What is Compound Interest?
Compound Interest is an interest earned on the original principal and the interest accumulated. Compound interest is like a snowball effect. In the snowball effect, a snowball size increases when more snow is added. Likewise, in compound interest, one earns interest on the initial principal and also interest on interest. C.I. denotes compound interest in mathematics. It finds its usage in finance and economics.
Introduction
C.I is the interest generated on a loan or deposit. Its calculation is based on both initial principal and collected interest. C.I is a result of reinvesting interest instead of paying it out. Interest for the next period is earned from the principal sum and previously accumulated interest.
Let’s know what compound interest is. Compound interest is defined as the interest calculated on the principal and the interest accumulated over the previous period of time. Compound interest is different from simple interest.
In simple interest, the interest is not added to the principal while calculating the interest during the next period while in the compound interest the interest is added to the principal to calculate the interest.
The formula for compound interest is,
where, P is equal to Principal , R is equal to Rate of
Interest, T is equal to Time (Period)
A Little More About Compound Interest
We will first understand the concept and what is compound interest and then move on to the compound interest formula. Now interest can be defined as the amount we calculate on the principal amount that is given to us. But in compound interest, we calculate the interest on the principal amount and the interest that has accumulated during the previous period. Essentially, compound interest is the interest on the interest! So in this method, rather than paying out the interest, it is reinvested and becomes a part of the principal.
As you will have noticed in simple interest, the interest amount remains the same for every period. This is not the case in compound interest. Since the previous interest amount is reinvested, the interest amount increases marginally every year. This is why we have a whole separate compound interest formula to help us calculate the compound interest of any given year.
The compound interest formula in maths is:
Where, P is equal to Principal, Rate is equal to Rate of
Interest, n is equal to the time (Period)
Compound Interest Formula Derivation
To better our understanding of the concept, let us take a look at the compound interest formula derivation. Here we will take our principal to be Rupee.1/- and work our way towards the interest amounts of each year gradually.
Year 1
The interest on Rupee 1/- for 1 year is equal to r/100 = i (assumed)
Interest after Year 1 is equal to Pi
FV (Final Value) after Year 1 is equal to P + Pi = P(1+i)
Year 2
Interest for Year 2 = P(1+i) × i
FV after year two is equal to P(1+i) + P(1+i) × i = P(1+i)2
Year ‘t’
Final Value (Amount) after year “t” is equal to P(1+i)t
Now substituting actual values we get Final Value is equal to ( 1 + R/100)t
CI = FV – P is equal to P ( 1 + R/100)t – P
This is the Compound Interest Formula Derivation
Applications of Compound Interest
Some of the applications of compound interest are:
1. Increase in population or decrease in population.
2. Growth of bacteria.
3. Rise in the value of an item.
4. Depreciation in the value of an item.
Now that we have some clarity about the concept and meaning of compound interest and compound interest formulae in maths, let us try some Compound interest problems with solutions to deepen our understanding of the subject.
Compound Interest Problems with Solutions
Q1) The count of a certain breed of bacteria was found to increase at the rate of 5% per hour. What will be the growth of bacteria at the end of 3 hours if the count was initially 6000?
Solution) Since the population of bacteria increases at the rate of 5% per hour,
We know the formula for calculating the amount, compound interest formula in maths
Amount= Principal(1 + R/100)n
Thus, the population at the end of 3 hours = 6000(1 + 3/100)3
= 6000(1 + 0.03)3
= 6000(1.03)3
= Rs 6556.36
Q2) Mr. A decided to open a bank account and opted for the Compound Interest Option at 10%. He invested 10,000 for 3 years. At the end of three years, how much money will he get, and what will be the interest amount. The interest is calculated annually.
Solution) As we already have a formula for future value amount, let us substitute the values in the compound interest formula in maths.
FV = ( 1 + R/100)t
FV = 10000 ( 1 + 10/100)5
FV = 10000 ( 1.1)5
FV = 16,105
CI = FV – P = 6,105/-
Q3) Mr. B lent money to his son at 8% CI calculated semi-annually. If he lent 1000/- for 2 years, how much will he get back at the end of the 2 years?
Solution) Since the CI is calculated semi-annually
t = 2t = 4
r = r/2 = 4
Final Value = P ( 1 + R/100)t
FV = 1000 ( 1 + 4/100)4
FV = 1000 (1.17)
FV or Amount = 1170/-
Q4. A town will have 10,000 residents in 2020. Its population reduces by 10% per annum. What would the population be in 2025?
Since the population of the town decreases by 10% per annum, it also experiences a new population surge every year.
The population of the upcoming year is calculated on the basis of the current year's population. So students need to use this formula.
A=P(1-r/100)n
Henceforth the population at the end of 5 years would be
10000(1-10/100)5
10000(1-0.1)5
5904.9
A= 5904
C.I Formula
As discussed, C. I findings are based on the initial principal amount and interest over a period of time. The compound interest formula is
C.I= Amount- Principal
Another formula for C.I
In this,
CI=P(1+r/n)nt-P
A= Amount
P= Principal
r=Rate of interest
n= number of interest compounded per year and the compounding frequency
t= time
The above formula is used for a number of times principal compounded in a year. For interest compounded annually, the amount is found through:
A=P(1+R/100)t
Evaluating the formula for the amount and interest calculation for different years
1 Year
Amount
P(1 + R/100)
Interest
PR/100
2 Year
Amount
P(1+R/100)2
Interest
3 Year
Amount
P(1+R/100)3
Interest
P(1+R/100)3-P
4 year
Amount
P(1+R/100)4
Interest
P(1+R/100)4-P
N year
Amount
P(1+R/100)n
Interest
P(1+R/100)n-P
Derivation of CI
To derive CI, students have to use simple interest formula. This is because SI for 1 year is equal to CI of 1 year.
Let’s assume P as the principal amount, n the time and rate be R.
SI for 1st year
SI1=PxRxT/100
Amount for 1st year=P+ SI
A=P+ PxRxT/100
P(1+R/100)=P2
SI for 2nd year
SI2=P2xRxT/100
Amount for 2nd year=P2+SI2
P2+P2xRxT/100
P2(1 + R/100)
P(1 + R/100)(1 + R/100)
For n years
A=P(1+R/100)n
CI=A-P=P((1+R/100)n-1))
Applications of CI
Compound interest finds its usage in different phases of life. Some of the common applications of the compound interest formula are as follows:
Helps in analyzing the growth and decay of the population
Finding the increase and decrease in commodity price
Finding the increase and decrease in the value of an item
Evaluation of inflation in profit and loss
Bank transactions
FAQs on Compound Interest Formula
1. What is the difference between Compound Interest and Simple Interest?
The difference between Compound Interest and Simple Interest are as follows:
Simple Interest | Compound Interest |
The simple interest is the same for all the number of years. | The compound interest is different for all the years. |
SI < CI. | CI > SI. |
Simple Interest (SI) = (P×R×T)/100 | CI = Principal (1+Rate/100)n - principal |
2. How do you calculate the Compound Interest?
We can calculate compound interest by multiplying the initial principal amount by adding one and the annual interest rate raised to the number of compound periods minus one.
3. How is Compound Interest used in real life?
The idea of compound interest is appealing only when you are on the earning side of the financial balance. Banks typically pay compounded interest on deposits, a benefit for depositors.Credit card companies charge interest on the principal amount and the accumulated interest.
4. Why is compound interest so powerful?
Compound interest basically makes a certain sum of money that grows at a faster rate than simple interest because, in addition to earning returns on the money you invest, you also earn returns on those returns at the end of each and every compounding period, which could be daily, monthly, quarterly or annually.
5. Why do banks use Compound Interest?
Almost all banks compound interest. Compounding means a financial institution pays you interest not only on the amount you originally deposited,but also on the interest your deposit has earned over time.So the more frequently the money of a person or a customer is compounded, the more interest you'll earn.
6. What is the disadvantage of C.I?
The primary disadvantage of C.I. is that it is expensive. The borrower needs to pay more interest to the lender in comparison to the borrowed amount. Another disadvantage of C.I. is that it is complicated as its calculation involves interest generated over different periods of time. Lenders and credit card suppliers can easily apply compound interest in the repayment of student loans and credit card debt. C.I. has a vicious cycle, and it can leave a borrower in debt.
7. What are the benefits of C.I?
C.I. is useful in the investment of large amounts such as in stocks, mutual funds. The application of C.I. in investments generates a large sum of wealth for the investor. In this case, the investor at the end of the tenure received both the principal amount and the interest earned on it. C.I. also helps in easy analyzing of the value of a product. C.I. finds its usage in broad areas of large-scale finances and the economy.