-yearly, quarterly, monthly, daily, etc. as per the requirement.
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.Â
Use following methods to find compound interest.
Step 1: Note, Principal, Rate, and Time period given
Step 2: Calculate amount using formula A = P(1 + r/100)^{n}Â
Step 3: Find Compound Interest using the formula CI = Amount – Principal
At regular intervals, the interest so far accumulated is clubbed with the existing principal amount and then the interest is calculated for the new principal. The new principal is equal to the sum of the Initial principal, and the interest accumulated so far.
Compound Interest = Interest on Principal + Interest on Interest of Principal(From second year and ownwards)
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 Formula – Derivation
Compound interest formula is a powerful tool used in finance to calculate the interest earned or paid on an initial principal amount, which includes both the initial principal and the interest accumulated over previous periods. The formula for compound interest is given by:
[Tex]A = P \left(1 + \frac{r}{n}\right)^{nt} [/Tex]
Where,
- A is Future Value of Investment or loan, including interest
- P is Principal Amount (initial investment or loan amount)
- r is Annual Interest Rate (as a decimal)
- n is Number of times that interest is compounded per year
- t is Time Money is invested or borrowed for, in years
Simple Interest Formula
Simple interest is calculated only on the principal amount. It can be represented by the formula,
[Tex] A = P + P \cdot r \cdot t [/Tex]
Compound Interest Formula with Continuous Compounding
When interest is compounded continuously (infinitely many times per year), the formula for compound interest is derived using the formula for continuous compounding:
[Tex]A = P \cdot e^{rt} [/Tex]
Where,
- e is Euler’s number (approximately 2.71828)
- P is Principal Amount
- r is Annual Interest Rate
- t is Time in Years
General Compound Interest Formula
To derive the general compound interest formula, let’s consider compounding interest n times per year.
If P is compounded n times per year at an annual interest rate r, the interest r is divided by n and applied n times per year. So, after t years, the formula becomes:
[Tex]A = P \left(1 + \frac{r}{n}\right)^{nt} [/Tex]
Where,
- [Tex] \frac{r}{n} [/Tex] represents the interest rate per compounding period.
- nt is the total number of compounding periods over t years.
This formula illustrates how the initial principal amount grows over time when interest is compounded at regular intervals. As n approaches infinity (i.e., continuous compounding), the formula converges toward the continuous compounding formula [Tex]A = P \cdot e^{rt} [/Tex].
In summary, the compound interest formula [Tex]A = P \left(1 + \frac{r}{n}\right)^{nt} [/Tex] is a result of the continuous compounding formula adapted for discrete compounding periods per year. It allows for the calculation of the future value of an investment or loan, factoring in compounded interest at regular intervals.
Half-yearly Compound Interest Formula
Let the principal invested be P and the interest rate is R % per annum which is compounded half-yearly for ‘t’ years
As it 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.
We know,
rate = R% per Annum Compounded Half Yearly
rate = (R/2) %
time is t years we know that t years have 2t half years.
Now,
A = P (1 + R/200)^{2t}
CI = A – P
Quarterly Compound Interest formula
Let the principal invested be P and the interest rate is R % per annum which is compounded quarterly for t years.
As it 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.
we know,
rate = R% per annum compounded quarterly
rate = (R/4)%
time is t years we know that t years have 4t quarters.
Now,
A = P(1 + R/400)^{4t}
CI = A – P
Monthly Compound Interest Formula
If the interest is compounded monthly then the number of times of compounding will be 12 and the interest each month will be 1/12 of annual compound interest. Hence, Monthly 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.
New Rate of interest will be R/365 %
n = 365
Hence, Daily Compound Interest Formula is given as,
A = P[1 + (R/36500)]^{365t}
CI = A – P
Periodic Compounding Rate Formula
Total amount, including 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 â€“ 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.
This calculation is also useful for calculating the inflated value of our money, i.e. it gives in how many years the value of our asset gets halved if it gets depreciated annually.
Rule of 72 formula
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 Rate at Which Our Money is Compounded Annually
Rule of 72 example
Suppose Kabir has invested 10,00,000 rupees in a debt fund which gives an 8% return. Find in how many years its money gets doubled 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 Previous Year. (CI of 3rd year is greater than CI of 2nd year). Difference between CI for any two consecutive years is interest of one year on C.I of preceding year.
C.I of 3rd year – C.I of 2nd year = C.I of 2nd year Ã— r Ã— 1/100Â
Difference between amounts of any two consecutive years is the interest of one year on amount of preceding year.
Amount of 3rd year – Amount of 2nd year = Amount of 2nd year Ã— r Ã— 1/100Â
Key Results
When we have 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
Continuous Compounding Interest Formula
Continuous Compounding Formula is used in Finance to calculate the final value of an investment which undergoes continuous compounding over different period and value is added over the time. The formula for continuous compounding is given as
Final Value = Present Value Ã— e^{rt}
where,
- r is rate of interest
- t is time
Learn More, Continuous Compounding Formula
Some Other Applications of Compound Interest
Growth: This is mainly used for growth if industries are related.
Production after n years = initial production Ã— (1 + r/100)^{n}
Depreciation: When the cost of a product depreciates by r% every year, then its value after n years isÂ
Present value Ã— (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 Ã— (1 + r/100)^{n}
Difference between Compound Interest and Simple Interest
The difference between Compound Interest and Simple Interest can be learned below in this article
Compound Interest vs Simple Interest | |
---|---|
Compound Interest (CI) | Simple Interest (SI) |
CI is interest that is calculated both on the principal and the previously earned interest. | SI is interest that is calculated only on the principal. |
For the same principle, Rate, and Time period CI > SI | For the same principle, Rate, and Time period SI < CI |
Formula for CI is A = P(1 + R/100) T CI = A – P | Formula for SI is SI = (PÃ—RÃ—T) / 100 |
Learn, Simple Interest
Compound Interest Examples
Some examples on compound interest formulas are,
Example 1: Find the Compound Interest when principal = Rs 6000, rate = 10% per annum and time = 2 years.
Solution:Â
Interest for first year = (6000 Ã— 10 Ã— 1)/100 = 600
Amount at the end of first year = 6000 + 600 = 6600
Interest for second year = (6600 Ã— 10 Ã— 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 Rs 8000 in two years when the rate of interest is 2% per annum?
Solution:Â
Given,
- Principal P = 8000
- Rate r = 2%Â
- Time = 2 yearsÂ
by formula
A = P (1 + R/100)^{n}
A = 8000 (1 + 2/100)^{2} = 8000 (102/100)^{2}
A = 8323
Compound interest = A – PÂ = 8323 – 8000 = Rs 323
Example 3: Hari deposited Rs. 4000 with a finance company for 2 years at an interest of 5% per annum. What is the compound interest that Rohit gets after 2 years?
Solution:Â
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 – PÂ = 4410 – 4000Â = 410
Example 4: Find the compound interest on Rs. 2000 at the rate of 4 % per annum for 1.5 years. When interest is 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)^{2n}
A = Â 2000 (1 + 4/200)^{3}
A = 2000 (204/200)^{3}
A = 2122
Compound Interest = A – PÂ = 2122 – 2000Â = 122
Example 5: What is the compound interest on 10000 for one year at the rate of 20% per annum, if the interest is compounded quarterly?
Solution:Â
Given,
- Principal P = Rs 10000
- Rate R = 12% (12/4 = 3 % per quarter year)
- Time = 1 year (1 Ã— 4 = 4 quarters)
By Formula,
A = P (1 + R/100)^{n}
A = 10000 (1 + 3/100)^{4}Â Â
A = 10000 (103/100)^{4}
Â AÂ = 11255
Compound Interest = A – P Â = 11255 – 10000Â = 1255
Example 6: Find the compound interest at the rate of 5% per annum for 2 years on that principal which in 2 years at the rate of 5% per annum given Rs. 400 as simple interest.
Solution:
Given,
- Simple Interest SI = 400
- Rate R = 5%
- Time T = 2 years
By Formula,
Simple Interest = (P Ã— T Ã— R)/100
â‡’ P = (SI Ã— 100)/T Ã— R
P = (400 Ã— 100)/2 Ã— 5 Rate of Compound Interest = 5%
P = 40000/10Â = Rs 4000
Time = 2 years
By Formula,
A = P (1 + R/100)
A = 4000 (1 + 5/100)
A = 4410
Compound Interest = A – P = 4410 – 4000 = 410
Example 7: Find the compound interest on Rs 30000 at 7% interest compounded annually for two years.
Solution:
- Principal P = Rs 30000
- Rate R = 7%
- Time = 2 year
By formula,
A = P (1 + R/100)^{n}
A = 30000 (1 + 7/100)^{2}
A = 30000 (107/100)^{2}
A = 34347
Compound Interest = A – P = 34347 – 30000 = 4347
Related Reads:
Compound Interest – Practice Questions
Various practice questions on compound interests are,
Q1. Find the Amount that need to be paid after 3 years if a sum of 10000 is lent at the rate of 4% compounded annually.
Q2. Find the interest need to be paid after 1.5 years if a sum of 2500 is lent at rate of 6% compounded half-yearly.
Q3. Calculate the compound interest for a amount of 9000 lent at the rate of 5% quarterly for 15 months.
Q4. Calculate the compound interest for a amount of 20000 lent at the rate of 12% for 3 months compounded monthly
Conclusion of Compound Interest
Compound interest is a powerful financial concept that allows investments or loans to grow or accumulate over time. Unlike simple interest, which only calculates interest on the initial principal amount, compound interest takes into account the interest earned on both the initial principal and any accumulated interest from previous periods.
Compound Interest – FAQs
What Compound Interest Meaning?
Compound Interest is the interest calculated on the principal as well as the previous interest earned over a fixed period of time
How to calculate Compound Interest?
To calculate Compound Interest first final amount is calculated then it is subtracted from the principal to get the final Compound Interest. The amount is calculated using the formula,
A = P(1 + R/100)^{t}
CI = A – P
Is Compound Interest better than Simple Interest for investors?
Yes, Compound Interest is far better than Simple Interest for investors.
What is Compound Interest Formula if it is compounded daily?
Suppose the given principal is P, the rate is R, and the time interval is T years then the compound interest formula when it is compounded daily is:
A = P(1 + R/365)^{{365 Ã— T}}
What is Difference between CI and SI?
Basic difference between CI and SI is SI is interest charged on Principal Amount while CI is interest charged on Principal amount as well as on the interest accumulated on the principal