# Simple vs Compound Interest

Interest is the income earned or expense incurred on a loan or other investment that pays a fixed profit. There are two variants of interest: simple interest and compound interest. In case of simple interest, interest is accrued only to the extent of the principal balance only while in case of compound interest, interest is accrued on the total outstanding amount (i.e. principal plus any interest already earned).

## Simple Interest

Simple interest is where the relationship between the present value and future value is such that interest is earned or charged only on the principal amount and there is no interest on interest.

This can be expressed mathematically as follows:

$$ FV=PV\times\left(1+i\times\frac{A}{B}\right) $$

Simple interest for a period can be calculated using the following formula:

$$ Simple\ Interest=PV\times i\times\frac{A}{B} $$

Where i is the annual simple interest rate, A is the total number of days for which interest is applicable and B is the total number of days in a year.

Alternatively, simple interest can be calculated using the INTRATE function in Excel.

## Compound Interest

Compound interest is where interest for a period is worked out based on the loan or investment value at the beginning of the period inclusive of the interest accumulated to that date.

Let’s you have an investment with principal value PV and annual compound interest rate r, the value of investment after first year will be PV × (1 + r), after second year it will be PV × (1 + r) × (1 + r), after third year, it will be PV × (1 + r) × (1 + r) × (1 + r) and so on. We can derive the following formula to calculate future value of a single sum under the compound interest:

$$ FV=PV\times{(1+r)}^n $$

Where there are more than one compounding periods per year, the above formula can be modified as follows:

$$ FV=PV\times\left(1+\frac{r}{m}\right)^{n\times m} $$

Where r is the annual percentage rate, n is the number of years and m is the number of compounding periods per year.

Interest earned between two periods under the compound interest equals the future value minus the initial principal amount.

Compound interest can also be calculated using the Microsoft Excel RATE function

## Example

The Wadiyan dictator Haffaz Aladeen allows its banks to charge only simple interest. Your company has raised $20 million worth of loan from one Wadiyan Bank at 10% per annum and invested the proceeds at 10% interest compounded twice a year. Find out your outstanding loan balance and investment value at the end of 10th year.

$$ Loan\ Balance \\ =\ $20,000,000 \times (1\ +\ 0.1\times\ 10) \\ =\ $40,000,000 $$

$$ Investment\ Value \\ =$20,000,000\times\left(1+\frac{0.1}{2}\right)^{10\times2} \\ =$53,065,954 $$

The difference between the investment value and loan balance of $13,065,954 is attributable to interest on interest. This difference between total interest earned or incurred increases with increase in interest rate, compounding periods per year and total number of years.

The following chart visualizes the difference between the loan balance under simple interest and investment value under compound interest over time.