# Simple and Compound Interest

Simple interest and compound interest are two different methods for calculating interest. Both methods involve the following three parameters.

1. Principal
2. Rate
3. Term
The principal is the original sum on which interest is calculated. The rate is the interest rate charged per period, and the term is the number of periods over which interest is charged. For example, a loan might be specified as 150,000 (Principal), taken out for 10 years (Term), at an interest rate of 6% per year (Rate).

How these three parameters are used determines whether the interest is simple interest or compound interest.

## How to Calculate Simple Interest

Simple interest is calculated on the original principal amount throughout the term.
The formula for simple interest is

`Simple interest = Principal x Rate x Term`

When calculating simple interest both the interest rate and the term should be specified using the same length of period. For example, if the interest rate is for a year then the term must be in years, if the interest rate is per month, then the term should be in months.

For example, if a loan of 120,000 has an annual interest rate of 8%, then the simple interest for 6 months is calculated as follows:

```Simple interest = Principal x Rate x Term
Principal = 120,000
Rate = 8% per year
Term = 6/12 years
Simple interest = 120,000 x 8% x 6/12
Simple interest = 4,800
```

Notice how the simple interest is calculated on the principal amount, and as the rate is an yearly interest rate, the term is expressed in years.

## How to Calculate Compound Interest

Compound interest is calculated on the principal amount plus any accumulated interest accrued at the start of a period. Interest is charged on interest.

The compound interest formula can be stated as follows.

`Compound interest = Principal x (1 + Rate)Term - Principal`

Again, the rate and the term must be for the same length of period.

For example, if a loan of 120,000 has a nominal annual interest rate of 8% compounded monthly, then the compounding interest for a period of 6 months is calculated as follows.

```Compound interest = Principal x (1 + Rate)Term - Principal
Principal = 120,000
Rate = 8%/12 per month
Term = 6 months
Compound interest = 120,000 x (1 + 8%/12)6 - 120,000
Compound interest = 124,880.71 - 120,000
Compound interest = 4,880.71
```

Notice in this case as the compounding period is monthly, the calculation must be carried out using a month as one period, and a monthly interest rate.

## Power of Compound Interest

In the interest examples above, the principal (120,000), rate (8%), and term (6 months) were the same, however the simple interest was calculated as 4,800, and the compound interest was 4,880.71.

If we look at the detailed calculations behind the interest formulas we can see the following.

### Simple Interest

For the simple interest calculations, interest accrues only on the principal amount as shown below.

Simple Interest Calculations
MonthOpening LoanInterest calculationInterestClosing Loan
Month 1120,000.00120,000.00 x 8%/12800.00120,800.00
Month 2120,800.00120,000.00 x 8%/12800.00121,600.00
Month 3121,600.00120,000.00 x 8%/12800.00122,400.00
Month 4122,400.00120,000.00 x 8%/12800.00123,200.00
Month 5123,200.00120,000.00 x 8%/12800.00124,000.00
Month 6124,000.00120,000.00 x 8%/12800.00124,800.00
Total4,800.00

In the case of simple interest, the original principle balance forms the basis of the interest calculation. No account is taken of the accumulated interest to date when calculating interest.

### Compound Interest

However, in contrast for the compound interest calculations, interest accrues on the principal plus the accumulated interest, represented by the closing loan balance at the end of each month as shown below.

Compound Interest Calculations
MonthOpening LoanInterest calculationInterestClosing Loan
Month 1120,000.00120,000.00 x 8%/12800.00120,800.00
Month 2120,800.00120,800.00 x 8%/12805.33121,605,33
Month 3121,605,33121,605.33 x 8%/12810.70122,416.04
Month 4122,416.04122,416.04 x 8%/12816.11123,232.14
Month 5123,232.14123,232.14 x 8%/12821.55124,053.69
Month 6124,053.69124,053.69 x 8%/12827.02124,880.71
Total4,880.71

You can see that the simple and compound interest calculations give different answers. Simple interest (4,800) is always lower, than compound interest (4,880.71), which is the effect of charging interest on interest when using the compound interest method.

The interest on the interest is in fact given by the future value of an annuity formula.

```FV = Pmt x ( (1 + i)n - 1 ) / i
Pmt = 800 regular simple interest payment
n = 6 months
i = 8%/12 per month
FV = 800 x ( (1 + 8%/12)6 - 1 ) / (8%/12)
FV =  4,880.71
and
Interest on interest =  FV - PV
Interest on interest = 4,880.71 - 800 x 6
Interest on interest = 80.71
```

The difference between simple interest and compound interest is the interest on interest.

So for example after 10 years (120 months) the difference between the accumulated simple and compound interest would be as follows.

```FV = Pmt x ( (1 + i)n - 1 ) / i
Pmt = 800 regular simple interest payment
n = 120 months
i = 8%/12 per month
FV = 800 x ( (1 + 8%/12)120 - 1 ) / (8%/12)
FV = 146,356.83
and
Interest on interest = FV - PV
Interest on interest = 146,356.83 - 800 x 120
Interest on interest = 50,356.83
```

After 10 years an additional 50,256.83 interest would have been paid on the loan using compound interest instead of simple interest.