What is banking?

  • The business of receiving money from depositors (or account holders), safeguarding, and lending money to businesses or individuals is called banking.
  • Therefore Banks are institutions that carry out the business of taking deposits and lending money.
  • When people deposit money, based on the scheme under which they deposit money, they get a return on their money.
  • Similarly, when the bank lends money to people or businesses, banks charge a rate of interest on the amount of money given to us.

The difference between the two is what the bank earns adjusted to their operational costs. In a very simple way we could say:

Banks Earnings = Earning on Money Invested – Interest paid to account holders  –  Operational cost

Off-course, banks have to take a banking license to start bank operations.

Types of Accounts: There are many types of accounts but from a course perspective, we will look at the following accounts:

  1. Savings Bank Account
  2. Recurring Deposit Accounts

Saving Bank Account

A person can deposit and withdraw money at will. The person gets a certain interest on the deposits, which could change with a change in market conditions.

How do we calculate the interest on the deposit?

Nowadays, because of the powerful computers, banks are able to calculate interest on a day-to-day basis. However, for our syllabus, we would calculate interest on a monthly basis. The concept is the same though. Here is how we will do it.

  1. Find the minimum balance on the \displaystyle 10^{th} day and up to the last day of each month. This minimum balance becomes the principal of the month.
  2. Add all such Principal amounts obtained for different months of a particular period in consideration.
  3. Now calculate the simple interest on the Principal obtained in Step 2 for one month at the prevailing rate of interest at that time.

Recurring Deposit Account

In this type of deposit, the account holder deposits a specified amount in the account every month for a fixed period of time. It could be three months to say 10 years. The time period is decided by the bank.

At the expiry of the period, the person gets a lump sum of money which includes the money that was deposited and the interest (compounded quarterly) that the money has earned over a period of time.

The formula that we use for calculating the maturity value of the recurring deposit is:

Maturity Amount  =  Total Sum Deposited  +  Interest Earned

If \displaystyle P is deposited every month in the bank for \displaystyle n months and \displaystyle r\% is the rate of interest per year, then

\displaystyle I=P \times \frac{n(n+1)}{2 \times 12} \times \frac{r}{100}  

 Total Sum Deposited \displaystyle = P \times n  

\displaystyle \text{Maturity Value }= P \times n + P \times \frac{n(n+1)}{2 \times 12} \times \frac{r}{100}  

Proof: If \displaystyle P is deposited every month in the bank for \displaystyle n months and \displaystyle r\% is the rate of interest per year, then

Maturity amount for \displaystyle P deposited in the \displaystyle 1^{st} month after \displaystyle n months

\displaystyle = P + P \times \frac{r}{100} \times \frac{n}{12}  

Maturity amount for \displaystyle P deposited in the \displaystyle 2^{nd} month after \displaystyle (n-1) months

\displaystyle = P + P \times \frac{r}{100} \times \frac{(n-1)}{12}  

Maturity amount for \displaystyle P deposited in the \displaystyle 3^{rd} month after \displaystyle (n-1) months

\displaystyle = P + P \times \frac{r}{100} \times \frac{(n-2)}{12}  

\displaystyle \cdots

Maturity amount for \displaystyle P deposited in the \displaystyle (n-1)^{th} month after \displaystyle (n-1) months

\displaystyle = P + P \times \frac{r}{100} \times \frac{2}{12}  

Maturity amount for \displaystyle P deposited in the \displaystyle (n)^{th} month after \displaystyle (n-1) months

\displaystyle = P + P \times \frac{r}{100} \times \frac{1}{12}  

Therefore

\displaystyle \text{Maturity amount } = P \times n + P \frac{r}{100} \Big( \frac{n}{12} + \frac{n-1}{12} + \frac{n-2}{12} + \cdots + \frac{2}{12} + \frac{1}{12} \Big)

\displaystyle = P \times n + P \frac{r}{100 \times 12 } \Big( n + (n-1) + (n-2) + \cdots + 2 + 1 \Big)

\displaystyle = P \times n + P \frac{n(n+1)}{2 \times 12} \frac{r}{100}