What is banking?

• Business of receiving money from depositors (or account holders), safe guarding and lending money to business or individuals is called banking.
• Therefore Banks are institutions which 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 the money to people or business, banks charge a rate of interest on the amount of money given our.

The difference in 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 change in market conditions.

How do we calculate the interest on the deposit?

Now a days, because of the powerful computers, the 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 $10^{th}$ 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 deposits, 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 $P$ is deposited every month in the bank for $n$ months and $r\%$ is the rate of interest per year, then $I=P \times$ $\frac{n(n+1)}{2 \times 12}$ $\times$ $\frac{r}{100}$

Total Sum Deposited $= P \times n$

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

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

Maturity amount for $P$ deposited in the $1^{st}$ month after $n$ months $= P + P \times$ $\frac{r}{100}$ $\times$ $\frac{n}{12}$

Maturity amount for $P$ deposited in the $2^{nd}$ month after $(n-1)$ months $= P + P \times$ $\frac{r}{100}$ $\times$ $\frac{(n-1)}{12}$

Maturity amount for $P$ deposited in the $3^{rd}$ month after $(n-1)$ months $= P + P \times$ $\frac{r}{100}$ $\times$ $\frac{(n-2)}{12}$ $\cdots$

Maturity amount for $P$ deposited in the $(n-1)^{th}$ month after $(n-1)$ months $= P + P \times$ $\frac{r}{100}$ $\times$ $\frac{2}{12}$

Maturity amount for $P$ deposited in the $(n)^{th}$ month after $(n-1)$ months $= P + P \times$ $\frac{r}{100}$ $\times$ $\frac{1}{12}$

Therefore

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