## Interest & Growth

### 1. Introduction

When a certain amount is lent, the lender needs to be compensated for parting with the money lent for the time period, possible inflation or even the risk of the borrower defaulting. Therefore, the borrower is required to pay an interest as a compensation. This interest is almost always applied as a percentage on the principal or amount and directly varies with the time period. The following are the key components in the interest calculation.**Principal**: The amount lent or invested

**Interest Rate**: Rate or percentage of the principal at which the borrower needs to pay as interest for a given time period. Interest rate is typically written as, say, $12 \%$ per annum, which means interest is to be paid at $12 \%$ of the principal or amount (as the case maybe) every year.

**Time**: This is the duration of the loan or investment. In the case of compound interest, time is broken down into time periods for compounding. This is detailed in later sections.

**Type of Interest**: There are typically two forms of interest – Simple Interest and Compound Interest. Simple Interest is wherein the interest rate is applied on the principal component only. Compound Interest applies the interest rate on the principal component as well as the interest accrued till the previous time period (when interest was compounded).

**Compounding Period**: This is applicable only in the case of Compound Interest. Interest can be compounded for any defined time period. Banks in India typically compound their loans on a monthly basis. For example, an interest rate of $12 \%$ per annum compounded semi-annually for a $2$ year period, effectively means an interest of $6 \%$ per semi annum for 4 time periods of $6$ months each. This is detailed in later sections.

**Amount**: This is the sum of principal and interest that the borrower needs to pay to the lender at the end.

### Example 1

Ram borrowed Rs. $1,000$ from Shyam on $1^{\text{st}}$ January $2018$. This sum is to be repaid with interest on $31^{\text{st}}$ December $2020$. Interest to be paid is $12 \%$ per annum of the amount initially borrowed. How much did Ram pay Shyam on $31^{\text{st}}$ December $2020$?

(1) Rs. $1,120$ (2) Rs. $1,240$ (3) Rs. $1,360$ (4) Rs. $1,480$

Note that the amount was borrowed at the start of $2018$ till the end of $2020$.

∴ The amount has been borrowed for

Total interest $= 120 \times 3 =$

Total Amount repaid = Principal + Interest

$= 1,000 + 360 =$

(1) Rs. $1,120$ (2) Rs. $1,240$ (3) Rs. $1,360$ (4) Rs. $1,480$

### Solution

Interest due per year $= 1000 \times \dfrac{12}{100}$**Rs. 120**Note that the amount was borrowed at the start of $2018$ till the end of $2020$.

∴ The amount has been borrowed for

**3 years**.Total interest $= 120 \times 3 =$

**Rs. 360**Total Amount repaid = Principal + Interest

$= 1,000 + 360 =$

**Rs. 1,360****Answer**: (3) $1360$The above example pertains to Simple Interest as interest is a percentage of the initial principal borrowed and not the accumulated interest. We can directly calculate the interest as follows.

Total Interest $= 1000 \times 12 \% \times 3 \text{years} = 1000 \times \dfrac{12}{100} \times 3 = \text{Rs.} 360$