Let us consider that we deposit certain sum \$ 1000 in a bank at 10% rate of interest. The bank allows its customer to withdraw their money at regular interval of time. Instead of receiving simple interest on the money for 10 years, we will now play a trick to multiply our money quickly. So, we deposit sum \$ 1000 in the bank on the first year and receive simple interest \$ 100 at the end of the year. Then, we withdraw the money. So, we have \$ 1100 now. Then, we re-deposit the money on the bank on the second year and receive simple interest \$ 110 at the end of the year. Then, we withdraw the money. So, we have \$ 1210 now. But, you know what? This sum \$ 1210 is more than the Amount we would receive at the end of 2 years if we had left it for Simple Interest. We would only receive \$ 1200 if allowed for Simple Interest. So, you repeat the process time and again. As you plot the amount on a graph, you will observe a curve as shown in the graph below.
Simple Interest Vs Compound Interest
Introduction to Compound Interest
Calculation of Compound Interest is a little different than the Simple Interest.
Calculating the Interest on the Amount to be received, in certain interval of time is called Compound Interest. This means, compound interest is calculated on the sum of Principal and Interest, of previous time, which is our new Principal.
Let's say, a person A deposits some sum 'x' in a bank and receives 'y' interest after one year. He doesn't want to take the Interest out rather he wants the bank to pay him interest on a new Principal 'z' that is 'x+y'. If he continues doing this in a regular interval, it is said to be compounding. In other words, he is not linearly increasing his money but exponentially increasing his money.
Here all three variables, Principal, Rate of Interest and Duration of time can be variables.
Things to Remember
- While calculating the Interest for the First time, value of Principal is equal to the initial Principal.
- In Compound Interest, the value of the Principal of current time is equal to the value of Amount of previous time.
- The duration of Time can be daily, monthly, quarterly, half-yearly, yearly or other time intervals.
- In Compound Interest, the Rate of Interest might also vary in different time interval.
Formulae for Compound Interest
Let us consider that we have certain principal Rs. P deposited in the bank at R% interest per annum. The rate of interest R does not change over time. The principal is compounded in a regular interval of T years. In the following derivation section, we shall write R/100 as R%. Then, we have,
For First Year
Principal = Rs. P
Simple Interest = PTR/100 = PTR%
Amount = P + PTR% = P (1 + TR%) ---- (1)
For Second Year
Principal = Rs. P (1 + TR%)
Simple Interest = {P (1 + TR%) } T R%
Amount = P ( 1 + TR%) + { P (1 + TR%) } TR%
= {P (1 + TR%)}(1 + TR%)
= P (1 + TR%)2 ---- (2)
For Third Year
Principal = Rs. P (1 + TR%)2
Simple Interest = {P (1 + TR%)2 } T R%
Amount = P ( 1 + TR%)2 + { P (1 + TR%)2 } TR%
= {P (1 + TR%)2}(1 + TR%)
= P (1 + TR%)3 ---- (3)
What we can analyse from the result of the cases (1), (2) and (3) are the following:
- The amount on the principal P each year is dependent upon the initial principal amount P.
- The amount on the principal P each year is dependent upon the nth power of the quantity (1 + TR%) where n is the number of years.
Thus, we can generalize a formula for the Compound Amount at a fixed rate of interest and compounded yearly as:
$$\text{A = P}\left [1 \ + \ \dfrac{\text{R}}{100} \right ]^\text{T}$$
Similarly, we can deduce the formula for the Compound Interest at a fixed rate of interest and compounded yearly as:
$$\text{CI = P}\left [1 \ + \ \dfrac{\text{R}}{100} \right ]^\text{T} \ - \ \text{P}$$
$$\text{CI = P}\left [ \left ( 1 \ + \ \dfrac{\text{R}}{100} \right )^\text{T} \ - \ 1 \right ]$$
Other Important Formulae
Let, in each cases, the compound amount be A and compound interest be CI.
1. When the interest is compounded annually but the rate being different in different years:
$$\text{A = P}\left (1 \ + \ \dfrac{\text{R}_1}{100} \right )\left (1 \ + \ \dfrac{\text{R}_2}{100} \right )\left (1 \ + \ \dfrac{\text{R}_3}{100} \right ) \ .... \ \left (1 \ + \ \dfrac{\text{R}_n}{100} \right )$$
$$\text{CI = P}\left [ \left (1 \ + \ \dfrac{\text{R}_1}{100} \right )\left (1 \ + \ \dfrac{\text{R}_2}{100} \right )\left (1 \ + \ \dfrac{\text{R}_3}{100} \right ) \ .... \ \left (1 \ + \ \dfrac{\text{R}_n}{100} \right ) \ - \ 1 \right ]$$
where R1, R2, R3, ... , Rn are the rate of interests at Years T1, T2, T3, ...., Tn.
2. When the interest is compounded annually but the time is given in 'T' years and 'M' months:
$$\text{A = P}\left [1 \ + \ \dfrac{\text{R}}{100} \right ]^\text{T}\left [ 1 + \dfrac{\text{MR}}{1200} \right ]$$
$$\text{CI = P}\left [ \left ( 1 \ + \ \dfrac{\text{R}}{100} \right )^\text{T}\left ( 1 + \dfrac{\text{MR}}{1200} \right ) \ - \ 1 \right ]$$
3. When the interest is compounded half-yearly
$$\text{A = P}\left [1 \ + \ \dfrac{\text{R}}{2 \times 100} \right ]^\text{2T}$$
$$\text{A = P}\left [ \left (1 \ + \ \dfrac{\text{R}}{2 \times 100} \right )^\text{2T} \ - \ 1 \right ]$$
4. When the interest is compounded quarter-yearly
$$\text{A = P}\left [1 \ + \ \dfrac{\text{R}}{4 \times 100} \right ]^\text{4T}$$
$$\text{A = P}\left [ \left ( 1 \ + \ \dfrac{\text{R}}{4 \times 100} \right )^\text{4T} \ - \ 1 \right ]$$
5. When the interest is compounded n times a year
$$\text{A = P}\left [1 \ + \ \dfrac{\text{R}}{\text{n} \times 100} \right ]^\text{nT}$$
$$\text{A = P}\left [ \left ( 1 \ + \ \dfrac{\text{R}}{\text{n} \times 100} \right )^\text{nT} \ - \ 1 \right ]$$