Compound interest: Basic Concepts

Understanding Compound Interest
Let us try to understand basics of compound interest and how it differs from Simple Interest. In Simple Interest, the Principal remains constant throughout the whole term. However, in practice, different financial institutions like banks, post offices, insurance companies etc, do not take the principal to be constant throughout the term. After every fixed interval of time, the interest is added to the principal. The combined total of the Principal and the Interest is taken as the new principal. This process is continued until the interest for the full term is calculated.
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So, we can say that in the Compound Interest formula, interest is periodically calculated and converted to principal. “Converting interest to Principal” means that the interest is added to the principal and is thereafter treated as Principal.
Suppose, you invest Rs. 1,00,000 at 10% compounded annually for 3 years. “Compounded annually” means that “Interest is compounded once per year.” Therefore, the compounding period is one year. Let us understand this complete process in an easy way:
FIRST YEAR, Principal is Rs. 1,00,000. You will earn Rs. 10,000 interest (10 % of Rs. 1,00,000).
At the end of the first year, this Rs. 10,000 will be added to the principal and balance of your account will show Rs. 1,10,000.
SECOND YEAR, new Principal is Rs. 1,10,000. You will earn Rs. 11,000 interest (10 % of Rs. 1,10,000). At the end of the second year, this Rs. 11,000 will be added to the principal and balance of your account will show Rs. 1,21,000.
THIRD YEAR, new Principal is Rs. 1,21,000. You will earn Rs. 12,100 interest (10 % of Rs. 1,21,000). At the end of the third year, this Rs. 12,100 will be added to the principal and balance of your account will show Rs. 1,33,100.
Let us now have the formula for calculation of Compound Interest.
When Interest is compounded yearly:
  • In case of Compound Interest, firstly Amount is calculated. Then, we get the Compound Interest by subtracting the Principal from Amount.
The formula used to calculate Amount is as follows:
 
Compound interest formula: A = P (1 + R/100)T
Where A is the Amount,
P is the Principal,
R is the rate of Interest, and
T is the time period for which the money is invested.
Hence, to calculate compound interest= Amount – Principal.
When Interest is compounded half-yearly:
  • When the interest is compounded half-yearly, the amount after the first half year becomes the principal for the next half-year and so on.
Compound Interest examples
Let us compare the case of two persons, one is investing at the rate which is compounded annually and the other person who is investing at the rate which is compounded half yearly:
Sunil invested Rs. 10,000 in a bank at an interest rate of 10% p.a. compounded annually whereas his friend Rishab invested the same amount with a private finance company at the same rate, but compounded half-yearly.
Compounded Annually Compounded Half Yearly
P = Rs. 10,000
R = 10% p.a.,
T = 1 year
C.I. = RS. (1000 X 10 X 1)/100 = RS. 1000
In this case, the interest has been calculated
directly by applying the simple interest
formula. Moreover, it is a case of one year
only, thus it can be found by applying the
simple interest formula as well.
P = Rs. 10,000
R = 10% p.a.,
T = 1 year = 2 half years
For first half year:
I = RS. (10000 X 10 X 1/2)/100
= RS. (10000 X 10 X 1)/200 = RS.500
A = P + I = RS.10000 + RS.500 = RS.10500
Now, for next half year:
P = RS.10,500 So I=
I = RS. (10500 X 10 X 1/2)/100 = RS. (10500 X 10 X 1)/200 = RS.525
A = Rs. 10500 + Rs. 525 = Rs. 11025
C.I. = Final amount – P
= Rs. 11025 – Rs. 10000 = Rs. 1025
From the above illustration, we observe the following:
  1. The compound interest when compounded annually is Rs. 11000 – Rs. 10000 = Rs. 1000
  2. The compound interest when compounded half-yearly is Rs. 11025 – Rs. 10000 = Rs. 1025
  3. When the interest is compounded half-yearly, we compute interest after every six months. Hence, we compute the interest two times a year. So, the time period becomes twice and the rate is taken to be half.
  4. More interest is generated when the interest is compounded half-yearly
    In the above example, the interest is compounded half-yearly.
    Compounded interest = Rs. 500 + Rs. 525 = Rs. 1025
Let us go through all the compound interest formulas, when interest is compounded half yearly, quarterly or time is given to be a fraction of a year:
  • Let P be the Principal and the rate of interest be R % per annum. If the interest is compounded half yearly, then the amount A and Compound Interest at the end of n years are given by
    A = P(1 + R/200)2n and C.I = {P(1 + R/200)2n - 1}
  • Let P be the Principal and the rate of interest be R % per annum. If the interest is compounded quarterly, then the amount A and Compound Interest at the end of n years are given by
    A = P(1 + R/400)4n and C.I = {P(1 + R/400)4n - 1}
The concepts learned here are useful in compound interest questions and answers. Now, let us solve some compound interest sums.
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